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Variational Approach to Closure of Nonlinear Dynamical Systems: Autonomous Case

Abstract

A general approach for the derivation of nonlinear parameterizations of neglected scales is presented for nonlinear systems subject to an autonomous forcing. In that respect, dynamically-based formulas are derived subject to a free scalar parameter to be determined per mode to parameterize. For each high mode, this free parameter is obtained by minimizing a cost functional—a parameterization defect—depending on solutions from direct numerical simulation (DNS) but over short training periods of length comparable to a characteristic recurrence or decorrelation time of the dynamics. An important class of dynamically-based formulas, for our parameterizations to optimize, are obtained as parametric variations of manifolds approximating the invariant ones. To better appreciate the origins of the modified manifolds thus obtained, the standard approximation theory of invariant manifolds is revisited in Part I of this article. A special emphasis is put on backward–forward (BF) systems naturally associated with the original system, whose asymptotic integration provides the leading-order approximation of invariant manifolds. Part II presents then (i) the modifications of these approximating manifolds based also on integration of the same BF systems but this time over a finite time \(\tau \), and (ii) the variational approach aimed at making an efficient selection of \(\tau \) per mode to parameterize. The parametric class of leading interaction approximation (LIA) of the high modes obtained this way, is completed by another parametric class built from the quasi-stationary approximation (QSA); close to the first criticality, the QSA is an approximation to the LIA, but it differs as one moves away from criticality. Rigorous results are derived that show that—given a cutoff dimension—the best manifolds that can be obtained through our variational approach, are manifolds which are in general no longer invariant. The minimizers are objects, called the optimal parameterizing manifolds (PMs), that are intimately tied to the conditional expectation of the original system, i.e. the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh–Bénard convection are then discussed. The approach is finally illustrated—in the context of the Kuramoto–Sivashinsky turbulence—as providing efficient closures without slaving for a cutoff scale \(k_\mathfrak {c}\) placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by the LIA or the QSA parameterizations in their standard forms, when they are used at this cutoff wavelength.

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Notes

  1. 1.

    Such as “cutting” within the inertial range of turbulence.

  2. 2.

    With respect to the probability measure \(\mathfrak {m}\) obtained as a projection of \(\mu \) onto \(E_\mathfrak {c}\).

  3. 3.

    As provided for instance by a center manifold or the unstable manifold of the origin.

  4. 4.

    According to this theorem, a candidate to a (truncated) Taylor expansion has to be first determined, and then it has to be checked to satisfy the invariance equation up to some order to ensure to be a genuine Taylor approximation; see also [88, Thm. 6.2.3].

  5. 5.

    Note however that other cost functionals may be considered at this stage; see Sect. 4.4 below.

  6. 6.

    i.e. up to an exceptional set of null measure with respect to \(\mathfrak {m}\).

  7. 7.

    While maximizing, in certain circumstances, the parameterization correlation, c(t), given by (3.6); see Sect. 5.2.

  8. 8.

    For instance this issue is not encountered for the chaotic regime analyzed in Sect. 5.3.

  9. 9.

    These parameters become \(\alpha =4000\), \(\overline{\delta t}=10^{-7}\) and \(N_x=256\) when scaling (6.3) is applied; see Remark 9.

  10. 10.

    Obtained by counting the number of (distinct) monomials \(\xi _i^{\ell } \xi _j^{\ell '}\), with \(i,j \in \{1,\ldots , 31\}\), and \(\ell , \ell ' \in \{0,1\}\).

  11. 11.

    Note that by taking A to be given by \(\nu \partial _{x}^4\) the resulting coefficients are bounded by \(\lambda _n^{-1}\), and since \(\lambda _n^{-1}<-\beta _n^{-1}\) the optimized \(K_\tau \) is not a priori of comparable parameterization defects, and in fact leads to less efficient closures.

  12. 12.

    Driven by the optimal QSA(\({\varvec{\tau }}^*\)) with \({\varvec{\tau }}^*\) minimizing the \(J_n\)’s.

