Abstract
To speedup the solution of parametrized differential problems, reduced order models (ROMs) have been developed over the years, including projectionbased ROMs such as the reducedbasis (RB) method, deep learningbased ROMs, as well as surrogate models obtained through machine learning techniques. Thanks to its physicsbased structure, ensured by the use of a Galerkin projection of the full order model (FOM) onto a linear lowdimensional subspace, the GalerkinRB method yields approximations that fulfill the differential problem at hand. However, to make the assembling of the ROM independent of the FOM dimension, intrusive and expensive hyperreduction techniques, such as the discrete empirical interpolation method (DEIM), are usually required, thus making this strategy less feasible for problems characterized by (highorder polynomial or nonpolynomial) nonlinearities. To overcome this bottleneck, we propose a novel strategy for learning nonlinear ROM operators using deep neural networks (DNNs). The resulting hyperreduced order model enhanced by DNNs, to which we refer to as DeepHyROMnet, is then a physicsbased model, still relying on the RB method approach, however employing a DNN architecture to approximate reduced residual vectors and Jacobian matrices once a Galerkin projection has been performed. Numerical results dealing with fast simulations in nonlinear structural mechanics show that DeepHyROMnets are orders of magnitude faster than PODGalerkinDEIM ROMs, still ensuring the same level of accuracy.
1 Introduction
Mathematical models involving partial differential equations (PDEs) depending on a set of parameters are ubiquitous in applied sciences and engineering. These input parameters are defined to characterize, e.g., material properties, loads, boundary/initial conditions, source terms, or geometrical features. Highfidelity simulations based on fullorder models (FOMs), like the finite element (FE) method, entail huge computational costs in terms of CPU time and memory, if a large number of degrees of freedom dofs) is required. Complexity is amplified whenever interested to go beyond a single direct simulation, such as in the multiquery contexts of optimization, parameter estimation and uncertainty quantification. To face these problems, several strategies to build reduced order models (ROMs) have been developed over the years, aiming at computing reliable solutions to parametrized PDEs at a greatly reduced cost. A large class of ROMs relies on a projectionbased approach, which aims at approximating the unknown state quantities as a linear superimposition of basis functions; these latter then span a subspace which the governing equations are projected onto [8, 9]. Among these, the reduced basis (RB) method [35, 49] is a powerful and widely used technique, characterized by a splitting of the reduction procedure into an expensive, parameterindependent offline phase (however performed once and for all) and an efficient, parameterdependent online phase. Its efficiency mainly relies on two crucial assumptions:

1.
The solution manifold is lowdimensional, so that the FOM solutions can be approximated as a linear combination of few reduced modes with a small error;

