Energy Dissipation Preserving Feature-based DNN Galerkin Methods for Gradient Flows

TL;DR

This paper introduces a feature-based DNN Galerkin framework that preserves energy dissipation in gradient flow simulations, offering improved accuracy and stability over traditional methods in high-dimensional PDE problems.

Contribution

It proposes a novel structure-preserving neural Galerkin method that employs neural features as trial spaces, ensuring energy stability and enhancing accuracy in PDE solutions.

Findings

Preserves energy dissipation in high-dimensional gradient flows.

Achieves higher accuracy than classical spectral methods with same degrees of freedom.

Effectively captures complex topological transitions.

Abstract

In recent years, deep learning methods, exemplified by Physics-Informed Neural Networks (PINNs), have been widely applied to the numerical solution of differential equations. However, these methods may suffer from limited accuracy, high training costs, and lack of robustness, particularly their inability to preserve the intrinsic physical structures of continuous PDE models, such as the energy dissipation property in gradient flow systems. To address these challenges, we propose a feature-based Deep Neural Network Galerkin (DNN-G) framework designed for structure-preserving simulations of gradient flows. Instead of treating neural networks merely as optimization-driven solvers, we employ them as adaptive feature generators that define nonlinear trial spaces within a Galerkin projection formulation.This formulation guarantees semi-discrete energy dissipation and can be naturally combined with energy stable time integration schemes. Several strategies for constructing neural basis functions are investigated, including random features, structured initialization, and problem-informed pre-training. Numerical experiments demonstrate that the proposed method preserves robust energy stability in high-dimensional settings and accurately captures complex topological transitions. With equivalent degrees of freedom, the DNN-G framework achieves higher accuracy than classical spectral methods, highlighting the effectiveness of neural feature representations for the numerical solution of partial differential equations.