Parameter Identification for Partial Differential Equations with Spatiotemporal Varying Coefficients
This addresses the challenge of parameter identification in complex nonlinear systems for researchers in computational physics and engineering, though it appears incremental as it builds on existing physics-informed neural network methods.
The authors tackled the problem of identifying spatiotemporally varying parameters in multi-state systems governed by partial differential equations, achieving precise parameter inversion as demonstrated on numerical cases like the 1D Burgers' equation and 2D wave equation.
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.