  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.

Optimization and Control;Dynamical Systems