One-shot Learning of Surrogates in PDE-constrained Optimization Under Uncertainty

TL;DR

This paper introduces a unified machine learning framework for PDE-constrained optimization under uncertainty, replacing complex models with surrogates learned simultaneously during the optimization process.

Contribution

It presents a novel one-shot surrogate learning approach integrated into PDE-constrained optimization, with theoretical consistency analysis and convergence guarantees.

Findings

Effective surrogate models for linear and nonlinear cases

Convergence of the stochastic gradient method with adaptive penalty control

Numerical experiments validate the approach's accuracy and robustness

Abstract

We propose a general framework for machine learning based optimization under uncertainty. Our approach replaces the complex forward model by a surrogate, which is learned simultaneously in a one-shot sense when solving the optimal control problem. Our approach relies on a reformulation of the problem as a penalized empirical risk minimization problem for which we provide a consistency analysis in terms of large data and increasing penalty parameter. To solve the resulting problem, we suggest a stochastic gradient method with adaptive control of the penalty parameter and prove convergence under suitable assumptions on the surrogate model. Numerical experiments illustrate the results for linear and nonlinear surrogate models.