To comprehend complex systems with multiple states, it is imperative to reveal the identity of these states by system outputs. Nevertheless, the mathematical models describing these systems often exhibit nonlinearity so that render the resolution of the parameter inverse problem from the observed spatiotemporal data a challenging endeavor. Starting from the observed data obtained from such systems, we propose a novel framework that facilitates the investigation of parameter identification for multi-state systems governed by spatiotemporal varying parametric partial differential equations. Our framework consists of two integral components: a constrained self-adaptive physics-informed neural network, encompassing a sub-network, as our methodology for parameter identification, and a finite mixture model approach to detect regions of probable parameter variations. Through our scheme, we can precisely ascertain the unknown varying parameters of the complex multi-state system, thereby accomplishing the inversion of the varying parameters. Furthermore, we have showcased the efficacy of our framework on two numerical cases: the 1D Burgers' equation with time-varying parameters and the 2D wave equation with a space-varying parameter.

翻译：为了理解具有多种状态的复杂系统，必须通过系统输出揭示这些状态的本质。然而，描述这些系统的数学模型通常呈现非线性，使得从观测的时空数据中求解参数逆问题成为一项艰巨的任务。基于从这类系统获得的观测数据，我们提出了一种新框架，以促进由时空变化参数偏微分方程支配的多状态系统的参数识别研究。我们的框架包含两个核心组成部分：一个受约束的自适应物理信息神经网络（包含一个子网络），作为参数识别方法；以及一个有限混合模型方法，用于检测参数可能变化的区域。通过我们的方案，可以精确确定复杂多状态系统的未知变化参数，从而实现变化参数的逆推。此外，我们通过两个数值案例展示了该框架的有效性：具有时变参数的一维Burgers方程和具有空间变化参数的二维波动方程。