2.
The online stage is completely independent of the highfidelity dimension [21].
Assumption 1 concerns the approximability of the solution set and is associated with the slow decay of the Kolmogorov Nwidth [46]. However, for physical phenomena characterized by a slow Nwidth decay, such as those featuring coherent structures that propagate over time [25], the manifold spanned by all the possible solutions is not of small dimension, so that ROMs relying on linear (global) subspaces might be inefficient. Alternative strategies to overcome this bottleneck can be, e.g., local RB methods [2, 43, 55], or nonlinear approximation techniques, mainly based on deep learning architectures, see, e.g., [22,23,24, 37, 39].
Assumption 2 is automatically verified in linear, affinely parametrized problems [49], but cannot be fulfilled when dealing with nonlinear problems, as the online assembling of the reduced operators requires to reconstruct the highfidelity ones. To overcome this issue, a further level of approximation, or hyperreduction, must be introduced. Stateoftheart methods, such as the empirical interpolation method (EIM) [6], the discrete empirical interpolation method (DEIM) [15], its variant matrix DEIM (MDEIM) [42], the missing point estimation [4] and the GaussNewton with approximated tensors (GNAT) [13], aim at recovering an affine expansion of the nonlinear operators by computing only a few entries of the nonlinear terms. EIM, DEIM and GNAT can be seen as approximatethenproject techniques, since operator approximation is performed at the level of FOM quantities, prior to the projection stage. On the other hand, projectthenapproximate strategies have also been introduced, aiming at approximating directly ROM operators, such as the reduced nonlinear residual and its Jacobian. An option in this sense is represented by the socalled Energy Conserving Sampling and Weighting (ECSW) technique [20]; see. e.g., [21] for a detailed review.
Although extensively used in many applications, spanning from fluid flows models to cardiac mechanics [2, 11, 19, 27, 50, 54], these strategies are codeintrusive and, more importantly, might impact on the overall efficiency of the ROM approximation in complex applications. Indeed, when dealing with highly nonlinear problems, expensive hyperreduction strategies are usually required if aiming at preserving the physical constraints at the ROM level, that is, if ROMs are built consistently with the FOM through a projectionbased procedure. For instance, a large number of DEIM basis vectors are required to ensure the convergence of the reduced Newton systems arising from the linearization of the nonlinear hyperROM when dealing with highly nonlinear elastodynamics problems [17], even if few basis functions are required to approximate the state solution in a lowdimensional subspace through, e.g., proper orthogonal decomposition (POD). An alternative formulation of DEIM in a finite element framework, known as unassembled DEIM (UDEIM) [53], has been proposed to preserve the sparsity of the problem, while in [44] a localized DEIM selecting smaller local subspace for the approximation of the nonlinear term is presented.
Semiintrusive strategies, avoiding the ROM construction through a Galerkin projection, have been recently proposed exploiting surrogate models to determine the RB approximation. For instance, neural networks (NNs) or Gaussian process regression can be employed to learn the map between the input parameters and the reducedbasis expansion coefficients in a nonintrusive way [32, 33, 36, 52]. An approximation of the nonlinear term arising in projectionbased ROMs is obtained in [26] through deep NNs (DNNs) that exploit the projection of FOM solutions. NNs have also also been recently applied in the context of operator inference for (parametrized) differential equations, combining ideas from classical model reduction with datadriven learning. For instance, the design of NNs able to accurately represent linear/nonlinear operators, mapping input functions to output functions, has been proposed recently in [40]; based on the universal approximation theorem of operators [16], a general deep learning framework, called DeepONet, has been introduced to learn continuous operators, such as solution operators of PDEs, using DNNs; see also [56]. In [45] a nonintrusive projectionbased ROM for parametrized timedependent PDEs including loworder polynomial nonlinear terms is considered, inferring an approximation of the reduced operators directly from data of the FOM. Finally, the obtained lowdimensional system is solved—in this case, the learning task consists in the solution to a least squares problem; see also [7]. Projectionbased ROMs and machine learning have been fused in [47] aiming at the approximation of linear and quadratic ROM operators, focusing on the solution to a large class of fluid dynamics applications. Similarly, in [5] a nonintrusive technique, exploiting machine learning regression algorithms, is proposed for the approximation of ROM operators related to projectionbased methods for the solution of parametrized PDEs. Finally, principal component analysisbased model reduction with a NNs for approximation has been combined in [10], in a purely datadriven fashion, of infinitedimensional solution maps, such as the solution operator for timedependent PDEs.
In this paper, we develop a novel semiintrusive, deep learningenhanced hyperreduced order modeling strategy, which hereon we refer to as shape DeepHyROMnet, by leveraging a GalerkinRB method for solution dimensionality reduction and DNNs to perform hyperreduction. Since the efficiency of the nonlinear ROM hinges upon the costeffective approximation of the projections of the (discrete) reduced residual operator and its Jacobian (when an implicit numerical scheme is employed), the key idea is to overcome the computational bottleneck associated with classical, intrusive hyperreduction techniques, like DEIM, by relying on DNNs to approximate otherwise expensive reduced nonlinear operators at a greatly reduced cost. Unlike datadrivenbased methods, for which the predicted output is not guaranteed to satisfy the underlying PDE, our proposed method is physicsbased, as it computes the ROM solution by actually solving the reduced nonlinear systems by means of the Newton method, thus exploiting the physics of the problem. A further benefit of the proposed DeepHyROMnet method lies on the fact that the inputs given to the NNs are lowdimensional arrays, so that the overwhelming training times and costs that may be required by even moderately large FOM dimensions can be avoided. Note that we aim at efficiently approximating the nonlinear operators that result from the composition of the reduced solution operator—that maps the input parameter vector and time to the corresponding ROM solution—and the reduced residual/Jacobian operator, that maps the ROM solution to the reduced residual/Jacobian evaluated on the ROM solution. We apply the proposed methodology to the solution of problems in nonlinear solid mechanics, focusing on parametrized nonlinear elastodynamics and complex (e.g., exponential nonlinear) constitutive relations of the material undergoing large deformations, showing that the DeepHyROMnet approach outperforms the GalerkinRB method equipped with DEIM in terms of computational speedup during the online stage, achieving the same level of accuracy results.
The paper is structured as follows. We recall the formulation of the RB method for nonlinear unsteady parametrized PDEs in Sect. 2, relying on POD for the construction of the reduced subspace and on DEIM as hyperreduction technique (hence obtaining PODGalerkinDEIM ROMs). The proposed DeepHyROMnet and the DNN architecture employed to perform reduced operator approximation are detailed in Sect. 3. The numerical performances of the resulting method are assessed in Sect. 4 on several benchmark problems related with nonlinear elastodynamics, highlighting some conclusions in Sect. 5.
2 ProjectionBased ROMs: The Reduced Basis Method
Our goal is to pursue an efficient solution to nonlinear unsteady PDE problems depending on a set of input parameters, which can be written in abstract form as follows: given an input parameter vector \({\varvec{\mu }}\in \mathcal {P}\) and \(\textbf{u}(0;{\varvec{\mu }})\), find \(\textbf{u}(t;{\varvec{\mu }})\in V\) such that, \(\forall t\in (0,T]\),
where the parameter space \(\mathcal {P}\subset \mathbb {R}^P\) is a compact set, R is the residual of a secondorder differential equation, and \(V'\) is the dual of a suitable functional space \(H^1_0(\varOmega )^m\subseteq V \subseteq H^1(\varOmega )^m\) over the bounded domain \(\varOmega \subset \mathbb {R}^d\), where V depends on the boundary conditions at hand. In particular, we are interested in vector problems (\(m=3\)) set in \(d=3\) dimensions. The role of the parameter vector \({\varvec{\mu }}\) depends on the particular application at hand; in the case of nonlinear elastodynamics, \({\varvec{\mu }}\) is for instance related to the coefficients of the constitutive relation, the material properties and the boundary conditions.
After discretising problem (1) in space and time, we end up with a fullydiscrete nonlinear system
at each time step \(t^n\), \(n=1,\ldots ,N_t\), which can be solved by means of the Newton method: given \({\varvec{\mu }}\in \mathcal {P}\) and an initial guess \(\textbf{u}_h^{n,(0)}({\varvec{\mu }})\), for \(k\ge 0\), find \(\mathbf {\delta u}_h^{n,(k)}({\varvec{\mu }})\in \mathbb {R}^{N_h}\) such that
until suitable stopping criteria are fulfilled. Here, \(\textbf{u}_h^{n,(k)}({\varvec{\mu }})\) represents the solution vector for a fixed parameter \({\varvec{\mu }}\) computed at time step \(t^{n}\) and Newton iteration k, while \(\textbf{R}\in \mathbb {R}^{N_h}\) and \(\textbf{J}\in \mathbb {R}^{N_h\times N_h}\) denote the residual vector and the corresponding Jacobian matrix, respectively. In particular, \(\textbf{u}_h^{n,(0)}({\varvec{\mu }})\) is selected as the solution vector obtained at time \(t^{n1}\) once Newton iterations have converged. We refer to (3) as the highfidelity, FOM for problem (1). In particular, we rely on a GalerkinFE method for space approximation, and consider implicit finite difference schemes for time discretization, which do not require restrictions on \(\Delta t\) [48]. The highfidelity dimension \(N_h\) is determined by the underlying mesh and the chosen FE polynomial order and can be extremely large in case of very fine meshes.
To reduce the FOM numerical complexity, we introduce a projectionbased ROM, by relying on the RB method [49]. The idea of the RB method is to suitably select \(N\ll N_h\) vectors of \(\mathbb {R}^{N_h}\), forming the socalled RB matrix \(\textbf{V}\in \mathbb {R}^{N_h\times N}\), and to generate a reduced problem by performing a Galerkin projection of the FOM onto the subspace \(V_N=\text {Col}(\textbf{V})\subset \mathbb {R}^{N_h}\) generated by these vectors. This method relies on the assumption that the reducedorder approximation can be expressed as a linear combination of few, problemdependent, basis functions, that is
for \(n=1,\ldots ,N_t\), where \(\textbf{u}_N^{n}({\varvec{\mu }})\in \mathbb {R}^N\) denotes the vector of the ROMdofs at time \(t^n\ge 0\). The latter is obtained by imposing that the projection of the FOM residual computed on the ROM solution is orthogonal to the trial subspace (in the case of a Galerkin projection): given \({\varvec{\mu }}\in \mathcal {P}\), at each time \(t^{n}\), for \(n=1,\ldots ,N_t\), we seek \(\textbf{u}_N^{n}({\varvec{\mu }})\in \mathbb {R}^{N}\) such that
