Parameter Identification for Partial Differential Equations with Spatiotemporal Varying Coefficients

TL;DR

This paper introduces a novel framework combining physics-informed neural networks and mixture models to accurately identify spatiotemporal varying parameters in complex multi-state PDE systems from observed data.

Contribution

It presents a new integrated approach for parameter inversion in multi-state systems governed by PDEs with varying coefficients, enhancing accuracy and regional detection capabilities.

Findings

Successfully identified time-varying parameters in 1D Burgers' equation.

Accurately detected space-varying parameters in 2D wave equation.

Demonstrated effectiveness on numerical simulations.

Abstract

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.