DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting

TL;DR

This paper introduces a neural operator framework combining energy splitting and physics-informed learning to efficiently simulate phase-field models, reducing computational costs and improving generalization.

Contribution

A novel neural operator approach integrating classical energy splitting schemes with physics-informed training for phase-field models.

Findings

Successfully applied to Allen-Cahn and dendritic growth models.

Achieved faster inference than Fourier spectral methods.

Provided better out-of-distribution generalization with physics-informed training.

Abstract

The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods.