🤖 AI Summary
This work addresses the high computational cost of traditional numerical methods for phase-field models, which arise from their multiscale and nonlinear nature requiring fine spatiotemporal discretization. The authors propose a novel approach that integrates convex–concave energy splitting with physics-informed learning, embedding energy dissipation constraints directly into neural operator training for the first time. They introduce a Reaction-Diffusion Neural Operator (RDNO) architecture tailored to reaction-diffusion equations and combine it with the Deep Ritz method to solve the associated variational problems. Demonstrated on both isotropic Allen–Cahn dynamics and anisotropic dendritic growth, the method exhibits superior generalization over purely data-driven models, achieves faster inference than conventional Fourier spectral methods, and rigorously preserves the energy dissipation property inherent to the physical system.
📝 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.