  13. 13.

    Note that a blind regression would lead in this case to \(89\times 45 \times 3=12015\) coefficients to estimate for each high mode; a number of coefficients comparable to the number of snapshots making thus the estimated coefficients by regression non-robust. Instead, one benefits here again greatly from the parametric (and dynamically-based) form of QSA\(({\varvec{\tau }})\) and only 2 scalar parameters (\(\tau _n^\ell \), \(\ell =0,1\)) need to be determined, for each high mode.

  14. 14.

    In that respect, we may mention the variational normal mode initialization in Meteorology, pioneered by Daley [45], who combined the Machenhauer [127] non-linear normal-mode initialization within a variational procedure allowing for the adjustment of confidence weights arising in a fidelity functional I; see also [168]. In these works, the manifold \(\mathcal {M}\) is fixed a priori and it is the point on \(\mathcal {M}\) nearest to the observation using the “metric” defined by I, that is sought.

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Acknowledgements

MDC wishes to acknowledge David Neelin for the stimulating discussions on the closure problem of convective processes in the tropical atmosphere. MDC and JCM are also thankful to Darryl Holm for his constructive comments at the beginning of this work. Finally, MDC and HL are greatly indebted to Shouhong Wang for the numerous and stimulating discussions about this work over the years, and it is a pleasure to express our gratitude to Shouhong for his constant encouragement. This work has been partially supported by the National Science Foundation grants DMS-1616981 (MDC) and DMS-1616450 (HL).

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Communicated by Valerio Lucarini.

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Appendix: Parameterization Defect Minimization Algorithm

Appendix: Parameterization Defect Minimization Algorithm

We present in this Appendix a simple gradient-descent method to solve efficiently the minimization problem (4.35) in order to determine the optimal \(\tau \)-value, \(\tau ^*\), for the parameterization, \(\varPhi _n(\tau ,\varvec{\beta }, \xi )\), given by (4.34). As shown below, the method allows furthermore for making apparent the dependence of the parameterization defect on statistical moments (up to order 4) of the original system’s solution.

To present the method, we first recast the parameterization defect associated with \(\varPhi _n\),

$$\begin{aligned} \mathcal {Q}_n(\tau ,T)= \frac{1}{T} \int _0^T \big | \varPi _n y(t)- \varPhi _n(\tau , \varvec{\beta }, \varPi _{\mathfrak {c}}y(t))\big |^2 \,\mathrm {d}t, \end{aligned}$$
(A.1)

into a matrix format. For this purpose, we arrange the coefficients \(D^{n}_{i, j}(\tau ,\varvec{\beta })B_{i, j}^n\) involved in the expression of \(\varPhi _n(\tau ,\varvec{\beta }, \xi )\) into an \(m^2\times 1\) vector \(\varvec{d}(\tau )\) so that the indices (ij)’s are arranged in lexicographical order; namely the \(k^{\mathrm {th}}\) component of \(\varvec{d}(\tau )\) is given by

$$\begin{aligned} d_{k}(\tau ) = D^{n}_{i, j}(\tau ,\varvec{\beta })B_{i, j}^n, \quad k = 1, \ldots , m^2, \end{aligned}$$
(A.2)

where (ij) is the unique low-mode pair of indices satisfying

$$\begin{aligned} (i-1)m + j = k, \quad \text { with } i, j \in \{1,\ldots , m\}. \end{aligned}$$
(A.3)

More precisely, the index pair (ij) in (A.2) is determined by:

$$\begin{aligned} {\left\{ \begin{array}{ll} {\displaystyle i = \frac{k-\mathrm {mod}(k,m)}{m}+1} \;\text { and }\; j = \mathrm {mod}(k,m), &{} \text { if } \mathrm {mod}(k,m) \ne 0,\\ {\displaystyle i = \frac{k}{m}} \;\text { and }\; j = m, &{}\text { otherwise}. \end{array}\right. } \end{aligned}$$
(A.4)

