Optimization with learning-informed differential equation constraints and its applications

TL;DR

This paper develops analysis and numerical methods for optimization problems constrained by differential equations with components learned from data, focusing on neural network approximations in control and imaging applications.

Contribution

It introduces error analysis and numerical techniques for data-driven differential equation constrained optimization, emphasizing neural network-based components.

Findings

Error bounds for neural network approximations in PDE-constrained optimization

Numerical results demonstrating effectiveness in control and imaging applications

Analytical insights into neural network integration in differential equation problems

Abstract

Inspired by applications in optimal control of semilinear elliptic partial differential equations and physics-integrated imaging, differential equation constrained optimization problems with constituents that are only accessible through data-driven techniques are studied. A particular focus is on the analysis and on numerical methods for problems with machine-learned components. For a rather general context, an error analysis is provided, and particular properties resulting from artificial neural network based approximations are addressed. Moreover, for each of the two inspiring applications analytical details are presented and numerical results are provided.