Energy stable neural network for gradient flow equations

TL;DR

The paper introduces EStable-Net, a neural network architecture that ensures energy stability when solving gradient flow equations, aligning with physical properties and applicable even with limited data.

Contribution

It presents a novel energy stable neural network architecture that enforces energy decrease, applicable to unknown equations and data-driven scenarios.

Findings

Successfully applied to Allen-Cahn and Cahn-Hilliard equations

Maintains energy stability during evolution

Effective with limited or unknown data

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.