Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Tao Tang, Jiang Yang, Yuxiang Zhao, Quanhui Zhu

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.

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

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.
Paper Structure (20 sections, 2 theorems, 54 equations, 13 figures, 1 table, 1 algorithm)

This paper contains 20 sections, 2 theorems, 54 equations, 13 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Structure-Preserving DNN Galerkin Scheme
  3. Neural basis functions
  4. Semi-discrete DNN-Galerkin System
  5. Energy Dissipation Preservation
  6. Choice of Neural Basis Functions
  7. Static Neural Basis Functions
  8. Random Features
  9. Structured First-Layer Initialization (SFLI)
  10. Offline Pre-training Basis
  11. Initial-Conditions-Motivated Basis
  12. Adaptive Basis Strategy
  13. Numerical Experiments
  14. Settings
  15. Heat Equation
  16. ...and 5 more sections

Key Result

Theorem 1

Assume that the operator $\mathcal{G}$ is symmetric negative semi-definite and that the neural basis functions are linearly independent so that the mass matrix $\mathbf M(\theta)$ is symmetric positive definite. Then the semi-discrete solution satisfies

Figures (13)

  • Figure 1: Construction of the neural basis functions.
  • Figure 2: Temporal convergence test for the 5D heat equation (Example \ref{['eg:heat_high_d']}, $d=5$). (a) The second-order DIRK scheme (DIRK2) exhibits an $O(\Delta t^2)$ convergence rate. (b) The third-order DIRK scheme (DIRK3) achieves $O(\Delta t^3)$ accuracy, eventually reaching a plateau due to spatial discretization limits.
  • Figure 3: Quantitative comparison for the 10D heat equation (Example \ref{['eg:heat_high_d']}, $d=10$). (a) Relative $L^2$ error evolution. (b) Energy dissipation. The standard PINN (green dotted line) is compared against the DNN-G scheme, where the integrals are approximated using varying numbers of Monte Carlo points ($N_{\text{MC}} = 10^4$ to $10^7$).
  • Figure 4: Temporal convergence test for the 1D Allen--Cahn equation (Example \ref{['eg:1d_AC']}) comparing the SSI1 and IMEX-RK2 schemes with adaptive DNN Galerkin method.
  • Figure 5: Numerical results for the 1D Allen--Cahn equation (Example \ref{['eg:1d_AC']}). (a)--(b) Comparison of solution profiles $u(x,t)$. (c) Temporal evolution of the relative $L^2$ error. (d) Free energy dissipation $E(u)$ curves.
  • ...and 8 more figures

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Example 1: Fully discrete energy-stable scheme for $L^2$ gradient flow
  • Remark 1
  • Theorem 2: Global discrete energy dissipation of the adaptive DNN-G scheme
  • proof
  • Example 2: High-dimensional Heat Equation
  • Example 3: 1D Allen--Cahn equation
  • Example 4: The evolution of two circular bubbles
  • Example 5: Non-uniform curvature-driven motion
  • ...and 2 more