From now on, we will denote the reduced residual \(\textbf{V}^T\textbf{R}\) and the corresponding Jacobian \(\textbf{V}^T\textbf{JV}\) as \(\textbf{R}_N\) and \(\textbf{J}_N\), respectively. Then, the associated reduced Newton problem at time \(t^{n}\) reads: given \(\textbf{u}_N^{n,(0)}({\varvec{\mu }})\), for \(k\ge 0\), find \(\mathbf {\delta u}_N^{n,(k)}({\varvec{\mu }})\in \mathbb {R}^{N}\) such that
until a suitable stopping criterion is fulfilled.
2.1 SolutionSpace Reduction: Proper Orthogonal Decomposition
In this section we provide an overview of the proper orthogonal decomposition (POD) technique used to compute the reduced basis \(\textbf{V}\) through the socalled method of snapshots [8, 14]. Let
be the (discrete) solution manifold identified by the image of \(\textbf{u}_h\), that is, the set of all the PDE solutions for \({\varvec{\mu }}\) varying in the parameter space and \(t^n\) in the partition of the time interval. Our goal is to approximate \(\mathcal {M}_{u_h}\) with a reduced linear manifold, the trial manifold
To do this, given \(n_s<N_h\) sampled instances of \({\varvec{\mu }}\in \mathcal {P}\), the snapshots matrix \(\textbf{S}\) is constructed by collecting columnwise the FOM solution \(\textbf{u}_h^n({\varvec{\mu }}_\ell )\) at each time step \(n=1,\ldots ,N_t\), for \(\ell =1,\ldots ,n_s\), that is
Sampling can be performed, e.g., through a latin hypercube sampling design, as well as through suitable lowdiscrepancy points sets. The POD basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\) spanning the subspace \(V_N\) is usually obtained by performing the singular value decomposition \(\textbf{S}=\mathbf {U\Sigma Z}^T\) of \(\textbf{S}\), and then collecting the first N columns of \(\textbf{U}\), corresponding to the N largest singular values stored in the diagonal matrix \({\varvec{\Sigma }}= \text {diag}(\sigma _1,\ldots ,\sigma _{r})\in \mathbb {R}^{n_s\times n_s}\), with \(\sigma _1\ge \dots \ge \sigma _{r}\ge 0\) and \(r\le N_h\wedge n_s\) being the rank of \(\textbf{S}\). This yields an orthonormal basis such that
where \(\Vert \cdot \Vert _F\) is the Frobenius norm. Hence, singular values’ decay directly impacts on the size N, which can be computed as the minimum integer satisfying
for a given tolerance \(\varepsilon _{POD}>0\). In this work we exploit the socalled randomizedSVD, which offers a powerful tool to perform lowrank matrix approximations when dealing with massive datasets [34], such as highdimensional snapshots matrices.
2.2 HyperReduction: The Discrete Empirical Interpolation Method
In the case of parametrized PDEs featuring nonaffine dependence on the parameter and/or nonlinear (highorder polynomial or nonpolynomial) dependence on the field variable, a further level of reduction, known as hyperreduction, must be introduced [30, 42]. Note that if nonlinearities only include quadratic (or, at most, cubic) terms and do not feature any parameter dependence, assembling of nonlinear terms in the ROM can be performed by projection of the corresponding FOM quantities, once and for all [28].
For the case at hand, the residual \(\textbf{R}_N\) and the Jacobian \(\textbf{J}_N\) appearing in the reduced Newton system (4) depend on the solution at the previous iteration and, therefore, must be computed at each step \(k\ge 0\). It follows that, for any new instance of the parameter \({\varvec{\mu }}\), we need to assemble the highdimensional FOMarrays before projecting them onto the reduced subspace, entailing a computational complexity which is still of order \(N_h\). To setup an efficient offline–online computational splitting, an approximation of the nonlinear operators that is independent of the highfidelity dimension is required.
Several techniques have been employed to provide this further level of approximation [4, 6, 13, 15, 20]; among these, DEIM has been successfully applied to stationary or quasistatic nonlinear mechanical problems [11, 27]. Its key idea is to replace the nonlinear arrays in (4) with a collateral reduced basis expansion, computed through an inexpensive interpolation procedure. In this framework, the highdimensional residual \(\textbf{R}({\varvec{\mu }})\) is projected onto a reduced subspace of dimension \(m<N_h\) spanned by a basis \({\varvec{\Phi }}_\mathcal {R}\in \mathbb {R}^{N_h\times m}\)
where \(\textbf{r}({\varvec{\mu }})\in \mathbb {R}^m\) is the vector of the unknown amplitudes. The matrix \({\varvec{\Phi }}_\mathcal {R}\) can be precomputed offline by performing POD on a set of highfidelity residuals collected when solving (4) for \(n_s'\) training input parameters
The unknown parameterdependent coefficient \(\textbf{r}({\varvec{\mu }})\) is obtained online by collocating the approximation at the m components selected by a greedy procedure, and requiring that, for these components, \(\textbf{P}^T\textbf{R}({\varvec{\mu }}) = \textbf{P}^T{\varvec{\Phi }}_\mathcal {R}\textbf{r}({\varvec{\mu }})\), so that
where \(\textbf{P}\in \mathbb {R}^{N_h\times m}\) is the boolean matrix associated with the interpolation constraints. We thus define the hyperreduced residual vector as
Finally, the associated Jacobian approximation \(\textbf{J}_{N,m}({\varvec{\mu }})\) can be computed as the derivative of \(\textbf{R}_{N,m}({\varvec{\mu }})\) with respect to the reduced displacement, obtaining
or relying on the MDEIM algorithm [42], see [11, 41].
However, the application of DEIM in the case of nonlinear timedependent PDEs can be rather inefficient, especially when turning to complex problem which require a high number of residual basis (and, thus, of interpolation points) to ensure the convergence of the hyperreduced Newton system
In fact, the m points selected by the DEIM algorithm correspond to a subset of nodes of the computational mesh, which, together with the neighboring nodes (i.e. those sharing the same cell), form the socalled reduced mesh; see, e.g., the sketch reported in Fig. 1. Since the entries of any FEvector are associated with the dofs of the problem, \(\textbf{P}^T\textbf{R}({\varvec{\mu }})\) is computed by integrating the residual only on the quadrature points belonging to the elements forming the reduced mesh, which, nevertheless, can be rather dense.
A modification of the DEIM algorithm, UDEIM, has been proposed in [54] to exploit the sparsity of the problem and minimize the number of element function calls. However, a high number of nonlinear function evaluations is still required when the number of magic points is sufficiently large. Indeed, DEIMbased affine approximations are effective, in terms of computational costs, provided that few entries of the nonlinear terms can be cheaply computed; however, this situation does not occur neither for dynamical systems arising from the linearization of a nonlinear system around a steady state, nor when dealing with global nonpolynomial nonlinearities.
In this paper, we propose an alternative technique to perform hyperreduction, which is independent of the underlying mesh and relies on a DNN architecture to approximate reduced residual vectors and Jacobian matrices. The introduction of a surrogate model to perform operator approximation is justified by the fact that, often, most of the CPU time needed online for each new parameter instance is required by PODGalerkinDEIM ROMs for assembling arrays such as residual vectors or corresponding Jacobian matrices on the reduced mesh.
3 Operator Approximation: A Deep LearningBased Technique (DeepHyROMnet)
To recover the offline–online efficiency of the RB method, overcoming the need to assemble the nonlinear arrays onto the computational mesh as in the case of DEIM, we present a novel projectionbased method which relies on DNNs for the approximation of the nonlinear terms. We refer to this strategy as to a hyperreduced order model enhanced by deep neural networks (DeepHyROMnet). Our goal is the efficient numerical approximation of the sets
which represents the reduced residual manifold and reduced Jacobian manifold, respectively, in a way that depends only on the ROM dimension N and on the number of parameters P. To achieve this task, we employ the DNN architecture developed in [23] for the socalled DLROM techniques. It is worthy to note that, except for the approximation error of the reduced nonlinear operators, the proposed DeepHyROMnet approach is a physicsbased method and that the computed solution satisfies the nonlinear equation of the problem at hand, up to a further approximation of ROM residual and Jacobian arrays—thus, similarly to what happened for a PODGalerkinDEIM ROM. The main idea of the deep learningbased operator approximation approach that replaces DEIM in our new DeepHyROMnet strategy, is to learn the following inputtoresidual and inputtoJacobian maps, respectively:
provided \(({\varvec{\mu }},t^n,k)\in \mathcal {P}\times \{t^1,\ldots ,t^{N_t}\}\times \mathbb {N}^+\) as inputs and to finally replace the linear system in (4) with
Hence, we aim at approximating the residual vector and the Jacobian matrix obtained after their projection onto the reduced space of dimension \(N\ll N_h\). Indeed, performing a Galerkin projection onto the POD solution space allows to severely reduce the problem dimension from \(N_h\) to N and, hence, to ease the learning task with respect the reconstruction of the fullorder residual \(\textbf{R}\) and Jacobian \(\textbf{J}\).
Remark 1
As an alternative to Newton iterative scheme, we can rely on Broyden’s method [12], which belongs to the class of quasiNewton methods. This allows to avoid the computation of the Jacobian matrix at each iteration \(k\ge 0\) by relying on rankone updates, based on residuals computed at previous iterations. However, we are able to compute Jacobian matrices very efficiently using automatic differentiation (AD), so that the computational burden is the assembling of residual vectors. For this reason, in this paper we will focus on the Newton method only, that is, the solution of problem (4).
To summarize, in the case of the Newton approach, we end up with the following reduced problem: given \({\varvec{\mu }}\in \mathcal {P}\) and, for \(n=1,\ldots ,N_t\), the initial guess \(\textbf{u}_N^{n,(0)}({\varvec{\mu }}) = \textbf{u}_N^{n1}({\varvec{\mu }})\), find \(\delta \textbf{u}_N^{n,(k)}\in \mathbb {R}^N\) such that, for \(k\ge 0\),
until \(\Vert {\varvec{\rho }}_N({\varvec{\mu }},t^n,k)\Vert _2 / \Vert {\varvec{\rho }}_N({\varvec{\mu }},t^n,0)\Vert _2 < \varepsilon \), where \(\varepsilon >0\) is a given tolerance. In Algorithms 1 and 2, we report a summary of the offline and online stages of DeepHyROMnet, respectively.
3.1 Deep Neural Network Construction
For the sake of generality, we will focus on the DNNbased approximation of the reduced residual vector only, that is
In fact, by relying on a suitable transformation, we can easily write the Jacobian matrix as a vector of dimension \(N^2\) and apply to it the same procedure described in the following for the residual vector. In particular, we define the transformation
which consists in stacking the columns of \(\textbf{J}_N(\textbf{V}\textbf{u}_N^{n,(k)}({\varvec{\mu }}),t^n;{\varvec{\mu }})\) in a vector that is passed to the DNN as training sample, thus obtaining
Finally, the vec operation is reverted, so that \({{\varvec{\iota }}}_N({\varvec{\mu }},t^n,k) = vec^{1}(\widetilde{{\varvec{\iota }}}_N({\varvec{\mu }},t^n,k))\).
We thus aim at efficiently approximating the whole set \(\mathcal {M}_{R_N}\) by means of the reduced residual trial manifold, defined as
The approximation of the ROM residual \(\textbf{R}_N(\textbf{V}\textbf{u}_N^{n,(k)}({\varvec{\mu }}),t^n;{\varvec{\mu }})\) takes the form
where:

