MAD-NG: Meta-Auto-Decoder Neural Galerkin Method for Solving Parametric Partial Differential Equations
This work addresses the problem of solving parametric PDEs for modeling physical and engineering systems, offering a scalable solution with improved efficiency and generalization, though it appears incremental as it builds upon existing Neural Galerkin Methods.
The paper tackles the challenges of generalization and efficiency in neural network-based solvers for parametric PDEs by proposing a novel framework that enhances the Neural Galerkin Method with meta-learning and space-time decoupling, achieving physically consistent, long-horizon predictions with significantly lower computational overhead.
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.