Learning-Informed Parameter Identification in Nonlinear Time-Dependent PDEs

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 latter 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.

Automated summary

This study introduces and analyzes a learning-informed parameter identification method for partial differential equations (PDEs) in an innovative framework. The nonlinearity is approximated using a neural network, with its parameters being learned from measurement data. The unknown state is assumed to be observed with noise, and both the state and physical parameters are identified together with the neural network's parameters. By treating the state as an additional variable, this all-at-once setting avoids explicitly constructing the parameter-to-state map. The practical feasibility of this method is confirmed through experiments using two different algorithmic settings.