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Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-set Method

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Abstract

We present an error-neural-modeling-based strategy for approximating two-dimensional curvature in the level-set method. Our main contribution is a redesigned hybrid solver [Larios-Cárdenas and Gibou, J. Comput. Phys. (May 2022), 10.1016/j.jcp.2022.111291] that relies on numerical schemes to enable machine-learning operations on demand. In particular, our routine features double predicting to harness curvature symmetry invariance in favor of precision and stability. The core of this solver is a multilayer perceptron trained on circular- and sinusoidal-interface samples. Its role is to quantify the error in numerical curvature approximations and emit corrected estimates for select grid vertices along the free boundary. These corrections arise in response to preprocessed context level-set, curvature, and gradient data. To promote neural capacity, we have adopted sample negative-curvature normalization, reorientation, and reflection-based augmentation. In the same manner, our system incorporates dimensionality reduction, well-balancedness, and regularization to minimize outlying effects. Our training approach is likewise scalable across mesh sizes. For this purpose, we have introduced dimensionless parametrization and probabilistic subsampling during data production. Together, all these elements have improved the accuracy and efficiency of curvature calculations around under-resolved regions. In most experiments, our strategy has outperformed the numerical baseline at twice the number of redistancing steps while requiring only a fraction of the cost.

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Data availability

The training datasets generated during the current study are not publicly available due to space limitations but are available from the corresponding author on reasonable request. Our neural networks, their preprocessing objects, and the testing flower-interface data sets we used, however, are publicly available at https://github.com/UCSB-CASL/Curvature_ECNet_2D.

Notes

  1. We use the terms node, grid point, and vertex interchangeably.

  2. We will limit ourselves to \(n = 2\) dimensions in this manuscript. Also, we will omit the explicit time dependence of \(\phi \) and \(\Gamma \) for compactness.

  3. For us, \(h\kappa \) denotes the numerical approximation to dimensionless curvature at the interface, is the neurally corrected estimation, \(h\kappa ^\star \) is MLCurvature()’s output, and \(h\kappa ^*\) represents the exact value.

  4. For consistency, we represent one-element nodal variables as M-vectors in lowercase bold faces (e.g., \(\varvec{{\phi }} \)) and variables with \(d > 1\) values per node as d-by-M matrices in caps (e.g., \({\hat{N}}\)). M is the number of vertices, e.g., all independent nodes that a p4est [48] macromesh is aware of.

  5. This is a dictionary mapping coordinate codes to numerical values; for example, m maps to \(-1\).

  6. One should choose \(h\kappa _{\min }^* \leqslant h\kappa _{\min }^{low}\), where the latter is the lower bound from Algorithm 1.

  7. In homogenous coordinates, \([x,y,1]^T\) is a point, while \([x,y,0]^T\) is a vector. \(T(\cdot )\) and \(R(\cdot )\) are thus 3-by-3 matrices in \({\mathbb {R}}^2\).

  8. Element-wise division.

  9. Standardization or z-scoring transforms a vector \(\varvec{{\psi }} \) into \(\varvec{{\psi }} '\), which has mean 0 and variance 1.

  10. If \(D \in {\mathbb {R}}^{n \times p}\) is a data set, and \(M \in {\mathbb {R}}^{n \times p}\) contains D’s column-wise mean p-vector \(\varvec{{\mu }} \) stacked n times, then \(C = \frac{1}{n-1}(D-M)^T(D-M)\) is D’s p-by-p covariance matrix. Further, if \({\mathbf {x}}= \varvec{{d}} - \varvec{{\mu }} \) is a centered, feature p-vector, and \(C = U\Sigma V^T\) is C’s SVD decomposition, then \({\mathbf {x}}_k = V_k^T{\mathbf {x}}\) is the PCA-transformed k-vector, where \(V_k\) contains V’s first k columns. Lastly, whitening involves normalizing \({\mathbf {x}}_k\) so that \({\mathbf {x}}_k' = \Sigma _k^{-1/2} {\mathbf {x}}_k\), where \(\Sigma _k\) is \(\Sigma \)’s k-order leading principal sub-matrix.

  11. The correlation matrix is equivalent to the covariance matrix of the standardized data [52].

  12. A single-precision general matrix-matrix multiplication of the form \(C = \alpha AB+\beta C\), where \(\alpha = 1\) and \(\beta = 0\).

  13. Using C++14 compiling optimization enabled via -O2 -O3 -march=native.

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The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

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Authors and Affiliations

  1. Computer Science Department, University of California, Santa Barbara, CA, 93106, USA

    Luis Ángel Larios-Cárdenas & Frédéric Gibou

  2. Mechanical Engineering Department, University of California, Santa Barbara, CA, 93106, USA

    Frédéric Gibou

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  1. Luis Ángel Larios-Cárdenas
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Contributions

Luis Ángel Larios-Cárdenas: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Resources, Data curation, Writing - Original draft, Visualization. Frédéric Gibou: Conceptualization, Methodology, Resources, Writing - Review & editing, Supervision, Project administration, Funding acquisition.

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Correspondence to Luis Ángel Larios-Cárdenas.

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Larios-Cárdenas, L.Á., Gibou, F. Error-Correcting Neural Networks for Two-Dimensional Curvature Computation in the Level-set Method. J Sci Comput 93, 6 (2022). https://doi.org/10.1007/s10915-022-01952-2

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