Parameter Identification for Partial Differential Equation with Jump Discontinuities in Coefficients by Markov Switching Model and Physics-Informed Machine Learning

TL;DR

This work tackles inverse problems for PDEs with jump discontinuities in coefficients by fusing physics-informed neural networks with Bayesian inference and Markov-switching dynamics. The core framework, gws-PINNs, uses a main solution network and a gradient-adaptive coefficient sampler, whose outputs feed a Markov-switching Bayesian estimator that employs Gaussian mixtures and birth-death MCMC to automatically select the number of states. Across a suite of PDEs (wave, heat, Burgers, Navier–Stokes, Helmholtz), the method achieves accurate coefficient identification, robust solution reconstruction, and scalable performance, outperforming conventional PINNs, SBL/DSBL, and other clustering approaches in jump-regime scenarios. The approach demonstrates strong noise robustness, efficient computation, and practical potential for high-dimensional, non-stationary spatiotemporal systems, with applications in fluid dynamics, geophysics, and engineering diagnostics. The work contributes a generalizable, multi-level inference pipeline that bridges physics-based modeling and data-driven statistical learning for complex heterogeneous media.

Abstract

Inverse problems involving partial differential equations (PDEs) with discontinuous coefficients are fundamental challenges in modeling complex spatiotemporal systems with heterogeneous structures and uncertain dynamics. Traditional numerical and machine learning approaches often face limitations in addressing these problems due to high dimensionality, inherent nonlinearity, and discontinuous parameter spaces. In this work, we propose a novel computational framework that synergistically integrates physics-informed deep learning with Bayesian inference for accurate parameter identification in PDEs with jump discontinuities in coefficients. The core innovation of our framework lies in a dual-network architecture employing a gradient-adaptive weighting strategy: a main network approximates PDE solutions while a sub network samples its coefficients. To effectively identify mixture structures in parameter spaces, we employ Markovian dynamics methods to capture hidden state transitions of complex spatiotemporal systems. The framework has applications in reconstruction of solutions and identification of parameter-varying regions. Comprehensive numerical experiments on various PDEs with jump-varying coefficients demonstrate the framework's exceptional adaptability, accuracy, and robustness compared to existing methods. This study provides a generalizable computational approach of parameter identification for PDEs with discontinuous parameter structures, particularly in non-stationary or heterogeneous systems.

Figures (16)

• Figure 1: Schematic diagram of gradient-adaptive weighted sub-main physics-informed neural networks (gws-PINNs).
• Figure 2: Schematic diagram of PDEs governed by Markov switching models.
• Figure 3: Numerical results for wave equations with discontinuously time varying coefficient $\alpha(t)$.
• Figure 4: Numerical results for the wave equation with discontinuously space varying coefficient $\alpha(x,y)$.
• Figure 5: Numerical results for heat equations with discontinuously time varying coefficient $c(t)$.