Parameter Identification for Partial Differential Equations with
  Spatiotemporal Varying Coefficients
Guangtao Zhang Yiting Duan Guanyu Pan Qijing Chen Huiyu Yang Zhikun Zhang
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.