Large-scale Bayesian optimal experimental design with derivative-informed projected neural network

TL;DR

This paper introduces a neural network surrogate method for large-scale Bayesian optimal experimental design involving PDEs, significantly reducing computational costs while maintaining accuracy.

Contribution

It proposes a derivative-informed projected neural network (DIPNet) to efficiently approximate the parameter-to-observable map in PDE-based OED problems, enabling scalable sensor placement optimization.

Findings

Achieved up to three orders of magnitude speedup over traditional Monte Carlo methods.

Demonstrated effectiveness on inverse scattering and reactive transport problems with thousands of parameters.

Established theoretical error bounds linking surrogate accuracy to EIG approximation error.

Abstract

We address the solution of large-scale Bayesian optimal experimental design (OED) problems governed by partial differential equations (PDEs) with infinite-dimensional parameter fields. The OED problem seeks to find sensor locations that maximize the expected information gain (EIG) in the solution of the underlying Bayesian inverse problem. Computation of the EIG is usually prohibitive for PDE-based OED problems. To make the evaluation of the EIG tractable, we approximate the (PDE-based) parameter-to-observable map with a derivative-informed projected neural network (DIPNet) surrogate, which exploits the geometry, smoothness, and intrinsic low-dimensionality of the map using a small and dimension-independent number of PDE solves. The surrogate is then deployed within a greedy algorithm-based solution of the OED problem such that no further PDE solves are required. We analyze the EIG approximation error in terms of the generalization error of the DIPNet and show they are of the same order. Finally, the efficiency and accuracy of the method are demonstrated via numerical experiments on OED problems governed by inverse scattering and inverse reactive transport with up to 16,641 uncertain parameters and 100 experimental design variables, where we observe up to three orders of magnitude speedup relative to a reference double loop Monte Carlo method.