Energy stable neural network for gradient flow equations

TL;DR

The paper introduces EStable-Net, a physics-informed neural network for gradient-flow equations that enforces discrete energy dissipation along a block-structured Autoflow architecture. By integrating energy-dissipation losses and finite-block evolution steps, the network achieves stable, accurate predictions without requiring vanishing time steps ($\Delta t \to 0$) and can operate even when the governing PDE is unknown, given data. Validation on two-dimensional Allen–Cahn and Cahn–Hilliard equations demonstrates accurate evolution with energy decreasing block-by-block, and comparisons show the energy-stable design improves physical fidelity and intermediate-state reliability. The approach is broadly applicable to gradient-flow problems and can incorporate additional physical or numerical properties through the block design and auxiliary-variable formulations.

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.

Figures (23)

• Figure 1: Network architecture of EStable-Net. (a) The overall Autoflow structure of the network. (b) The structure within each energy stable block, which mimics the evolution from time $t_n$ to $t_{n+1}$, where $M$ is the number of blocks in the network. The input and output of each block are respectively $\phi^n$ and $\phi^{n+1}$, which mimic the intermediate states of the evolution process of the gradient flow equation with enforcement of the energy stable property. The neural network is trained via supervised learning at time $T$.
• Figure 2: Inference of EStable-Net.
• Figure 3: Allen-Cahn equation: (a) The train loss and the test loss (MSE). (b) Evolution of the energy of the EStable-Net solution and that of the exact solution. (c) An example of the prediction of the EStable-Net and the exact solution.
• Figure 4: Allen-Cahn equation: An example of the initial state and output of the 5 energy stable blocks in EStable-Net for the evolution from $0$ to $T=5$, from (a) to (f).
• Figure 5: Allen-Cahn equation: An example of snapshots in the evolution with time increment $T/5=1$ obtained by numerical method using a much smaller time step, from (a)-(f). The initial condition is the same as that for the example in Fig. \ref{['fig..AC2D_middle']}.

Theorems & Definitions (4)

• Theorem A.1: Energy stable property
• proof
• Theorem A.2: Boundness of $\{V^{n}\}$
• proof