Continuous nonlinear adaptive experimental design with gradient flow

TL;DR

This paper develops a continuous nonlinear adaptive experimental design method using gradient flow and optimal transport, improving measurement strategies in inverse problems like Lorenz 63 and Schrödinger equations.

Contribution

It introduces a novel continuous design framework with gradient-flow and optimal transport techniques for adaptive experimental design in inverse problems.

Findings

Effective identification of measurement times and locations.

Enhanced reconstruction accuracy of unknown parameters.

Demonstrated success on Lorenz 63 and Schrödinger systems.

Abstract

In computational inverse problems, the optimal experimental design (OED) problem seeks the best locations in time and space at which to take measurements. We investigate the nonlinear OED problem in the context of continuously-indexed design space for the measurements. In contrast to traditional approaches that select experiments from a finite measurement set, a continuous design space is often a better reflection of practical experimental options, where there is considerable flexibility concerning where and when to take measurements. The continuously-indexed space introduces computational challenges, and we address them by employing gradient-flow and optimal transport techniques, complemented by an adaptive strategy for bi-level optimization. Numerical results on the Lorenz 63 system and Schrodinger equation demonstrate that our solver identifies good measurement times / locations and achieves improved reconstruction of unknown parameters in inverse problems.