Energy-based physics-informed neural networks with stochastic minibatching for phase-field modeling of fracture

This work presents an efficient energy-based physics-informed neural network (PINN) framework for phase-field modeling of fracture, in which the potential energy is estimated from stochastic minibatches of the integration points to reduce training cost. Building on the deep energy method (DEM), the network is trained by directly minimizing the regularized potential energy of the body rather than the residual of the governing equations. At each optimization step, the energy integral is evaluated on a randomly drawn subset of the integration points and rescaled so that the estimate is unbiased with respect to the full-grid quadrature energy. The framework is verified on a set of established fracture benchmarks, reproducing the expected crack development, including interacting and curved crack paths. Benchmarking reveals minibatching reduces the total training time by up to 50% relative to full-batch training, while preserving the crack trajectory. Furthermore, we extend the proposed model to hydrogen-assisted fracture, and it captures the coupled stress-hydrogen interaction where the drop in fracture toughness caused by hydrogen promotes earlier crack initiation and reduced load-carrying capacity. The results indicate that energy-based PINNs, combined with minibatch estimation of the discretized energy, offer an efficient route to phase-field fracture simulation and its multiphysics extensions.