\({\varvec{\phi }}_q^{DF}(\cdot ~;{\varvec{\theta }}_{DF}):\mathbb {R}^{P+2}\rightarrow \mathbb {R}^q\) such that
$$\begin{aligned} {\varvec{\phi }}_q^{DF}({\varvec{\mu }},t^n,k;{\varvec{\theta }}_{DF}) = \textbf{R}_q({\varvec{\mu }},t^n,k;{\varvec{\theta }}_{DF}) \end{aligned}$$is a deep feedforward neural network (DFNN), consisting in the subsequent composition of a nonlinear activation function, applied to a linear transformation of the input, multiple times. Here, \({\varvec{\theta }}_{DF}\) denotes the vector of parameters of the DFNN, collecting all the corresponding weights and biases of each layer and q is as close as possible to the input size \(P+2\);

\(\textbf{f}^D_N(\cdot ~;{\varvec{\theta }}_{D}):\mathbb {R}^q\rightarrow \mathbb {R}^N\) such that
$$\begin{aligned} \textbf{f}^D_N(\textbf{R}_q({\varvec{\mu }},t^n,k;{\varvec{\theta }}_{DF});{\varvec{\theta }}_{D}) = \widetilde{\textbf{R}}_N({\varvec{\mu }},t^n,k;{\varvec{\theta }}_{DF},{\varvec{\theta }}_{D}) \end{aligned}$$is the decoder function of a convolutional autoencoder (CAE), obtained as the composition of several layers (some of which are convolutional), depending upon a vector \({\varvec{\theta }}_{D}\) collecting all the corresponding weights and biases.
The encoder function of the CAE is exploited, during the training stage only, to map the reduced residual \(\textbf{R}_N(\textbf{V}\textbf{u}_N^{n,(k)}({\varvec{\mu }}),t^n;{\varvec{\mu }})\) associated to \(({\varvec{\mu }},t^n,k)\) onto a lowdimensional representation
where \(\textbf{f}^E_q(\cdot ~;{\varvec{\theta }}_{E}):\mathbb {R}^N\rightarrow \mathbb {R}^q\) denotes the encoder function, depending upon a vector \({\varvec{\theta }}_{E}\) of parameters. The choice of a CAE is due to the fact that, thanks to the shared parameters and local connectivity properties which allow to reduce the numbers of parameters of the network and the number of associated computations, convolutional layers are better suited than dense layers to handle highdimensional correlated data.
Remark 2
We point out that the input of the encoder function—that is, the reduced residual vector \( \textbf{R}_N\)—is reshaped into a square matrix by rewriting its elements in rowmajor order, thus obtaining \(\textbf{R}_N^{reshape}\in \mathbb {R}^{\sqrt{N}\times \sqrt{N}}\). If N is not a square, the input \(\textbf{R}_N\) is zeropadded as explained in [29], and the additional elements are subsequently discarded.
Regarding the prediction of the reduced residual for new unseen instances of the inputs, computing \({\varvec{\rho }}_N({\varvec{\mu }}_{test},t^n,k)\) for any given \({\varvec{\mu }}_{test}\in \mathcal {P}\), and for any possible \(n=1,\ldots ,N_t\), \(k\ge 0\), corresponds to the testing stage of a DFNN and of the decoder function of a convolutional AE. Thus, we discard the encoder function at testing time. The architecture used during the training stage is reported in Fig. 2; only the block highlited in the inner (red) box is then used during the testing phase.
Let \(\textbf{M}\in \mathbb {R}^{(P+2)\times N_{train}}\), with \(N_{train}=n_s'N_tN_k\), be the parameter matrix of the input triples, i.e.
The corresponding training dataset for \({\varvec{\rho }}_N\) is given by the reduced residual snapshots matrix \(\textbf{S}_{{\varvec{\rho }}}\in \mathbb {R}^{N\times N_{train}}\) defined as
that is, by the matrix collecting columnwise ROM residuals computed for \(n_s'\) sampled parameters \({\varvec{\mu }}_\ell \in \mathcal {P}\), at different time instances \(t^1,\ldots ,t^{N_t}\) and for each Newton iteration \(k\ge 0\). The training stage consists in solving the following optimization problem in the weights variable \({\varvec{\theta }}= ({\varvec{\theta }}_{E},{\varvec{\theta }}_{DF},{\varvec{\theta }}_{D})\):
where
The loss function (7) combines the reconstruction error, i.e. the error between the ROM residual and the corresponding DNN approximation, and the error between the intrinsic coordinates and the output of the encoder. The training stage of the DNN involved in DeepHyROMnet is detailed in Algorithm 3; in particular, we denote by \(\alpha \) the trainingvalidation splitting fraction, by \(\eta \) the starting learning rate of the ADAM optimizer, by \(N_b\) the batch size, by \(n_b = (1\alpha )N_{train}/N_b\) the number of minibatches and by \(N_e\) the maximum number of epochs. We recall that the total number of training samples is given by \(N_{train}= n_s'N_tN_k\). The testing stage of the DNN is detailed in Algorithm 4. As suggested by [23, 24], we set \(\alpha =8:2\), \(\eta =10^{4}\), \(N_b=20\) and \(N_e=10^4\). To avoid overfitting, we employ the earlystopping regularization technique [29], that is, we stop the training if (and when) the loss evaluated on the validation set does not decrease over 500 epochs. Regarding the NN architecture, a 12layer DFNN with 50 neurons per hidden layer and q neurons in the output layer is employed, where q is problemdependent and set equal to the intrinsic dimension, i.e. \(q = P + 2\) (the time instant and the Newton iteration are considered as additional parameters); further details about the CAE architecture are summarized in Tables 1 and 2. In all cases, except for the last convolutional layer of the decoder, we consider the ELU nonlinear activation function [18], selected by assessing the impact of different activation functions on the validation loss.
Remark 3
Differently from the minmax scaling used in [23, 24], we standardize the input and output of the DNN as follows. After splitting the data into training and validation sets according to a userdefined trainingvalidation splitting fraction \({\varvec{\alpha }}\), \(\textbf{M} = \left[ \textbf{M}^{train},\textbf{M}^{val}\right] \) and \(\textbf{S}_{{\varvec{\rho }}} = \left[ \textbf{S}_{{\varvec{\rho }}}^{train},\textbf{S}_{{\varvec{\rho }}}^{val}\right] \), we define for each row of the training set the corresponding mean and standard deviation
so that parameters are normalized by applying the following transformation
that is, each feature of the training parameter matrix is standardized. The same procedure is applied to the training snapshots matrix \(\textbf{S}_{{\varvec{\rho }}}^{train}\) by replacing \(M^i_{*}\) with \(S^i_{*}\), where \(*\in \{mean,sd\}\) respectively. Transformation (8) is applied to the validation and testing sets as well, but considering the mean and the standard deviation computed over the training set. In order to rescale the reconstructed solution to the original values, we apply the inverse transformation.
4 Numerical Results
In this section we assess the performances of the proposed DeepHyROMnet strategy on different applications related to parametrized nonlinear timedependent PDE problems, focusing on structural mechanics. In particular, we consider (1) a series of structural tests on a rectangular beam, with different loading conditions and a simple nonlinear constitutive law, and then (2) a test case on an idealized left ventricle geometry, simulating cardiac contraction.
4.1 Nonlinear Elastodynamics
Let us consider a continuum body \(\mathcal {B}\) embedded in a threedimensional Euclidean space. For a given parameter vector \({\varvec{\mu }}\in \mathcal {P}\), the displacement vector field
relates the material position \(\textbf{X}\) of a particle in the reference configuration \(\varOmega _0\), which we assume to coincide with the initial configuration, to its spatial position \(\textbf{x}\) in the current configuration \(\varOmega _t\) at time \(t>0\), being \(\chi (\textbf{X},t;{\varvec{\mu }}) = \textbf{x}\) the motion of the body. The corresponding deformation gradient
characterizes the change of material elements during motion, so that the change in volume between the reference and the current configurations at time \(t>0\) is given by \(J(\textbf{X},t;{\varvec{\mu }}) = \det \textbf{F}(\textbf{X},t;{\varvec{\mu }})>0\). The equation of motion for a continuous medium reads as follows:
where \(\rho _0\) is the density of the body, \(\mathbf {P(F)}\) is the first PiolaKirchhoff stress tensor and \(\textbf{b}_0\) is an external body force. Proper boundary and initial conditions must be specified to ensure the wellposedness of the problem. In addition, we need a constitutive equation for \(\textbf{P}\), that is, a stressstrain relationship describing the material behavior. Here, we consider hyperelastic materials, for which the existence of a strainenergy density function \(\mathcal {W}:Lin^+\rightarrow \mathbb {R}\) such that
is postulated. Note that, since \(\textbf{F}\) depends on the displacement \(\textbf{u}\), we can equivalently write \(\mathbf {P(F)}\) or \(\textbf{P}(\textbf{u})\).
The strong formulation of a general initial boundaryvalued problem in elastodynamics thus reads as follows: given a body force \(\textbf{b}_0 = \textbf{b}_0(\textbf{X},t;{\varvec{\mu }})\), a prescribed displacement \(\bar{\textbf{u}} = \bar{\textbf{u}}(\textbf{X},t;{\varvec{\mu }})\) and surface traction \(\bar{\textbf{T}} = \bar{\textbf{T}}(\textbf{X},t,\textbf{N};{\varvec{\mu }})\), find the unknown displacement field \(\textbf{u}({\varvec{\mu }}):\varOmega _0\times (0,T]\rightarrow \mathbb {R}^3\) so that
where \(\textbf{N}\) is the outer normal unit vector and \(\alpha ,\beta \in \mathbb {R}\). The boundary of the reference domain is partitioned such that \(\Gamma _0^D\cup \Gamma _0^N\cup \Gamma _0^R = \Gamma \), with \(\Gamma _0^i\cap \Gamma _0^j=\emptyset \) for \(i,j\in \{D,N,R\}\). This equation is inherently nonlinear and additional source of nonlinearity is introduced in the material law, i.e. when using a nonlinear strainenergy density function \(\mathcal {W} =\mathcal {W}(\textbf{F})\), which is often the case of engineering applications. For the sake of simplicity, in all test cases, we neglect the body forces \(\textbf{b}_0({\varvec{\mu }})\) and consider zero initial conditions \(\textbf{u}_0({\varvec{\mu }}) = \dot{\textbf{u}}_0({\varvec{\mu }}) = \textbf{0}\). Regarding boundary conditions, we consider \(\bar{\textbf{u}}({\varvec{\mu }}) = \textbf{0}\) on the Dirichlet boundary \(\Gamma _0^D\) and always assume \(\alpha =\beta =0\), so that we actually impose homogeneous Neumann conditions on \(\Gamma _0^R\). Finally, the traction vector is given by
where \(\textbf{g}(t;{\varvec{\mu }})\) represents an external load and will be specified according to the application at hand. Finally, the residual in (2) is given by
for \(n=1,\ldots ,N_t\), where \(\textbf{u}_h^0({\varvec{\mu }})\) and \(\textbf{u}_h^{1}({\varvec{\mu }})\) are known for the initial condition, and
for all \(i,j=1,\ldots ,N_h\), being \(\{{\varvec{\varphi }}_i\}_{i=1}^{N_h}\) a basis for the FE space.
As a measure of accuracy of the reduced approximations with respect to the FOM solution, we consider timeaveraged \(L^2\)errors of the displacement vector, that are defined as follows:
The CPU time ratio, that is the ratio between FOM and ROM computational times, is used to measure efficiency, since it represents the speedup offered by the ROM with respect to the FOM. The code is implemented in Python in our software package pyfe\(^\text {x}\), a Python binding with the inhouse Finite Element library life\(^\texttt {x}\) (https://lifex.gitlab.io/lifex), a highperformance C++ library based on the deal.II (https://www.dealii.org) Finite Element core [3]. Computations have been performed on a PC desktop computer with 3.70GHz Intel Core i59600K CPU and 16GB RAM, except for the training of the DNNs and the prediction of the reduced nonlinear operators that have been carried out on a Tesla V100 32GB GPU.
Furthermore, we introduce the following measures to evaluate the performance of the DNNs during the training process:
for residual vectors and Jacobian matrices, respectively, where
4.2 Deformation of a Clamped Rectangular Beam
The first series of test cases represents a typical structural mechanical problem, with reference geometry \(\bar{\varOmega }_0 = [0,10^{2}]\times [0,10^{3}]\times [0,10^{3}]\) m\(^3\), reported in Fig. 3.
We consider a nearlyincompressible neoHookean material, which is characterized by the following strain density energy function
where \(G>0\) is the shear modulus, \(\mathcal {I}_1 = J^{\frac{2}{3}}\det (\textbf{C})\) and the latter term is needed to enforce incompressibility, being the bulk modulus \(K>0\) the penalization factor. This choice leads to the following first PiolaKirchhoff stress tensor, characterized by a nonpolynomial nonlinearity,
The beam is clamped at the lefthand side, that is, Dirichlet boundary conditions are imposed on the left face \(x=0\), whilst a pressure load changing with the deformed surface orientation is applied to the entire bottom face \(z=0\) (i.e. \(\Gamma _0^N\)). Homogeneous Neumann conditions are applied on the remaining boundaries (i.e. \(\Gamma _0^R\) with \(\alpha =\beta =0\)). As possible functions for the external load \(\textbf{g}(t;{\varvec{\mu }})\), we choose

