DeepRitzSplit: A Variational Neural Operator Framework for Gradient Flows with Application to Dendritic Growth

Abstract

Phase-field models are widely used to simulate microstructure evolution but their numerical solution remains computationally demanding. Neural operator surrogates provide a promising alternative, yet existing approaches are data-driven, requiring large datasets and exhibiting limited robustness when extrapolating to unseen configurations. We introduce DeepRitzSplit, an unsupervised neural operator framework for gradient-flow systems based on a variational formulation of energy-stable splitting schemes. It learns the time evolution semigroup associated with a convex-concave free-energy decomposition by minimizing the underlying variational problem at each time step. It is designed to inherit the dissipative structure of the underlying gradient flow without requiring trajectory datasets. We further introduce a Reaction-Diffusion Neural Operator (RDNO) architecture motivated by splitting schemes, with separate reaction and diffusion components to improve training efficiency and interpretability. We assess the approach on the Allen-Cahn equation and apply it to a phase-field model of anisotropic dendritic growth during solidification. In the Allen-Cahn benchmark, DeepRitzSplit shows better generalization than the supervised baseline on the out-of-distribution configurations tested, using 180 training configurations. In dendritic growth, the method captures the overall phase morphology and dendrite tip velocity well, while the tip radius is more sensitive to the training strategy. Refinement based on the residual of the splitting scheme markedly improves this second-order morphological quantity. The trained operators retain good predictive capabilities on configurations substantially different from the training set. These results demonstrate the potential of variationally trained neural operators as physically informed surrogates for accelerated microstructure simulations.

arXiv comment36 pages