Learning-Informed Parameter Identification in Nonlinear Time-Dependent PDEs

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 rather general setting with three unknowns: physical parameter, state nonlinearity. Inspired by advances machine learning, we approximate the nonlinearity via neural network, whose parameters are learned from measurement data. latter assumed to be given as noisy observations unknown state, both identified simultaneously network. Moreover, diverging classical approach, proposed avoids constructing parameter-to-state map explicitly handling additional variable. practical feasibility confirmed experiments using two different algorithmic settings: A function-space algorithm based on analytic adjoints well purely discretized standard learning algorithms.