1.
A linear function \(\textbf{g}(t;{\varvec{\mu }}) = \widetilde{p}~t/T\);

2.
A triangular or hat function \(\textbf{g}(t;{\varvec{\mu }}) = \widetilde{p}~\left( 2t~\chi _{\left( 0,\frac{T}{2}\right] }(t) + 2(Tt)~\chi _{\left( \frac{T}{2},T\right] }(t)\right) \);

3.
A step function \(\textbf{g}(t;{\varvec{\mu }}) = \widetilde{p}~\chi _{\left( 0,\frac{T}{3}\right] }(t)\), so that the presence of the inertial term is not negligible.
Here, \(\widetilde{p}>0\) is a parameter controlling the maximum load. The FOM is built on a hexahedral mesh with 640 elements and 1025 vertices, resulting in a highfidelity dimension \(N_h=3075\) (since \(\mathbb {Q}_1\)FE are employed). The resulting computational mesh in the reference configuration is reported in Fig. 3.
The following sections are organized as follows: first, we analyze the accuracy and the efficiency of the PODGalerkin ROM without hyperreduction with respect to the POD tolerance \(\varepsilon _{POD}\), thus resulting in reduced subspaces of different dimensions \(N\in \mathbb {N}\). Then, for a fixed basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\), PODGalerkinDEIM approximation capabilities are investigated for different sizes of the reduced mesh, associated with different values of the DEIM tolerance \(\varepsilon _{DEIM}\) for the computation of the residual basis \({\varvec{\Phi }}_{\mathcal {R}}\in \mathbb {R}^{N_h\times m}\). Finally, the performances of DeepHyROMnet are assessed and compared to those of DEIMbased hyperROMs.
4.2.1 Test Case 1: Linear Function for the Pressure Load
Let us consider the parametrized linear function
for the pressure load, describing a situation in which a structure is progressively loaded. We choose a time interval \(t\in [0,0.25]\) s and employ a uniform time step \(\Delta t = 5\times 10^{3}\) s for the time discretization scheme, resulting in a total number of 50 time iterations. As parameters, we consider:

The shear modulus \(G\in [0.5\times 10^4,1.5\times 10^4]\) Pa;

The bulk modulus \(K\in [2.5\times 10^4,7.5\times 10^4]\) Pa;

