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MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations

Qiuqi Li, Yiting Liu, Jin Zhao, Wencan Zhu

TL;DR

This work addresses the challenge of efficiently solving parametric PDEs with uncertain initial data and domains. It introduces MAD-NGM, a two-stage framework that first builds a meta-learned nonlinear trial manifold via a decoder for initial conditions and then performs space-time decoupled Neural Galerkin time evolution, optionally enhanced by randomized sparse updates (MAD-RSNGS) to reduce cost. The approach achieves physically consistent, long-horizon predictions with strong generalization across unseen parameter configurations, outperforming MAD-PINN in long-time accuracy and robustness. The method is validated on KdV, Burgers, and Allen-Cahn equations, including high-dimensional 2D settings, showing practical potential for real-time simulation and uncertainty quantification in parameterized PDEs.

Abstract

Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods, often face challenges in generalization and long-time prediction efficiency due to their dependence on full space-time approximations. To address these issues, we propose a novel and scalable framework that significantly enhances the Neural Galerkin Method (NGM) by incorporating the Meta-Auto-Decoder (MAD) paradigm. Our approach leverages space-time decoupling to enable more stable and efficient time integration, while meta-learning-driven adaptation allows rapid generalization to unseen parameter configurations with minimal retraining. Furthermore, randomized sparse updates effectively reduce computational costs without compromising accuracy. Together, these advancements enable our method to achieve physically consistent, long-horizon predictions for complex parameterized evolution equations with significantly lower computational overhead. Numerical experiments on benchmark problems demonstrate that our methods performs comparatively well in terms of accuracy, robustness, and adaptability.

MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations

TL;DR

This work addresses the challenge of efficiently solving parametric PDEs with uncertain initial data and domains. It introduces MAD-NGM, a two-stage framework that first builds a meta-learned nonlinear trial manifold via a decoder for initial conditions and then performs space-time decoupled Neural Galerkin time evolution, optionally enhanced by randomized sparse updates (MAD-RSNGS) to reduce cost. The approach achieves physically consistent, long-horizon predictions with strong generalization across unseen parameter configurations, outperforming MAD-PINN in long-time accuracy and robustness. The method is validated on KdV, Burgers, and Allen-Cahn equations, including high-dimensional 2D settings, showing practical potential for real-time simulation and uncertainty quantification in parameterized PDEs.

Abstract

Parametric partial differential equations (PDEs) are fundamental for modeling a wide range of physical and engineering systems influenced by uncertain or varying parameters. Traditional neural network-based solvers, such as Physics-Informed Neural Networks (PINNs) and Deep Galerkin Methods, often face challenges in generalization and long-time prediction efficiency due to their dependence on full space-time approximations. To address these issues, we propose a novel and scalable framework that significantly enhances the Neural Galerkin Method (NGM) by incorporating the Meta-Auto-Decoder (MAD) paradigm. Our approach leverages space-time decoupling to enable more stable and efficient time integration, while meta-learning-driven adaptation allows rapid generalization to unseen parameter configurations with minimal retraining. Furthermore, randomized sparse updates effectively reduce computational costs without compromising accuracy. Together, these advancements enable our method to achieve physically consistent, long-horizon predictions for complex parameterized evolution equations with significantly lower computational overhead. Numerical experiments on benchmark problems demonstrate that our methods performs comparatively well in terms of accuracy, robustness, and adaptability.
Paper Structure (24 sections, 40 equations, 20 figures, 4 tables, 3 algorithms)

This paper contains 24 sections, 40 equations, 20 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Mathematical Formulation of Meta-Auto-Decoder Neural Galerkin Method
  3. Evolution equations
  4. Approximation of Random Initial Conditions
  5. Neural Galerkin Schemes
  6. Randomized Sparse Neural Galerkin Schemes
  7. Computational Framework and Implementation
  8. Treatment of Boundary Conditions
  9. Approximation of Random Initial Conditions
  10. Time Discretization
  11. For the Neural Galerkin Schemes
  12. For the Randomized Sparse Neural Galerkin Schemes
  13. Numerical Experiments
  14. Korteweg-de Vries Equation:
  15. Results of the Training Set Samples
  16. ...and 9 more sections

Figures (20)

  • Figure 1: Architecture of the MAD-NGM framework
  • Figure 2: Comparison of predicted and reference solutions for the KdV equation on the training set, showing that the proposed approach effectively captures both the nonlinear and dispersive characteristics of the wave evolution.
  • Figure 3: Temporal evolution of the reference and predicted solutions for the KdV equation on the training sample at $t = 0.0~,0.5~,1.0$, illustrating the high reconstruction accuracy of the MAD-NGM method for nonlinear dispersive wave dynamics.
  • Figure 4: Comparison of predicted and reference solutions for the KdV equation on the testing set, demonstrating the strong generalization and predictive capability of the proposed MAD-NGM framework.
  • Figure 5: Comparison of predicted and reference one-dimensional profiles of the KdV equation at $t = 0.0~,0.5~,1.0$ on the testing sample. The results confirm the reliability of the proposed MAD-NGM framework in reproducing the testing dynamics.
  • ...and 15 more figures