Learning-informed parameter identification in nonlinear time-dependent PDEs

TL;DR

This paper presents a novel all-at-once framework for identifying parameters in nonlinear time-dependent PDEs by integrating neural networks to learn nonlinearity directly from noisy measurement data, avoiding traditional parameter-to-state mappings.

Contribution

It introduces a learning-informed parameter identification method that handles the state, parameters, and nonlinearity simultaneously within an all-at-once approach, leveraging neural networks and multiple algorithmic settings.

Findings

Method successfully identifies parameters and nonlinearity from noisy data.

Both analytic adjoint and standard machine learning algorithms are effective.

The approach avoids explicit parameter-to-state map construction.

Abstract

We introduce and analyze a method of learning-informed parameter identification for partial differential equations (PDEs) in an all-at-once framework. The underlying PDE model is formulated in a rather general setting with three unknowns: physical parameter, state and nonlinearity. Inspired by advances in machine learning, we approximate the nonlinearity via a neural network, whose parameters are learned from measurement data. The later is assumed to be given as noisy observations of the unknown state, and both the state and the physical parameters are identified simultaneously with the parameters of the neural network. Moreover, diverging from the classical approach, the proposed all-at-once setting avoids constructing the parameter-to-state map by explicitly handling the state as additional variable. The practical feasibility of the proposed method is confirmed with experiments using two different algorithmic settings: A function-space algorithm based on analytic adjoints as well as a purely discretized setting using standard machine learning algorithms.