Similarly, we define an \(m^2\times 1\) vector \(\varvec{\gamma }(\tau )\), whose components are given by

$$\begin{aligned} \gamma _k(\tau ) = V_{i,j}^n(\tau , \varvec{\beta }) F_{j} (B_{i,j}^n + B_{j,i}^n), \quad k = 1, \ldots , m^2. \end{aligned}$$
(A.5)

Now, given the solution y(t) to the underlying N-dimensional ODE system (4.16) over [0, T], we introduce

$$\begin{aligned} u_{k}(t) = \varPi _k y(t),\qquad k =1,\ldots , m, \end{aligned}$$

where \(\varPi _k\) denotes the projection onto the mode \(\varvec{e}_k\); see (4.19).

We define next the column vectors \(\varvec{Q}_1\), \(\varvec{Q}_2\), \(\widehat{\varvec{Q}}_2\) and \(\varvec{Q}_3\) of size \(m^2 \times 1\) as well as the matrices \(\widetilde{\varvec{Q}}_2\), \(\widetilde{\varvec{Q}}_3\) and \(\varvec{Q}_4\) of size \(m^2 \times m^2\) as follows:

$$\begin{aligned} (\varvec{Q}_1)_{p}&= \langle { \overline{u}_{p_1}} \rangle _T, \quad p = 1,\ldots , m^2, \nonumber \\ (\varvec{Q}_2)_{p}&= \langle { \overline{u}_{p_1} \overline{u}_{p_2}} \rangle _T, \quad p = 1,\ldots , m^2, \nonumber \\ (\widehat{\varvec{Q}}_2)_{p}&= \langle u_{n} { \overline{u}_{p_1}} \rangle _T, \quad p = 1,\ldots , m^2, \nonumber \\ (\varvec{Q}_3)_{p}&= \langle u_{n} { \overline{u}_{p_1} \overline{u}_{p_2}} \rangle _T, \quad p = 1,\ldots , m^2,\nonumber \\ (\widetilde{\varvec{Q}}_2)_{pq}&= \langle { \overline{u}_{p_1}}u_{q_1} \rangle _T, \quad p,\, q = 1,\ldots , m^2, \nonumber \\ (\widetilde{\varvec{Q}}_3)_{pq}&= \langle { \overline{u}_{p_1}} u_{q_1}u_{q_2} \rangle _T, \quad p,\, q = 1,\ldots , m^2, \nonumber \\ (\varvec{Q}_4)_{pq}&= \langle { \overline{u}_{p_1} \overline{u}_{p_2}} u_{q_1} u_{q_2} \rangle _T, \quad p, \, q = 1,\ldots , m^2, \end{aligned}$$
(A.6)

where \(\overline{z}\) denotes the complex conjugate of z in \(\mathbb {C}\), \(\langle \cdot \rangle _T\) denotes the time average over [0, T], and the low-mode index pair \((p_1, p_2)\) (resp. \((q_1, q_2)\)) relates to p (resp. q) according to (A.4), namely where p (resp. q) plays the role of k and \((p_1, p_2)\) (resp. \((q_1, q_2)\)) that of (ij) in (A.4).

Besides, let us recall the constant terms given in the RHS of (4.33) for the parameterization, \(\varPhi _n(\tau ,\varvec{\beta }, \xi )\):

$$\begin{aligned} \alpha _n(\tau ) = \sum _{i, j = 1}^m U_{i, j}^n(\tau , \varvec{\beta }) B_{i, j}^n F_{i}F_{j} - \frac{1 - e^{\tau \beta _n}}{\beta _n} F_n. \end{aligned}$$
(A.7)