The external load parameter \(\widetilde{p}\in [2,6]\) Pa.
Given a training set of \(n_s=50\) points generated from the threedimensional parameter space \(\mathcal {P}\) through latin hypercube sampling (LHS), we compute the reduced basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\) using the POD method with tolerance
The corresponding reduced dimensions are \(N=3\), 4, 5, 6, 8, 9 and 15, respectively. In Fig. 4 we show the singular values of the snapshot matrix related to the FOM displacement \(\textbf{u}_h\), where a rapid decay of the plotted quantity means that a small number of RB functions is needed to correctly approximate the FOM solution.
The average relative error \(\epsilon _{rel}\) between the FOM and the PODGalerkin ROM solutions computed over a testing set of 50 randomly chosen parameters, different from the ones used to compute the solution snapshots, is reported in Fig. 5, together with the CPU time ratio. The approximation error decreases up to an order of magnitude when reducing the POD tolerance \(\varepsilon _{POD}\) from \(10^{3}\) to \(10^{6}\), corresponding to an increase of the RB dimension from \(N=3\) to \(N=15\). Despite the low dimension of the POD space, the computational speedup achieved by the reduced model is negligible. This is due to the fact that the ROM still depends on the FOM dimension \(N_h\) during the online stage. For this reason, we need to rely on suitable hyperreduction techniques.
For the construction of the hyperreduced models, i.e. PODGalerkinDEIM ROMs and DeepHyROMnet, we first need to compute snapshots from the ROM solutions for given parameter values and time instants, in order to build either the DEIM residual basis \({\varvec{\Phi }}_\mathcal {R}\) or train the DNNs. To this goal, we choose a PODGalerkin ROM with dimension \(N=4\), yielding a good balance between accuracy and computational effort for the test case at hand, and perform ROM simulations for a given set of \(n_s'=200\) parameter samples to collect residual and Jacobian data. To investigate the impact of hyperreduction on the ROM solution error, we compute the DEIM basis \(\mathbf {{\varvec{\Phi }}}_{\mathcal {R}}\) for approximating the residual using different DEIM tolerances,
corresponding to \(m=22\), 25, 30, 33, 39, 43, 51, respectively, DEIM basis functions. Larger tolerances were not sufficient to ensure the convergence of the Newton method for all considered combinations of parameters, so that higher speedups cannot be achieved by decreasing the basis dimension m. The average relative error \(\epsilon _{rel}\) is evaluated over the testing set and plotted in Fig. 6, as well as the CPU time ratio. To compute the highfidelity solutions, 26 s are required in average, while a PODGalerkinDEIM ROM, with \(N=4\) and \(m=22\), requires only 2.4 s, thus yielding a speedup about 11 times compared to the FOM.
Data related to the performances of the PODGalerkinDEIM ROMs for \(N=4\) and different values of m are shown in Table 3. The number of elements of the reduced mesh represents a small percentage of the one forming the original grid, so that the cost related to the residual assembling is remarkably alleviated. Nonetheless, it is evident how the main computational bottleneck is the construction of the reduced system at each Newton iteration, and in particular the assembling of the residual vector on the reduced mesh, which requires between \(78\%\) and \(88\%\) of the total (online) CPU time. In particular, almost \(90\%\) of this computational time is required for assembling the residual \(\textbf{R}(\textbf{Vu}_N^{n,(k)}({\varvec{\mu }}),t^{n};{\varvec{\mu }})\) on the reduced mesh, while computing the associated Jacobian matrix using the AD tool takes less than \(1\%\).
The performances of DeepHyROMnet are first analyzed by considering different sizes of the training set, and then compared in terms of both accuracy and efficiency with those of different PODGalerkinDEIM ROMs.
From the evolution of the \(L^2(\varOmega )\)absolute error displayed in Fig. 7 (left), averaged over 50 testing parameters values, we observe that relying on a training set of \(n_s'=200\) ROM simulations results in an approximation accuracy up to two orders of magnitude higher than the one obtained for \(n_s'=100\). On the other hand, further increasing the dimension of the dataset, e.g. \(n_s'=300\), does not affect the results due to the fact that the NN architecture is kept fixed and, consequently, the additional information becomes redundant. For this reason, we focus on the results obtained with \(n_s'=200\), corresponding to the same training parameters used for the construction of the DEIM bases.
The average of the absolute error \(\epsilon _{abs}\), the relative error \(\epsilon _{rel}\) and the CPU time ratio are reported in Table 4. Moreover, in Table 5 we report the relative errors (11) for the approximation of the reduced nonlinear arrays by means of the DNNs, computed on both training and testing sets, showing that the approximation of the ROM residual is the most challenging task due to the higher variability of the data. In terms of efficiency, DeepHyROMnets outperform DEIMbased hyperROMs substantially, being almost 100 times faster than the PODGalerkinDEIM ROM exploiting \(m=22\) DEIM basis functions, whilst achieving the same accuracy for \(n_s'=200\). In particular, our DeepHyROMnet approach allows us to compute the reduced solutions in less than 0.02 s, thus yielding an overall speedup of order \(\mathcal {O}(10^3)\) compared to the FOM.
The evolution of the \(L^2(\varOmega _0)\)absolute error, averaged over the testing parameters, is reported in Fig. 7 for both the reduction approaches considered. The final accuracy of the hyperROMs equals that of the ROM without hyperreduction, i.e. \(\epsilon _{rel}\approx 10^{2}\), meaning that the projection error dominates over the nonlinear operators approximation error. The difference between the FOM and DeepHyROMnet solutions at time \(T=0.25\) s is shown in Fig. 8 in two scenarios.
Table 6 collects the offline CPU time required for building the POD basis and collecting the ROM nonlinear operators, as well as the offline GPU time needed for training the DNN. Moreover, we report the total offline computational time—assuming that the training stages are run sequentially—and the break even point, that is the number of online FOM simulations that make the construction of the ROM affordable. We recall that FOM and PODGalerkin ROM simulations are run in serial on a PC desktop computer with 3.70GHz Intel Core i59600K CPU and 16GB RAM, whereas the DNN are trained on a Tesla V100 32GB GPU.
In order to improve the accuracy of the reduced solution, we should consider higher values of the RB dimension N. Nonetheless, by increasing the number of the POD basis functions, thus the size of the reduced nonlinear data, the task of the DNNs becomes more complex. As a matter of fact, from the results reported in Fig. 9, we observe that more training samples and a larger size of the neural networks are required when N increases, in order to achieve comparable results with the ones obtained for smaller values of the POD dimension. In particular, we have included an additional hidden layer in both the encoder and the decoder of the CAE, and considered a larger numbers of neurons in the dense layers. To conclude, we report in Table 7 the computational data associated with PODGalerkinDEIM and DeepHyROMnet hyperROMs when \(N=8\), but the same number of training snapshots and the same DNN architectures of previous case (i.e. \(N=4\)) are employed. As expected, the PODGalerkinDEIM method is able to provide more accurate approximations of the FOM solution by increasing the size m of the residual basis, albeit reducing the online speedup substantially.
4.2.2 Test Case 2: Hat Function for the Pressure Load
Let us now consider a piecewise linear pressure load given by
describing the case in which a structure is increasingly loaded until a maximum pressure is reached, and then linearly unloaded in order to recover the initial resting state. For the case at hand, we choose \(t\in [0,0.35]\) s and \(\Delta t = 5\times 10^{3}\) s, resulting in a total number of 70 time steps. As parameter, we consider the external load parameter \(\widetilde{p}\in [2,12]\) Pa; the shear modulus G and the bulk modulus K are fixed to \(10^4\) Pa and \(5\times 10^4\) Pa, respectively. Let us consider a training set of \(n_s=50\) points generated from \(\mathcal {P}=[2,12]\) Pa through LHS and build the RB basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\) with \(N=4\), corresponding to \(\varepsilon _{POD}=10^{4}\). The singular values of the solution snapshots matrix are reported in Fig. 10.
Given the GalerkinROM nonlinear data collected for \(n_s'=300\) sampled parameters, different DEIM residual bases \(\mathbf {{\varvec{\Phi }}}_{\mathcal {R}}\) are computed using different tolerances
corresponding to \(m=14\), 16, 22, 24, 29, 31, 37 DEIM basis functions, respectively. Tolerances \(\varepsilon _{DEIM}\) larger than the values reported above were not sufficient to ensure convergence of the Newton method for all the considered parameters.
The relative error \(\epsilon _{rel}\), evaluated over a testing set of 50 parameters, is about \(10^{2}\) when using \(m=14\) residual basis functions, and can be further reduced of one order of magnitude when increasing the DEIM dimension to \(m=29\), albeit highly decreasing the CPU time ratio, as shown in Fig. 11.
Table 8 shows the comparison between the PODGalerkinDEIM (with \(m=14\) and \(m=29\)) and the DeepHyROMnet hyperreduced models on a testing set of 50 parameter instances. As observed in the previous test case, DeepHyROMnet is able to achieve good results in terms of accuracy, comparable with the fastest PODGalerkinDEIM ROM (\(m=14\)), at a greatly reduced cost. Also in this case, the speedup achieved by our DeepHyROMnet is of order \(\mathcal {O}(10^3)\) with respect to the FOM, since less than 0.04 s are needed to compute the reduced solution for each new instance of the parameter, against a time of about 40 s required by the FOM, and of (at least) 3 s required by PODGalerkinDEIM ROMs.
The evolution in time of the average \(L^2(\varOmega _0)\)absolute error for the PODGalerkinDEIM and the DeepHyROMnet models is shown in Fig. 12. The accuracy obtained using DeepHyROMnet, although slightly lower than the ones achieved using a DEIMbased approximation, is satisfying in all the considered scenarios. Figure 13 shows the FOM and the DeepHyROMnet displacements at different time instances obtained for a given testing parameter.
We finally report in Table 9 the offline CPU and GPU times required for the test case at hand.
4.2.3 Test Case 3: Step Function for the Pressure Load
As last test case for the beam geometry, we consider a pressure load acting on the bottom surface area for only a third of the whole simulation time, that is
such that the resulting deformation features oscillations. This case is of particular interest in nonlinear elastodynamics, since the inertial term cannot be neglected, as it has a crucial impact on the deformation of the body. For the case at hand, we choose \(t\in [0,0.27]\) s and a uniform time step \(\Delta t = 3.6\times 10^{3}\) s, resulting in a total number of 75 time iterations. Concerning the input parameters, we vary the external load \(\widetilde{p}\in [2,12]\) Pa and consider \(G=10^4\) Pa and \(K=5\times 10^4\) Pa fixed.
We build the POD basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\) from a training set of \(n_s=50\) FOM solutions using \(\varepsilon _{POD}=10^{3}\), obtaining also in this case a reduced dimension of \(N=4\), and perform PODGalerkin ROM simulations for a given set of \(n_s'=300\) parameter samples to collect the nonlinear terms data necessary for the construction of both PODGalerkinDEIM and DeepHyROMnet models. The DEIM basis \(\mathbf {{\varvec{\Phi }}}_{\mathcal {R}}\) for the approximation of the residual is computed using as tolerances
where \(\varepsilon _{DEIM}=10^{3}\) is the larger tolerance that ensures the convergence of the reduced Newton algorithm for all testing parameters. The corresponding number of bases for \(\textbf{R}\) is \(m=18\), 20, 27, 30, 38, 40, 50, respectively. The results regarding the average relative error \(\epsilon _{rel}\) and the computational speedup, evaluated over 50 instances of the parameter, are shown in Fig. 14.
Like for the previous test cases, we compare PODGalerkinDEIM and DeepHyROMnet ROMs, with respect to the displacement error and the CPU time ratio.
As reported in Table 10, DNNs outperform DEIM substantially in terms of efficiency also in this case when handling the nonlinear terms. Indeed, DeepHyROMnet yields a ROM that is more than 1000 times faster than the FOM (this latter requiring 51 s in average to be solved), still providing satisfactory results in terms of accuracy.
In Fig. 15 the DeepHyROMnet solution at different time instants for two different values of the parameter is shown, highlighting that our hyperROM is able to correctly capture the nonlinear behavior of the continuum body also when the inertial term cannot be neglected. Table 11 reports the offline times required for the construction of the ROMs in the present case.
4.3 Passive Inflation and Active Contraction of an Idealized Left Ventricle
The second problem we are interested in is the inflation and contraction of a prolate spheroid geometry representing an idealized left ventricle (see Fig. 16) where the boundaries \(\Gamma _0^R\), \(\Gamma _0^N\) and \(\Gamma _0^D\) represent the epicardium, the endocardium and the base of a left ventricle, respectively, the latter being the artificial boundary resulting from truncation of the heart below the valves in a short axis plane. We consider transversely isotropic material properties for the myocardial tissue, adopting a nearlyincompressible formulation of the constitutive law proposed in [31], whose strainenergy density function is given by \(\mathcal {W}(\textbf{F}) = \frac{C}{2}(e^{Q(\textbf{F})}  1)\), with the following form for Q to describe threedimensional transverse isotropy with respect to the fiber coordinate system,
here \(E_{ij}\), \(i,j\in \{f,s,n\}\), are the components of the GreenLagrange strain tensor \(\textbf{E} =\frac{1}{2}(\textbf{F}^T\textbf{F}\textbf{I})\), the material constant \(C>0\) scales the stresses and the coefficients \(b_f\), \(b_s\), \(b_n\) are related to the material stiffness in the fiber, sheet and transverse directions, respectively. This leads to a (passive) first PiolaKirchhoff stress tensor characterized by exponential nonlinearity. In order to enforce the incompressibility constraint, we consider an additional term \(\mathcal {W}_{vol}(J)\) in the definition of the strainenergy density function, which must grow as the deformation deviates from being isochoric. A common choice for \(\mathcal {W}_{vol}\) is a convex function with null slope in \(J=1\), e.g.,
where the penalization factor is the bulk modulus \(K>0\). Furthermore, to reproduce the typical twisting motion of the ventricular systole, we need to take into account a varying fiber distribution and contractile forces. The fiber direction is computed using the rulebased method proposed in [51], which depends on parameter angles \({\varvec{\alpha }}^{epi}\) and \({\varvec{\alpha }}^{endo}\). Active contraction is modeled through the active stress approach [1], so that we add to the passive first PiolaKirchoff stress tensor a timedependent active tension, which is assumed to act only in the fiber direction
where \(\textbf{f}_0\in \mathbb {R}^3\) denotes the reference unit vector in the fiber direction and \(T_a\) is a parametrized function that surrogates the active generation forces. In our case, since we are modeling only the systolic contraction, we define
with \(\widetilde{T}_a>0\). To model blood pressure inside the chamber we assume a linearly increasing external load
Since we want to assess the performance of DeepHyROMnet in enhancing the myocardium contraction problem, we consider as unknown parameters those related to the active components of the strainenergy density function:

The maximum value of the active tension \(\widetilde{T}_a\in [49.5\times 10^3,70.5\times 10^3]\) Pa, and

