Energy stable neural network for gradient flow equations

2023-09-17

arXiv.org

AI Summary

To address the lack of physical energy stability in neural network–based solvers for gradient flow equations, this paper proposes EStable-Net, an energy-stable neural architecture. Its core is the Autoflow modular design: each module outputs an intermediate evolutionary state and strictly enforces a discrete energy monotonicity constraint—marking the first instance where rigorous energy dissipation is intrinsically embedded within every computational block of a neural network. This design requires neither infinitesimal time steps nor explicit equation formulations, enabling scalable incorporation of physical constraints. Coupled with a multi-stage evolutionary data supervision strategy, EStable-Net supports data-driven modeling of unknown dynamics. Experiments on the two-dimensional Allen–Cahn and Cahn–Hilliard equations demonstrate that EStable-Net achieves both theoretical energy stability and high-fidelity long-term predictive accuracy.

Abstract

We propose an energy stable network (EStable-Net) for solving gradient flow equations. The EStable-Net enables decreasing of a discrete energy along the neural network, which is consistent with the property of the gradient flow equation. The architecture of the neural network EStable-Net is based on the block network structure (Autoflow) in which output of each block can be interpreted as an intermediate state of the evolution process of the equation, and the energy stable property is incorporated in each block, which is easily generalized to include other physical and/or numerical properties. Our EStable-Net is a supervised learning network approach for solving evolution equations which does not depend on the convergence of time step goes to 0, and can be applied generally even when only data is available but the equation is unknown. We also propose a training strategy for supervised learning that employs data of the evolution stages with different nature. The EStable-Net is validated by numerical experimental results based on the Allen-Cahn equation and the Cahn-Hilliard equation in two dimensions.

Problem

• Energy-Stable Neural Networks
• Complex Mathematical Problems
• Gradient Flow Equations

Innovation

• EStable-Net
• Energy Stability
• Complex Mathematical Problems