Energetic Variational Neural Network Discretizations of Gradient Flows

TL;DR

This paper introduces neural network-based, energy-dissipation law-driven algorithms for solving gradient flows and diffusions, ensuring stability and accuracy in high-dimensional, mesh-free settings.

Contribution

It proposes novel neural network discretizations based on energy laws rather than PDE forms, enhancing stability and efficiency for high-dimensional gradient flow problems.

Findings

Algorithms preserve energy dissipation law ensuring stability.

Mesh-free neural network schemes work efficiently in high dimensions.

Numerical experiments confirm accuracy and energy stability.

Abstract

We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial discretization. Unlike most existing methods that construct numerical discretizations based on the strong or weak form of the underlying PDE, the proposed schemes are constructed based on the energy-dissipation law directly. This guarantees the monotonic decay of the system's free energy, which avoids unphysical states of solutions and is crucial for the long-term stability of numerical computations. To address challenges arising from nonlinear neural network discretization, we perform temporal discretizations on these variational systems before spatial discretizations. This approach is computationally memory-efficient when implementing neural network-based algorithms. The proposed neural network-based schemes are mesh-free, allowing us to solve gradient flows in high dimensions. Various numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed numerical schemes.