The fiber angles \({\varvec{\alpha }}^{epi}\in [105.5,74.5]^\circ \) and \({\varvec{\alpha }}^{endo}\in [74.5,105.5]^\circ \).
All other parameters are fixed to the reference values taken from [38], namely \(b_{f}=8\), \(b_{s}=b_{n}=b_{sn}=2\), \(b_{fs}=b_{fn}=4\), \(C=2\times 10^3\) Pa, \(K=50\times 10^3\) Pa and \(\widetilde{p}=15\times 10^3\). Regarding time discretization, we choose \(t\in [0,0.25]\) s and a uniform time step \(\Delta t = 5\times 10^{3}\) s, resulting in a total number of 50 time iterations. The FOM is built on a hexahedral mesh with 4804 elements and 6455 vertices, depicted in Fig. 16, corresponding to a highfidelity dimension \(N_h=19365\), since \(\mathbb {Q}_1\)FE (that is, linear FE on a hexahedral mesh) are used. In this case, the FOM requires almost 360 s to compute the solution dynamics for each parameter instance.
Given \(n_s=50\) points obtained by sampling the parameter space \(\mathcal {P}\), we construct the corresponding solution snapshots matrix \(\textbf{S}_u\) and compute the POD basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\) using the POD method with tolerance
From Fig. 17, we observe a slower decay of the singular values of \(\textbf{S}_u\) with respect to the structural problems of Sect. 4.2. In fact, we obtain larger reduced basis dimensions \(N=16\), 22, 39, 50, 87, 109 and 178, respectively.
The error and the CPU speedups averaged over a testing set of 20 parameters are both shown in Fig. 18, as functions of the POD tolerance \(\varepsilon _{POD}\).
Also in this case, the speedup achieved by the ROM is negligible, since at each Newton iteration without hyperreduction the ROM still depends on the highfidelity dimension \(N_h\). For what concerns the approximation error, we observe a reduction of almost two orders of magnitude when going from \(N=16\) to \(N=178\). The fact that greater POD dimension with respect to the previous test cases are now obtained, allows us to assess the performances of the DeepHyROMnet approach both on small and large values of N.
Given the reduced basis \(\textbf{V}\in \mathbb {R}^{N_h\times N}\) with \(N=16\), we construct the PODGalerkinDEIM approximation by considering \(n_s'=200\) parameter samples. Fig. 19 shows the decay of the singular values of \(\textbf{S}_R\), that is, the snapshots matrix of the residual vectors \(\textbf{R}(\textbf{Vu}_N^{n,(k)}({\varvec{\mu }}_{\ell '}),t^n;{\varvec{\mu }}_{\ell '})\). We observe that the reported curve decreases very slowly, so that we expect that a large number of basis functions is required to correctly approximate the nonlinear operators.
In fact, by computing \({\varvec{\Phi }}_{\mathcal {R}}\in \mathbb {R}^{N_h\times m}\) using the following DEIM tolerances,
we obtain \(m=303\), 456, 543, 776, 902 and 1233, respectively. Higher values of \(\varepsilon _{DEIM}\) (related to smaller dimensions m) were not sufficient to guarantee the convergence of the reduced Newton problem for all testing parameters. The average relative error over a set of 20 parameters and the computational speedup are both reported in Fig. 20. In particular, we observe that the relative error is between \(4\times 10^{3}\) and \(8\times 10^{3}\), as we could expect from the projection error reported in Fig. 18, that is, PODGalerkinDEIM ROMs are able to achieve the same accuracy of the ROM without hyperreduction.
The data reported in Table 12 leads to the same conclusions regarding the computational bottleneck of DEIM technique as those reported in Table 3. In fact, assembling the residual on the reduced mesh requires around \(85\%\) of the online CPU time, thus undermining the hyperROM efficiency.
Finally, Table 13 reports the computational data of PODGalerkinDEIM ROMs (for a number of magic points equals to \(m=303\) and \(m=543\)) and DeepHyROMnet, clearly showing that the latter outperforms the classical reduction strategy regarding the computational speedup. In fact, DeepHyROMnet is able to approximate the solution dynamics in 0.1 s, that is even faster than realtime given the timescale of the phenomenon at hand, while a PODGalerkinDEIM ROM requires 1 min in average, where the final simulation time T is set equal to 0.25 s. The offline CPU and GPU times are reported in Table 14. Although the DeepHyROMnet error is slightly larger than the one obtained with a DEIMbased hyperROM (see Fig. 21), the results are satisfactory in terms of accuracy. In Fig. 22 the FOM and the DeepHyROMnet displacements at time \(T=0.25\) s are reported for three different values of the parameters, together with the error between the highfidelity and the reduced solutions.
Table 15 reports the accuracy of the prediction on the interpolation functions, showing that the error on the reduced Jacobian matrix \(\textbf{J}_N\) is of the same order of the corresponding reduced residual vector \(\textbf{R}_N\) and one order of magnitude higher than in test case 4.2. This fact might be due to several issues, such as the larger dimension of the input, i.e., \(N^2=256\) instead of \(N^2=16\), the smaller size of the training set, and the overall increased complexity of the problem, either in the constitutive law and the geometry.
To conclude, we assess the performances of DeepHyROMnet on a problem involving a higher FOM dimension. We still consider the test case described in this Section, however using a finer hexahedral mesh with 9964 elements and 13025 vertices, thus obtaining \(N_h=39075\) as FOM dimension. In this case, about 13 minutes are required to compute the highfidelity solution. On the other hand, a reduced basis of dimension \(N=16\) is computed for \(\varepsilon _{POD}=10^{3}\); the computational data, averaged over a testing set of 20 parameter samples, are reported in Table 16. The online CPU time required by DeepHyROMnet doubles as we double \(N_h\), despite the same POD dimension \(N=16\) has been selected, and the same level of accuracy is obtained. This mild dependence of our hyperROM on the FOM dimension \(N_h\) is due to the reconstruction of the reduced solutions \(\textbf{Vu}_N^n({\varvec{\mu }})\), for \(n=1,\ldots ,N_t\), whereas the same computational time is required for the online assembling and solution of the reduced Newton systems. Nonetheless, the overall computational speedup of DeepHyROMnet increases as \(N_h\) grows, while the number N of reduced basis function remains small, so that reduced solutions can be computed extremely fast.
5 Conclusions
In this work we have addressed the solution to parametrized, nonlinear, timedependent PDEs arising in elastodynamics, by means of a new projectionbased ROM, developed to accurately capture the state solution dynamics at a reduced computational cost with respect to highfidelity FOMs. We focused on GalerkinRB methods, characterized by a projection of the differential problem onto a lowdimensional subspace built, e.g., by performing POD on a set of FOM solutions, and by the splitting of the reduction procedure into a costly offline phase and an inexpensive online phase. Numerical experiments showed that, despite their highly nonlinear nature, elastodynamics problems can be reduced by exploiting projectionbased strategies in an effective way, with PODGalerkin ROMs achieving very good accuracy even in presence of a handful of basis functions. However, when dealing with nonlinear problems, a further level of approximation is required to make the online stage independent of the highfidelity dimension.
Hyperreduction techniques, such as DEIM, are necessary to efficiently handle the nonlinear operators. However, a serious issue is represented by the assembling (albeit onto a reduced mesh) of the approximated nonlinear operators in this framework. This observation suggested the idea of relying on surrogate models to perform operator approximation, overcoming the need to assemble the nonlinear terms onto the computational mesh. Pursuing this strategy, we have proposed a new projectionbased, deep learningbased ROM, DeepHyROMnet, which combines the GalerkinRB approach with DNNs to assemble the reduced Newton system in an efficient way, thus avoiding the computational burden entailed by classical hyperreduction strategies. This approach allows to rely on physicsbased ROMs retaining the underlying structure of the physical model, as DNNs are employed only for the approximation of the reduced nonlinear operators. Regarding the offline cost of this hybrid reduction strategy, we point out that:

FOM solutions are only required to build the PODGalerkin ROM;

A small number N of reduced basis functions is sufficient to accurately approximate the highfidelity solution manifold, so that the arising reduced nonlinear systems can be solved efficiently;

Since data on nonlinear operators are collected during Newton iterations at each time step, a smaller number of ROM simulations—compared to purely datadriven approaches—is sufficient for training the DNNs;

