Uncertainty quantification for deeponets with ensemble kalman inversion

TL;DR

This work tackles uncertainty quantification for DeepONet operator learners under limited and noisy data by introducing an Ensemble Kalman Inversion (EKI)–based training framework. To scale to large DeepONet datasets, a minibatch EKI variant is employed together with a heuristic for adaptive artificial-dynamics covariance $Q$, enabling informative prediction intervals through ensemble means and variances. The authors validate the approach on three benchmark problems—the anti-derivative operator, a reaction-diffusion system, and a gravity pendulum—across small and large noise regimes, demonstrating strong correlations between predicted uncertainty and actual error, and high coverage of ground truth within two-standard-deviation bands. They also show that learning $Q$ (adaptive $Q$) yields more reliable uncertainty estimates than fixed $Q$, with potential extensions to very large networks via dimensionality reduction and localization. Overall, the method provides an efficient, scalable pathway for principled UQ in operator learning, with practical significance for safety-critical applications where data are scarce or noisy.

Abstract

In recent years, operator learning, particularly the DeepONet, has received much attention for efficiently learning complex mappings between input and output functions across diverse fields. However, in practical scenarios with limited and noisy data, accessing the uncertainty in DeepONet predictions becomes essential, especially in mission-critical or safety-critical applications. Existing methods, either computationally intensive or yielding unsatisfactory uncertainty quantification, leave room for developing efficient and informative uncertainty quantification (UQ) techniques tailored for DeepONets. In this work, we proposed a novel inference approach for efficient UQ for operator learning by harnessing the power of the Ensemble Kalman Inversion (EKI) approach. EKI, known for its derivative-free, noise-robust, and highly parallelizable feature, has demonstrated its advantages for UQ for physics-informed neural networks [28]. Our innovative application of EKI enables us to efficiently train ensembles of DeepONets while obtaining informative uncertainty estimates for the output of interest. We deploy a mini-batch variant of EKI to accommodate larger datasets, mitigating the computational demand due to large datasets during the training stage. Furthermore, we introduce a heuristic method to estimate the artificial dynamics covariance, thereby improving our uncertainty estimates. Finally, we demonstrate the effectiveness and versatility of our proposed methodology across various benchmark problems, showcasing its potential to address the pressing challenges of uncertainty quantification in DeepONets, especially for practical applications with limited and noisy data.

Figures (15)

• Figure 2.1: Schematic of the DeepONet architecture. The "branch" network, denoted as $b(u)$, encodes input functions evaluated at predetermined "sensor" locations. The "trunk" network, $t(y)$, encodes a "query" location where the output function is to be evaluated. DeepONet fuses the encoded information from both networks via a dot product to produce an approximate function output at the "query" location.
• Figure 2.2: Schematic of the EKI B-DeepONet architecture. Using the structure of a single DeepONet seen in Fig. \ref{['fig:nn']}, the ensemble consists of $J$ DeepONets with different sets of neural network parameters.
• Figure 3.1: Example \ref{['subsec:ex1']} small noise: (a) and (b) two sample output function approximations with two standard deviation confidence intervals; (c) worst case scenario approximation of the output samples.
• Figure 3.2: Example \ref{['subsec:ex1']} small noise: (Left) Uncertainty versus the relative error for outputs from the EKI DeepONet. (Right) Estimates of marginal densities of the relative error and uncertainties of the testing sample outputs.
• Figure 3.3: Example \ref{['subsec:ex1']} large noise: (a) and (b) two sample output approximations with two standard deviation confidence intervals; (c) worst case scenario approximation of the output samples.