Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structure-preserving learning for multi-symplectic PDEs

Süleyman Yıldız, Pawan Goyal, Peter Benner

TL;DR

The paper tackles preserving global and local conservation properties when learning reduced-order models for multi-symplectic PDEs from data. It introduces a non-intrusive operator inference framework that enforces structure by using POD bases respecting skew-symmetric K and L operators, yielding ROMs that satisfy semi-discrete multi-symplectic and local energy conservation. Through three canonical PDEs (linear wave, KdV, ZK), the approach demonstrates robust energy preservation and accurate dynamics outside the training interval, validating the method’s viability for structure-preserving, data-driven MOR. This yields practical impact for fast, reliable simulations of high-dimensional multi-symplectic systems where discrete operators are inaccessible or black-box solvers are used.

Abstract

This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving reduced-order methods use symplectic Galerkin projection to construct reduced-order Hamiltonian models by projecting the full models onto a symplectic subspace. However, symplectic projection requires the existence of fully discrete operators, and in many cases, such as black-box PDE solvers, these operators are inaccessible. In this work, we propose an energy-preserving machine learning method that can infer the dynamics of the given PDE using data only, so that the proposed framework does not depend on the fully discrete operators. In this context, the proposed method is non-intrusive. The proposed method is grey box in the sense that it requires only some basic knowledge of the multi-symplectic model at the partial differential equation level. We prove that the proposed method satisfies spatially discrete local energy conservation and preserves the multi-symplectic conservation laws. We test our method on the linear wave equation, the Korteweg-de Vries equation, and the Zakharov-Kuznetsov equation. We test the generalization of our learned models by testing them far outside the training time interval.

Structure-preserving learning for multi-symplectic PDEs

TL;DR

The paper tackles preserving global and local conservation properties when learning reduced-order models for multi-symplectic PDEs from data. It introduces a non-intrusive operator inference framework that enforces structure by using POD bases respecting skew-symmetric K and L operators, yielding ROMs that satisfy semi-discrete multi-symplectic and local energy conservation. Through three canonical PDEs (linear wave, KdV, ZK), the approach demonstrates robust energy preservation and accurate dynamics outside the training interval, validating the method’s viability for structure-preserving, data-driven MOR. This yields practical impact for fast, reliable simulations of high-dimensional multi-symplectic systems where discrete operators are inaccessible or black-box solvers are used.

Abstract

This paper presents an energy-preserving machine learning method for inferring reduced-order models (ROMs) by exploiting the multi-symplectic form of partial differential equations (PDEs). The vast majority of energy-preserving reduced-order methods use symplectic Galerkin projection to construct reduced-order Hamiltonian models by projecting the full models onto a symplectic subspace. However, symplectic projection requires the existence of fully discrete operators, and in many cases, such as black-box PDE solvers, these operators are inaccessible. In this work, we propose an energy-preserving machine learning method that can infer the dynamics of the given PDE using data only, so that the proposed framework does not depend on the fully discrete operators. In this context, the proposed method is non-intrusive. The proposed method is grey box in the sense that it requires only some basic knowledge of the multi-symplectic model at the partial differential equation level. We prove that the proposed method satisfies spatially discrete local energy conservation and preserves the multi-symplectic conservation laws. We test our method on the linear wave equation, the Korteweg-de Vries equation, and the Zakharov-Kuznetsov equation. We test the generalization of our learned models by testing them far outside the training time interval.
Paper Structure (8 sections, 2 theorems, 71 equations, 9 figures)

This paper contains 8 sections, 2 theorems, 71 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Multi-symplectic Full-order Model
  3. Structure-preserving model reduction
  4. Numerical Results
  5. Linear wave equation
  6. Korteweg--de Vries equation
  7. Zakharov-Kuznetsov equation
  8. Conclusions

Key Result

Theorem 1

The spatially-discrete equation eqn:Ms-proj yields a semi-discrete multi-symplectic conservation law

Figures (9)

  • Figure 4.1: Linear wave equation: discrete energy comparison between full and reduced-order models.
  • Figure 4.2: Linear wave equation: a comparison of the reduced coefficients obtained by the full and the reduced-order models.
  • Figure 4.3: Linear wave equation: a comparison between the full and reduced-order models in full model dimension.
  • Figure 4.4: KdV: discrete energy comparison between full and reduced-order models.
  • Figure 4.5: KdV: a comparison of the reduced coefficients obtained by the full and the reduced-order models.
  • ...and 4 more figures

Theorems & Definitions (6)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 1