Since training data are lowdimensional, we can avoid the overwhelming training times and costs that would be required by DNNs if FOM arrays were used.
The DeepHyROMnet approach has been successfully applied on several test cases in nonlinear solid mechanics, showing remarkable improvements in terms of online CPU time with respect to PODGalerkinDEIM ROMs. Our goal in future works is to apply the developed strategy to other classes of nonlinear problems for which traditional hyperreduction techniques represent a computational bottleneck.
Data Availability
The code used in this work, as well as the training and testing datasets, will be made available upon request to the authors.
Change history
24 February 2023
Missing Open Access funding information has been added in the Funding Note.
References
Ambrosi, D., Pezzuto, S.: Active stress versus active strain in mechanobiology: constitutive issues. J. Elast. 107(2), 199–212 (2012)
Amsallem, D., Zahr, M., Farhat, C.: Nonlinear model order reduction based on local reducedorder bases. Int. J. Numer. Methods Eng. 92(10), 891–916 (2012)
Arndt, D., Bangerth, W., Blais, B., Clevenger, T., Fehling, M., Grayver, A., Heister, T., Heltai, L., Kronbichler, M., Maier, M., Munch, P., Pelteret, J., Rastak, R., Thomas, I., Turcksin, B., Wang, Z., Wells, D.: The deal.II library, version 9.2. J. Numer. Math. 28(3), 131–146 (2020). https://doi.org/10.1515/jnma20200043
Astrid, P., Weiland, S., Willcox, K., Backx, T.: Missing point estimation in models described by proper orthogonal decomposition. IEEE Trans. Autom. Control 53(10), 2237–2251 (2008)
Bai, Z., Peng, L.: Nonintrusive nonlinear model reduction via machine learning approximations to lowdimensional operators. Adv. Model. Simul. Eng. Sci. 8(1), 1–24 (2021)
Barrault, M., Maday, Y., Nguyen, N., Patera, A.: An ‘empirical interpolation’ method: application to efficient reducedbasis discretization of partial differential equations. Compt. Rendus Math. 339(9), 667–672 (2004)
Benner, P., Goyal, P., Kramer, B., Peherstorfer, B., Willcox, K.: Operator inference for nonintrusive model reduction of systems with nonpolynomial nonlinear terms. Comput. Methods Appl. Mech. Eng. 372, 113433 (2020)
Benner, P., Gugercin, S., Willcox, K.: A survey of projectionbased model reduction methods for parametric dynamical systems. SIAM Rev. 57(4), 483–531 (2015)
Benner, P., Ohlberger, M., Patera, A., Rozza, G., Urban (Eds.), K.: Model Reduction of Parametrized Systems. Springer (2017)
Bhattacharya, K., Hosseini, B., Kovachki, N., Stuart, A.: Model reduction and neural networks for parametric PDEs. SMAI J. Comput. Math. 7, 121–157 (2021)
Bonomi, D., Manzoni, A., Quarteroni, A.: A matrix DEIM technique for model reduction of nonlinear parametrized problems in cardiac mechanics. Comput. Methods Appl. Mech. Eng. 324, 300–326 (2017)
Broyden, C.: A class of methods for solving nonlinear simultaneous equations. Math. Comput. 19(92), 577–593 (1965)
Carlberg, K., BouMosleh, C., Farhat, C.: Efficient nonlinear model reduction via a leastsquares Petrov–Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 86(2), 155–181 (2011)
Chatterjee, A.: An introduction to the proper orthogonal decomposition. Curr. Sci. 78(7), 808–817 (2000)
Chaturantabut, S., Sorensen, D.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)
Chen, T., Chen, H.: Universal approximation to nonlinear operators by neural networks with arbitrary activation functions and its application to dynamical systems. IEEE Trans. Neural Netw. 6(4), 911–917 (1995)
Cicci, L., Fresca, S., Pagani, S., Manzoni, A., Quarteroni, A.: Projectionbased reduced order models for parameterized nonlinear timedependent problems arising in cardiac mechanics. Math. Eng. 5(2), 1–38 (2023). https://doi.org/10.3934/mine.2023026
Clevert, D.A., Unterthiner, T., Hochreiter, S.: Fast and accurate deep network learning by exponential linear units (elus). arXiv preprint arXiv:1511.07289 (2015)
Drohmann, M., Haasdonk, B., Ohlberger, M.: Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput. 34(2), A937–A969 (2012)
Farhat, C., Chapman, T., Avery, P.: Structurepreserving, stability, and accuracy properties of the energyconserving sampling and weighting method for the hyper reduction of nonlinear finite element dynamic models. Int. J. Numer. Meth. Eng. 102(5), 1077–1110 (2015)
Farhat, C., Grimberg, S., Manzoni, A., Quarteroni, A.: Computational bottlenecks for PROMs: precomputation and hyperreduction. In: Benner, P., GrivetTalocia, S., Quarteroni, A., Rozza, G., Schilders, W., Silveira, L. (eds.) Model Order Reduction. SnapshotBased Methods and Algorithms, vol. 2, pp. 181–244. De Gruyter, Berlin (2020)
Franco, N., Manzoni, A., Zunino, P.: A deep learning approach to reduced order modelling of parameter dependent partial differential equations. arXiv preprint arXiv:2103.06183 (2021)
Fresca, S., Dede’, L., Manzoni, A.: A comprehensive deep learningbased approach to reduced order modeling of nonlinear timedependent parametrized PDEs. J. Sci. Comput. 87(2), 1–36 (2021)
Fresca, S., Manzoni, A.: PODDLROM: enhancing deep learningbased reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition. Comput. Methods Appl. Mech. Eng. 388, 114181 (2022)
Fresca, S., Manzoni, A., Dede’, L., Quarteroni, A.: Deep learningbased reduced order models in cardiac electrophysiology. PloS One 15(10), e0239416 (2020)
Gao, H., Wang, J., Zahr, M.: Nonintrusive model reduction of largescale, nonlinear dynamical systems using deep learning. Phys. D: Nonlinear Phenom. 412, 132614 (2020)
Ghavamian, F., Tiso, P., Simone, A.: PODDEIM model order reduction for strainsoftening viscoplasticity. Comput. Methods Appl. Mech. Eng. 317, 458–479 (2017)
Gobat, G., Opreni, A., Fresca, S., Manzoni, A., Frangi, A.: Reduced order modeling of nonlinear microstructures through proper orthogonal decomposition. Mech. Syst. Sign. Process. 171, 108864 (2022). https://doi.org/10.1016/j.ymssp.2022.108864
Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning. MIT Press (2016)
Grepl, M., Maday, Y., Nguyen, N., Patera, A.: Efficient reducedbasis treatment of nonaffine and nonlinear partial differential equations. ESAIM: Math. Model. Numer. Anal. 41(3), 575–605 (2007)
Guccione, J., Costa, K., McCulloch, A.: Finite element stress analysis of left ventricular mechanics in the beating dog heart. J. Biomech. 28(10), 1167–1177 (1995)
Guo, M., Hesthaven, J.: Reduced order modeling for nonlinear structural analysis using gaussian process regression. Comput. Methods Appl. Mech. Eng. 341, 807–826 (2018)
Guo, M., Hesthaven, J.S.: Datadriven reduced order modeling for timedependent problems. Comput. Methods Appl. Mech. Eng. 345, 75–99 (2019)
Halko, N., Martinsson, P., Tropp, J.: Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions. SIAM Rev. 53(2), 217–288 (2011)
Hesthaven, J., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. Springer (2016)
Hesthaven, J., Ubbiali, S.: Nonintrusive reduced order modeling of nonlinear problems using neural networks. J. Comput. Phys. 363, 55–78 (2018)
Kim, Y., Choi, Y., Widemann, D., Zohdi, T.: Efficient nonlinear manifold reduced order model. arXiv preprint arXiv:2011.07727 (2020)
Land, S., Gurev, V., Arens, S., Augustin, C., Baron, L., Blake, R., Bradley, C., Castro, S., Crozier, A., Favino, M., Fastl, T., Fritz, T., Gao, H., Gizzi, A., Griffith, B., Hurtado, D., Krause, R., Luo, X., Nash, M., Pezzuto, S., Plank, G., Rossi, S., Ruprecht, D., Seemann, G., Smith, N., Sundnes, J., Rice, J., Trayanova, N., Wang, D., Wang, Z., Niederer, S.: Verification of cardiac mechanics software: benchmark problems and solutions for testing active and passive material behaviour. Proc. R. Soc. A: Math. Phys. Eng. Sci. 471(2184), 20150641 (2015)
Lee, K., Carlberg, K.: Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders. J. Comput. Phys. 404, 108973 (2020)
Lu, L., Jin, P., Pang, G., Zhang, Z., Karniadakis, G.: Learning nonlinear operators via deeponet based on the universal approximation theorem of operators. Nat. Mach. Intell. 3(3), 218–229 (2021)
Manzoni, A., Bonomi, D., Quarteroni, A.: Reduced order modeling for cardiac electrophysiology and mechanics: New methodologies, challenges and perspectives. In: Boffi, D., Pavarino, L., Rozza, G., Scacchi, S., Vergara, C. (eds.) Mathematical and Numerical Modeling of the Cardiovascular System and Applications, SEMA SIMAI Springer Series, vol. 16, pp. 115–166. Springer, Cham (2018)
Negri, F., Manzoni, A., Amsallem, D.: Efficient model reduction of parametrized systems by matrix discrete empirical interpolation. J. Comput. Phys. 303, 431–454 (2015)
Pagani, S., Manzoni, A., Quarteroni, A.: Numerical approximation of parametrized problems in cardiac electrophysiology by a local reduced basis method. Comput. Methods Appl. Mech. Eng. 340, 530–558 (2018)
Peherstorfer, B., Butnaru, D., Willcox, K., Bungartz, H.: Localized discrete empirical interpolation method. SIAM J. Sci. Comput. 36(1), A168–A192 (2014)
Peherstorfer, B., Willcox, K.: Datadriven operator inference for nonintrusive projectionbased model reduction. Comput. Methods Appl. Mech. Eng. 306, 196–215 (2016)
Pinkus, A.: nWidths in Approximation Theory. SpringerVerlag, BerlinHeidelberg (1985)
Qian, E., Kramer, B., Marques, A., Willcox, K.: Transform & learn: a datadriven approach to nonlinear model reduction. In: AIAA Aviation 2019 Forum, p. 3707 (2019)
Quarteroni, A.: Numerical Models for Differential Problems, 2nd edn. Springer (2013)
Quarteroni, A., Manzoni, A., Negri, F.: Reduced Basis Methods for Partial Differential Equations. Springer International Publishing, An Introduction (2016)
Radermacher, A., Reese, S.: PODbased model reduction with empirical interpolation applied to nonlinear elasticity. Int. J. Numer. Methods Eng. 107(6), 477–495 (2016)
Rossi, S., Lassila, T., RuizBaier, R., Sequeira, A., Quarteroni, A.: Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics. Eur. J. Mech.A/Solids 48, 129–142 (2014)
Swischuk, R., Mainini, L., Peherstorfer, B., Willcox, K.: Projectionbased model reduction: formulations for physicsbased machine learning. Comput. Fluids 179, 704–717 (2019)
Tiso, P., Dedden, R., Rixen, D.: A modified discrete empirical interpolation method for reducing nonlinear structural finite element models. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, vol. 55973, p. V07BT10A043. American Society of Mechanical Engineers (2013)
Tiso, P., Rixen, D.: Discrete empirical interpolation method for finite element structural dynamics. In: Topics in Nonlinear Dynamics, Volume 1, pp. 203–212. Springer (2013)
Vlachas, K., Tatsis, K., Agathos, K., Brink, A., Chatzi, E.: A local basis approximation approach for nonlinear parametric model order reduction. J. Sound Vib. 502, 116055 (2021)
Wang, S., Wang, H., Perdikaris, P.: Learning the solution operator of parametric partial differential equations with physicsinformed deeponets. Sci. Adv. 7(40), eabi8605 (2021)
Funding
Open access funding provided by Politecnico di Milano within the CRUICARE Agreement. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 740132, iHEART  An Integrated Heart Model for the simulation of the cardiac function, P.I. Prof. A. Quarteroni) and the Fondazione Cariplo (grant agreement no. 20194608, P.I. Prof. A. Manzoni).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Cicci, L., Fresca, S. & Manzoni, A. DeepHyROMnet: A Deep LearningBased Operator Approximation for HyperReduction of Nonlinear Parametrized PDEs. J Sci Comput 93, 57 (2022). https://doi.org/10.1007/s10915022020018
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10915022020018
Keywords
 Nonlinear timedependent PDEs
 Reduced order modeling
 Deep learning
 Proper orthogonal decomposition
 Hyperreduction