Fast PDE-constrained optimization via self-supervised operator learning

TL;DR

This paper introduces a self-supervised deep operator network framework that efficiently solves PDE-constrained optimization problems, significantly speeding up computations without requiring paired training data.

Contribution

It presents a novel physics-informed DeepONet approach for fast, differentiable surrogates in PDE-constrained optimization, applicable without paired data and scalable to high-dimensional controls.

Findings

DeepONets achieve rapid minimization of high-dimensional cost functionals.

The framework outperforms traditional PDE solvers in speed, often by orders of magnitude.

Applications include heat transfer control and drag minimization in fluid flow.

Abstract

Design and optimal control problems are among the fundamental, ubiquitous tasks we face in science and engineering. In both cases, we aim to represent and optimize an unknown (black-box) function that associates a performance/outcome to a set of controllable variables through an experiment. In cases where the experimental dynamics can be described by partial differential equations (PDEs), such problems can be mathematically translated into PDE-constrained optimization tasks, which quickly become intractable as the number of control variables and the cost of experiments increases. In this work we leverage physics-informed deep operator networks (DeepONets) -- a self-supervised framework for learning the solution operator of parametric PDEs -- to build fast and differentiable surrogates for rapidly solving PDE-constrained optimization problems, even in the absence of any paired input-output training data. The effectiveness of the proposed framework will be demonstrated across different applications involving continuous functions as control or design variables, including time-dependent optimal control of heat transfer, and drag minimization of obstacles in Stokes flow. In all cases, we observe that DeepONets can minimize high-dimensional cost functionals in a matter of seconds, yielding a significant speed up compared to traditional adjoint PDE solvers that are typically costly and limited to relatively low-dimensional control/design parametrizations.