Conformalized-DeepONet: A Distribution-Free Framework for Uncertainty Quantification in Deep Operator Networks

TL;DR

The paper tackles uncertainty quantification for neural operator learning by applying split conformal prediction to existing DeepONet uncertainty frameworks (B-DeepONet and Prob-DeepONet) and by introducing Quantile-DeepONet. It develops Conformalized-DeepONet to produce distribution-free, finite-sample coverage for operator predictions, and proposes Conformal Quantile-DeepONet (CQR) to estimate conditional quantiles and enable adaptive intervals. Through three numerical experiments (nonlinear pendulum, diffusion–reaction, and viscous Burgers’) plus a multi-fidelity illustration, the authors demonstrate robust coverage guarantees and adaptive interval lengths, often outperforming baseline UQ methods. The approach is model-agnostic and readily extensible to other DeepONet variants, offering practical, rigorous uncertainty quantification for scientific machine learning surrogates.

Abstract

In this paper, we adopt conformal prediction, a distribution-free uncertainty quantification (UQ) framework, to obtain confidence prediction intervals with coverage guarantees for Deep Operator Network (DeepONet) regression. Initially, we enhance the uncertainty quantification frameworks (B-DeepONet and Prob-DeepONet) previously proposed by the authors by using split conformal prediction. By combining conformal prediction with our Prob- and B-DeepONets, we effectively quantify uncertainty by generating rigorous confidence intervals for DeepONet prediction. Additionally, we design a novel Quantile-DeepONet that allows for a more natural use of split conformal prediction. We refer to this distribution-free effective uncertainty quantification framework as split conformal Quantile-DeepONet regression. Finally, we demonstrate the effectiveness of the proposed methods using various ordinary, partial differential equation numerical examples, and multi-fidelity learning.

Figures (8)

• Figure 1: Stacked version DeepONet $G_{\theta}$. $\bigotimes$ denotes the inner product in $\mathbb{R}^K$.
• Figure 2: Confidence intervals for a random test trajectory of the nonlinear pendulum experiment given a miscoverage rate $\alpha=0.05$. (a) Confidence intervals for the Conformalized Prob-DeepONet and the baseline Prob-DeepONet. (b) Confidence intervals for the Conformalized Quantile-DeepONet and the baseline Quantile-DeepONet. Please refer to Table \ref{['table:pendulum-coverage']} for the summary of the average coverage rate for all testing trajectories with different approaches.
• Figure 3: The distribution of predicted confidence intervals' coverages (in percentages), obtained by using the conformalized Quantile-DeepONet on the 100 test trajectories of the nonlinear pendulum experiment, with a specified miscoverage rate of $\alpha=0.05$.
• Figure 4: The distribution of the lengths of the confidence intervals (given a miscoverage rate $\alpha=0.05$) predicted using the proposed conformalized Prob-DeepONet for the $100$ test trajectories of the nonlinear pendulum experiment.
• Figure 5: The empirical distribution of coverages $C_k$ for different values of the calibration dataset size $n \in \{500, 1000, 5000, 10000\}$. We observe that for $n>500$, the distribution, as expected, centers around $95$% ($\alpha=0.05$). Notably, for $n=1000$ and this nonlinear pendulum dataset, we achieve a satisfactory distribution, and the gain from increasing $n$ is minimal.

Theorems & Definitions (1)

• Theorem 3.1