PINNs-MPF: A Physics-Informed Neural Network framework for Multi-Phase-Field simulation of interface dynamics

We present PINNs-MPF framework, an application of Physics-Informed Neural Networks (PINNs) to handle Multi-Phase-Field (MPF) simulations of microstructure evolution. A combination of optimization techniques within PINNs and in direct relation to MPF method are extended and adapted. The numerical resolution is realized through a multi-variable time-series problem by using fully discrete resolution. Within each interval, space, time, and phases/grains are treated separately, constituting discrete subdomains. PINNs-MPF is equipped with an extended multi-networking (parallelization) concept to subdivide the simulation domain into multiple batches, with each batch associated with an independent NN trained to predict the solution. To ensure continuity across the spatio-temporal-phasic subdomains, a Master NN efficiently is to handle interactions among the multiple networks and facilitates the transfer of learning. A pyramidal training approach is proposed to the PINN community as a dual-impact method: to facilitate the initialization of training when dealing with multiple networks, and to unify the solution through an extended transfer of learning. Furthermore, a comprehensive approach is adopted to specifically focus the attention on the interfacial regions through a dynamic meshing process, significantly simplifying the tuning of hyper-parameters, serving as a key concept for addressing MPF problems using machine learning. We perform a set of systematic simulations that benchmark foundational aspects of MPF simulations, i.e., the curvature-driven dynamics of a diffuse interface, in the presence and absence of an external driving force, and the evolution and equilibrium of a triple junction. The proposed PINNs-MPF framework successfully reproduces benchmark tests with high fidelity and Mean Squared Error (MSE) loss values ranging from 10 −6 to 10 −4 compared to ground truth solutions.