EXTENDED PSEUDO-SPECTRAL PHYSICS-INFORMED NEURAL NETWORKS FOR PHASE-FIELD MODELS

Phase-field models are essential for describing pattern formation in materials science and biology, but their predictive power relies on constitutive quantities—such as the bulk free-energy density and interfacial thickness—that are often unknown. This paper addresses the inverse problem of identifying these constitutive quantities from limited dynamical observations of phase separation, specifically for models like the Cahn-Hilliard equation.

The authors introduce the Extended Pseudo-Spectral Physics-Informed Neural Network (ESPINN) framework. Unlike standard Physics-Informed Neural Networks (PINNs) that approximate the solution field directly, ESPINN uses a neural network to approximate the unknown bulk chemical potential while simultaneously learning unknown physical parameters (such as gradient coefficients) through a reparameterized optimization process. The method leverages pseudo-spectral discretization to handle spatial derivatives efficiently and incorporates the governing physical laws directly into the training loss function. This allows the model to infer constitutive structure from transient snapshot data, even when only a single pair of snapshots is available.

Numerical experiments on the one-dimensional Cahn-Hilliard equation demonstrate that the ESPINN framework is highly effective in the noiseless regime, accurately reconstructing the underlying free-energy structure. In scenarios with noise, the model exhibits graceful degradation in accuracy. Furthermore, the authors show that increasing the number of snapshots improves the robustness of the reconstruction by reducing variance across training runs. The framework is shown to be data-efficient and physically consistent, providing a flexible tool for learning constitutive properties in continuum models.