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

2025-10-16

AI Summary

This work addresses the high-dimensional, nonlinear, and discontinuous inverse problem of parameter identification for partial differential equations (PDEs) with jump-discontinuous coefficients. We propose a novel physics-informed deep learning framework integrated with Bayesian inference. Methodologically, we design a dual-network architecture coupled with a gradient-adaptive weighting scheme and explicitly incorporate a Markov switching model to capture latent state transitions and abrupt structural changes in the parameter space. Compared to conventional numerical methods and purely data-driven approaches, our framework achieves accurate reconstruction of parameter variability regions and solution fields in nonstationary heterogeneous systems. Extensive validation on multiple benchmark PDEs with discontinuous coefficients demonstrates significant improvements in estimation accuracy, robustness to noise, and precise localization of discontinuities. The proposed approach establishes a new paradigm for interpretable, uncertainty-aware parameter inversion in complex spatiotemporal systems.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Probabilistic Inference
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• User Modeling, Personalization and Recommendation: Psychology-informed user models and recommender systems
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Identifying parameters in PDEs with discontinuous coefficients
• Overcoming limitations of traditional numerical and machine learning methods
• Developing physics-informed Bayesian framework for parameter-varying regions

Innovation

• Dual-network architecture with gradient-adaptive weighting strategy
• Markovian dynamics methods capture hidden state transitions
• Physics-informed deep learning integrated with Bayesian inference