# Distribution-Free Robust Predict-Then-Optimize in Function Spaces

arXiv: https://arxiv.org/abs/2602.08215 · HTML: https://mlsanity.com/paper/2602.08215/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 28 of its 67 references that are in the map.
- First posted on arXiv 9 Feb 2026.

## Builds on (most cited first, 28 of 28)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [A Gentle Introduction to Conformal Prediction and Distribution-Free Uncertainty Quantification](https://mlsanity.com/paper/2107.07511/) (2021)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Handbook of Convergence Theorems for (Stochastic) Gradient Methods](https://mlsanity.com/paper/2301.11235/) (2023)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [A Survey of Contextual Optimization Methods for Decision Making under Uncertainty](https://mlsanity.com/paper/2306.10374/) (2023)
- [Deep UQ: Learning deep neural network surrogate models for high dimensional uncertainty quantification](https://mlsanity.com/paper/1802.00850/) (2018)
- [Conformal Decision Theory: Safe Autonomous Decisions from Imperfect Predictions](https://mlsanity.com/paper/2310.05921/) (2023)
- [Decision Theoretic Foundations for Conformal Prediction: Optimal Uncertainty Quantification for Risk-Averse Agents](https://mlsanity.com/paper/2502.02561/) (2025)
- [Conformal Prediction Sets Improve Human Decision Making](https://mlsanity.com/paper/2401.13744/) (2024)
- [L-HYDRA: Multi-Head Physics-Informed Neural Networks](https://mlsanity.com/paper/2301.02152/) (2023)
- [Calibrated Uncertainty Quantification for Operator Learning via Conformal Prediction](https://mlsanity.com/paper/2402.01960/) (2024)
- [Conformal Contextual Robust Optimization](https://mlsanity.com/paper/2310.10003/) (2023)
- [End-to-end Conditional Robust Optimization](https://mlsanity.com/paper/2403.04670/) (2024)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [Utility-Directed Conformal Prediction: A Decision-Aware Framework for Actionable Uncertainty Quantification](https://mlsanity.com/paper/2410.01767/) (2024)
- [Deep Operator Learning-based Surrogate Models with Uncertainty Quantification for Optimizing Internal Cooling Channel Rib Profiles](https://mlsanity.com/paper/2306.00810/) (2023)
- [Calibrated Physics-Informed Uncertainty Quantification](https://mlsanity.com/paper/2502.04406/) (2025)
- [Guaranteed prediction sets for functional surrogate models](https://mlsanity.com/paper/2501.18426/) (2025)
- [The Open DAC 2025 Dataset for Sorbent Discovery in Direct Air Capture](https://mlsanity.com/paper/2508.03162/) (2025)
- [A Physics-Guided Bi-Fidelity Fourier-Featured Operator Learning Framework for Predicting Time Evolution of Drag and Lift Coefficients](https://mlsanity.com/paper/2311.03639/) (2023)
- [Conformal Robust Control of Linear Systems](https://mlsanity.com/paper/2405.16250/) (2024)
- [Valid Error Bars for Neural Weather Models using Conformal Prediction](https://mlsanity.com/paper/2406.14483/) (2024)

## Abstract (arXiv)

The need to rapidly solve PDEs in engineering design workflows has spurred the rise of neural surrogate models. In particular, neural operator models provide a discretization-invariant surrogate by retaining the infinite-dimensional, functional form of their arguments. Despite improved throughput, such methods lack guarantees on accuracy, unlike classical numerical PDE solvers. Optimizing engineering designs under these potentially miscalibrated surrogates thus runs the risk of producing designs that perform poorly upon deployment. In a similar vein, there is growing interest in automated decision-making under black-box predictors in the finite-dimensional setting, where a similar risk of suboptimality exists under poorly calibrated models. For this reason, methods have emerged that produce adversarially robust decisions under uncertainty estimates of the upstream model. One such framework leverages conformal prediction, a distribution-free post-hoc uncertainty quantification method, to provide these estimates due to its natural pairing with black-box predictors. We herein extend this line of conformally robust decision-making to infinite-dimensional function spaces. We first extend the typical conformal prediction guarantees over finite-dimensional spaces to infinite-dimensional Sobolev spaces. We then demonstrate how such uncertainty can be leveraged to robustly formulate engineering design tasks and characterize the suboptimality of the resulting robust optimal designs. We then empirically demonstrate the generality of our functional conformal coverage method across a diverse collection of PDEs, including the Poisson and heat equations, and showcase the significant improvement of such robust design in a quantum state discrimination task.