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Gradient Preserving Operator Inference: Data-Driven Reduced-Order Models for Equations with Gradient Structure

Yuwei Geng, Jasdeep Singh, Lili Ju, Boris Kramer, Zhu Wang

TL;DR

The paper addresses learning reduced-order models for gradient-structure PDEs while preserving the underlying energy/gradient properties. It introduces Gradient-Preserving Operator Inference (GP-OPINF), a nonintrusive framework that learns a reduced operator $\mathbf{D}_r$ under antisymmetry (conservative) or negative semidefiniteness (dissipative) from snapshot data and the Hamiltonian gradient. A rigorous a priori error bound is derived, showing that ROM error is bounded by the sum of projection, data, and optimization errors, and is analyzed in the preasymptotic regime. Extensive numerical experiments on conservative problems (parameterized wave, KdV, 3D elastic plate) and dissipative problems (Allen-Cahn in 1D and 2D) demonstrate that GP-OPINF achieves structure-preserving, accurate, and predictive ROMs, with strong parametric generalization and stable long-time behavior compared to non-structure-preserving approaches.

Abstract

Hamiltonian Operator Inference has been introduced in [Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022] to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an $a\ priori$ error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg-de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.

Gradient Preserving Operator Inference: Data-Driven Reduced-Order Models for Equations with Gradient Structure

TL;DR

The paper addresses learning reduced-order models for gradient-structure PDEs while preserving the underlying energy/gradient properties. It introduces Gradient-Preserving Operator Inference (GP-OPINF), a nonintrusive framework that learns a reduced operator under antisymmetry (conservative) or negative semidefiniteness (dissipative) from snapshot data and the Hamiltonian gradient. A rigorous a priori error bound is derived, showing that ROM error is bounded by the sum of projection, data, and optimization errors, and is analyzed in the preasymptotic regime. Extensive numerical experiments on conservative problems (parameterized wave, KdV, 3D elastic plate) and dissipative problems (Allen-Cahn in 1D and 2D) demonstrate that GP-OPINF achieves structure-preserving, accurate, and predictive ROMs, with strong parametric generalization and stable long-time behavior compared to non-structure-preserving approaches.

Abstract

Hamiltonian Operator Inference has been introduced in [Sharma, H., Wang, Z., Kramer, B., Physica D: Nonlinear Phenomena, 431, p.133122, 2022] to learn structure-preserving reduced-order models (ROMs) for Hamiltonian systems. This approach constructs a low-dimensional model using only data and knowledge of the Hamiltonian function. Such ROMs can keep the intrinsic structure of the system, allowing them to capture the physics described by the governing equations. In this work, we extend this approach to more general systems that are either conservative or dissipative in energy, and which possess a gradient structure. We derive the optimization problems for inferring structure-preserving ROMs that preserve the gradient structure. We further derive an error estimate for the reduced-order approximation. To test the algorithms, we consider semi-discretized partial differential equations with gradient structure, such as the parameterized wave and Korteweg-de-Vries equations, and equations of three-dimensional linear elasticity in the conservative case and the one- and two-dimensional Allen-Cahn equations in the dissipative case. The numerical results illustrate the accuracy, structure-preservation properties, and predictive capabilities of the gradient-preserving Operator Inference ROMs.
Paper Structure (38 sections, 1 theorem, 71 equations, 21 figures, 1 table)

This paper contains 38 sections, 1 theorem, 71 equations, 21 figures, 1 table.

Table of Contents

  1. Introduction
  2. Full-Order Model and Structure-Preserving Reduced-Order Model
  3. Full-Order Model
  4. Intrusive Structure-Preserving ROMs
  5. Gradient-Preserving Operator Inference
  6. Proposed GP-OPINF Framework
  7. Solution of the GP-OpInf Optimization for the Conservative Case
  8. Solution of the GP-OpInf Optimization for the Dissipative Case
  9. Error Estimation
  10. Numerical Experiments
  11. Error Measures
  12. Conservative PDEs
  13. Parameterized wave equation
  14. Computational setting
  15. ROM construction
  16. ...and 23 more sections

Key Result

Theorem 1

Let ${\mathbf y}(t)$ be the solution of the FOM eq:ham_fom on the time interval $[0, T]$ and ${{\mathbf y}_r}(t)$ be the solution of the ROM eq:ham_rom_opinf on the same interval. Suppose $\nabla_{{\mathbf y}} H({\mathbf y})$ is Lipschitz continuous, then the ROM approximation error satisfies where $C(T)= \max\{1+C_2^2, 2\} T \alpha(T)$, $\alpha(T)= 2\int_0^T e^{2C_1 (T-\tau)}\,{ \rm d} \tau$, an

Figures (21)

  • Figure 1: (Wave Equation) Full-order model simulation when $T_{\text{FOM}} = 100$: time evolution of ${\mathbf u}$ (left), ${\mathbf v}$ (middle), and energy $\overline{H}(t)$ (right).
  • Figure 2: (Wave Equation) ROMs of $2r$-dimensions when $T_{\text{FOM}} = 5$ and $T_{\text{ROM}} = 5$: (left) maximum magnitude of $|{\mathbf D}_r+{\mathbf D}_r^\intercal|$ for the inferred ${\mathbf D}_r$ when $r$ varies; (right) ROM approximation error $\mathcal{E}$ from \ref{['eq:e_approx']} when $r$ varies from 5 to 200 in increments of five. The inserted plots cover the portion of $r$ between 75 and 175.
  • Figure 3: (Wave Equation) ROMs of $2r$-dimensions when $T_{\text{FOM}} = 10$ and $T_{\text{ROM}} = 10$: (left) maximum magnitude of $|{\mathbf D}_r+{\mathbf D}_r^\intercal|$ for the inferred ${\mathbf D}_r$ when $r$ varies; (right) numerical error $\mathcal{E}$ from \ref{['eq:e_approx']} when $r$ varies from 5 to 200 in increments of five.
  • Figure 4: (Wave Equation) Numerical errors of the $2r$-dimensional GP-OpInf ROM when $T_{\text{FOM}}=T_{\text{ROM}}=10$ and $\Delta x = \Delta t = 2\times 10^{-4}$: (left) the ROM approximation error \ref{['eq:e_approx']} together with POD projection error \ref{['eq:e_proj']} and optimization error \ref{['eq:e_opt']}; (right) comparison of the ROM approximation error for the GP-OpInf and SP-G ROM.
  • Figure 5: (Wave Equation) Numerical accuracy of the $2r$-dimensional GP-OpInf: (left) time evolution of the summand of the ROM approximation error \ref{['eq:e_approx']}; (right) time evolution of the reduced-order approximate Hamiltonian energy. The vertical dashed line indicates the end of the training interval.
  • ...and 16 more figures

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • proof
  • Remark 2
  • Remark 3
  • Remark 4