Debiased neural operators for estimating functionals

TL;DR

This paper introduces DOPE, a debiased neural operator estimator designed to accurately estimate scalar functionals of solution trajectories, overcoming bias issues in naive methods and applicable to complex physical systems.

Contribution

The paper develops a novel Neyman-orthogonal estimator for neural operators that removes bias in estimating target functionals, extending automatic debiased machine learning to operator-valued nuisances.

Findings

DOPE outperforms naive plug-in estimators in numerical experiments.

The method effectively handles irregular observation designs.

DOPE reduces bias in estimating functionals of solution trajectories.

Abstract

Neural operators are widely used to approximate solution maps of complex physical systems. In many applications, however, the goal is not to recover the full solution trajectory, but to summarize the solution trajectory via a scalar target quantity (e.g., a functional such as time spent in a target range, time above a threshold, accumulated cost, or total energy). In this paper, we introduce DOPE (debiased neural operator): a semiparametric estimator for such target quantities of solution trajectories obtained from neural operators. DOPE is broadly applicable to settings with both partial and irregular observations and can be combined with arbitrary neural operator architectures. We make three main contributions. (1) We show that, in contrast to DOPE, naive plug-in estimation can suffer from first-order bias. (2) To address this, we derive a novel one-step, Neyman-orthogonal estimator that treats the neural operator as a high-dimensional nuisance mapping between function spaces, and removes the leading bias term. For this, DOPE uses a weighting mechanism that simultaneously accounts for irregular observation designs and for how sensitive the target quantity is to perturbations of the underlying trajectory. (3) To learn the weights, we extend automatic debiased machine learning to operator-valued nuisances via Riesz regression. We demonstrate the benefits of DOPE across various numerical experiments.