Thus, we rewrite the parameterization defect \(\mathcal {Q}(\tau , T)\) recalled in (A.1) as follows:

$$\begin{aligned} \mathcal {Q}_n(\tau ,T)&= \varvec{d}(\tau )^{*} \varvec{Q}_4 \varvec{d}(\tau ) - 2 \mathrm {Re} \big ( \varvec{Q}_3^{*} \varvec{d}(\tau ) \big ) + 2 \mathrm {Re} \big ( \varvec{\gamma }(\tau )^{*} \widetilde{\varvec{Q}}_3 \varvec{d}(\tau ) \big ) + \varvec{\gamma }(\tau )^{*} \widetilde{\varvec{Q}}_2 \varvec{\gamma }(\tau ) \nonumber \\&\quad - 2 \mathrm {Re} \big (\widehat{\varvec{Q}}_2^{*} \varvec{\gamma }(\tau ) \big ) + 2 \mathrm {Re} \big ( \overline{\alpha }_n(\tau ) \varvec{Q}_2^{*} \varvec{d}(\tau ) \big ) + 2 \mathrm {Re} \big ( \overline{\alpha }_n(\tau ) \varvec{Q}_1^{*} \varvec{\gamma }(\tau ) \big ) \nonumber \\&\quad + \langle u_n \overline{u}_n \rangle _T - 2 \mathrm {Re} \big (\overline{\alpha }_n(\tau ) \langle u_n \rangle _T \big ) + \alpha _n(\tau ) \overline{\alpha }_n(\tau ), \end{aligned}$$
(A.8)

where \(M^*\) denotes the conjugate transpose of a given vector or matrix M.

Note also

$$\begin{aligned} \frac{\,\mathrm {d}}{\,\mathrm {d}\tau }\mathcal {Q}_n(\tau ,T)&= 2 \mathrm {Re} \Big ( \varvec{d}(\tau )^{*} \varvec{Q}_4 \varvec{d}'(\tau ) - \varvec{Q}^{*}_3 \varvec{d}'(\tau ) + \varvec{\gamma }'(\tau )^{*} \widetilde{\varvec{Q}}_3 \varvec{d}(\tau ) + \varvec{\gamma }(\tau )^{*} \widetilde{\varvec{Q}}_3 \varvec{d}'(\tau ) \nonumber \\&\quad + \varvec{\gamma }(\tau )^{*} \widetilde{\varvec{Q}}_2 \varvec{\gamma }'(\tau ) - \widehat{\varvec{Q}}^{*}_2 \varvec{\gamma }'(\tau ) + \overline{\alpha }'_n(\tau ) \varvec{Q}^{*}_2 \varvec{d}(\tau ) + \overline{\alpha }_n(\tau ) \varvec{Q}^{*}_2 \varvec{d}'(\tau ) \nonumber \\&\quad +\overline{\alpha }'_n(\tau ) \varvec{Q}^{*}_1 \varvec{\gamma }(\tau ) +\overline{\alpha }_n(\tau ) \varvec{Q}^{*}_1 \varvec{\gamma }'(\tau ) - \overline{\alpha }'_n(\tau ) \langle u_n \rangle _T + \alpha '_n(\tau ) \overline{\alpha }_n(\tau ) \Big ). \end{aligned}$$
(A.9)

With the above expression of \(\mathcal {Q}_n(\tau ,T)\) and of its derivative, the minimization of \(\mathcal {Q}_n(\tau ,T)\) in the \(\tau \)-variable can now be performed efficiently by application of a gradient-descent method as described in Algorithm 1. Note that if the first moments up to the 4th order are known, then the determination of \(\tau ^*\) by Algorithm 1 does not require any data from direct integration of the full system. There is a vast literature about moment closure techniques and we refer to [110] for a recent survey on the topic.

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Chekroun, M.D., Liu, H. & McWilliams, J.C. Variational Approach to Closure of Nonlinear Dynamical Systems: Autonomous Case. J Stat Phys 179, 1073–1160 (2020). https://doi.org/10.1007/s10955-019-02458-2

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Keywords

  • Approximate invariance formulas
  • Backward–forward systems
  • Dynamical closure
  • Optimization
  • Parameterizing manifold