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

TL;DR

MAD-NG introduces a scalable neural Galerkin framework that combines meta-learning and space-time decoupling to efficiently solve parametric PDEs with improved generalization, stability, and reduced computational costs.

Contribution

The paper presents a novel Meta-Auto-Decoder enhanced Neural Galerkin Method that improves stability, efficiency, and adaptability in solving parametric PDEs compared to existing neural network approaches.

Findings

Achieves stable, long-horizon predictions for complex PDEs.

Reduces computational costs via randomized sparse updates.

Demonstrates strong accuracy and robustness on benchmark problems.

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.