标题： 基于马尔可夫切换模型和物理信息机器学习的系数中存在跳跃不连续性的偏微分方程参数识别 显示英文标题

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

摘要： 涉及具有不连续系数的偏微分方程（PDEs）的反问题是在具有异质结构和不确定动力学的复杂时空系统建模中的基本挑战。 传统的数值和机器学习方法由于高维性、固有的非线性和不连续参数空间，在处理这些问题时常常面临限制。 在本工作中，我们提出了一种新颖的计算框架，该框架协同集成物理信息深度学习与贝叶斯推断，以在具有系数跳跃不连续性的PDEs中进行准确的参数识别。 我们框架的核心创新在于一种双网络架构，采用梯度自适应加权策略：一个主网络近似PDE解，而一个子网络采样其系数。 为了有效识别参数空间中的混合结构，我们采用马尔可夫动力学方法来捕捉复杂时空系统的隐藏状态转换。 该框架在解决方案的重建和参数变化区域的识别中有应用。 对各种具有跳跃变化系数的PDEs的全面数值实验表明，与现有方法相比，该框架表现出卓越的适应性、准确性和鲁棒性。 本研究为具有不连续参数结构的PDEs提供了可推广的参数识别计算方法，特别是在非平稳或异质系统中。 显示英文摘要

摘要： 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.