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.

翻译：系数具有间断性的偏微分方程反问题，是建模具有异质结构和不确定动态的复杂时空系统时的基本挑战。传统数值方法与机器学习方法在处理此类问题时，常因高维性、固有非线性及参数空间的不连续性而面临局限。本研究提出一种新颖的计算框架，将物理信息深度学习与贝叶斯推断协同融合，以精确辨识系数具有跳跃间断的偏微分方程参数。该框架的核心创新在于采用梯度自适应加权策略的双网络架构：主网络逼近偏微分方程解，而子网络对其系数进行采样。为有效识别参数空间中的混合结构，我们采用马尔可夫动态方法来捕捉复杂时空系统的隐藏状态转移。该框架在解的重构与参数变化区域的辨识中具有应用价值。在多种系数跳跃变化的偏微分方程上进行的大量数值实验表明，相较于现有方法，该框架具有卓越的适应性、准确性和鲁棒性。本研究为具有不连续参数结构的偏微分方程，特别是在非平稳或异质系统中，提供了一种可推广的参数辨识计算方法。