Inverse Problem for Partial Differential Equations with Jump Discontinuities in Coefficients by Two-stage Physics-Informed Deep Learning and Statistical Mixture Models

This work proposes a two-stage physics-informed deep learning framework that combines neural-network-based sampling with statistical inference and constrained parameter refinement. In the first stage, a dual-network physics-informed architecture is used, where a main-network approximates the PDE solution and an auxiliary coefficient sub-network provides a relaxed continuous soft approximation of the true discontinuous coefficient field. A gradient-adaptive weighting strategy is incorporated to improve residual training and enhance sampling reliability near possible discontinuity regions. The sampled coefficient values are then analyzed using Bayesian learning for Gaussian mixture models and birth-death Markov chain model selection, which identify the number of coefficient regimes and provide candidate intervals for coefficient values and transition regions. In the second stage, the inverse problem is reformulated as a constrained physics-informed estimator, in which the coefficient is replaced by a form-consistent hard approximation explicitly represented as a piecewise-constant function over the spatiotemporal domain. Comprehensive numerical experiments on PDEs with jump-discontinuous coefficients demonstrate that the proposed framework achieves adaptability and accurate parameter identification with acceptable computational costs compared to existing methods. Applications to solution reconstruction further illustrate its practical potential. This work provides a generalizable computational approach for inverse problems governed by PDEs with discontinuous parameter structures, particularly in non-stationary and heterogeneous systems.