Apr 20, 2026 · math.APJ/K move · Enter open · S save
Chih-Kang Huang, Ludovick Gagnon, Miha Založnik, Benoît Appolaire
1Universit´e de Lorraine, CNRS, IJL, F-54000 Nancy, France · 2Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France
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.