Guaranteed confidence-band enclosures for PDE surrogates

2025-01-30

AI Summary

Existing PDE surrogate models—particularly function-space mapping models such as neural operators—lack statistically guaranteed predictive confidence. Method: We propose the first conformalized confidence band construction method for such infinite-dimensional models. Our approach embeds the surrogate’s error into a low-dimensional SVD subspace, models the truncation error explicitly, and leverages zonotope-based set representation and set propagation to yield model-agnostic, provably valid functional confidence bands. Contribution/Results: This work pioneers the extension of conformal inference to infinite-dimensional function spaces; ensures theoretical soundness via SVD-based error decomposition; and delivers tight, computationally tractable, black-box-compatible confidence bands with guaranteed coverage. Experiments across diverse neural operators demonstrate strict adherence to user-specified coverage levels, significantly enhancing the reliability and interpretability of scientific machine learning predictions.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Uncertainty Representations
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

We propose a method for obtaining statistically guaranteed confidence bands for functional machine learning techniques: surrogate models which map between function spaces, motivated by the need build reliable PDE emulators. The method constructs nested confidence sets on a low-dimensional representation (an SVD) of the surrogate model's prediction error, and then maps these sets to the prediction space using set-propagation techniques. The result are conformal-like coverage guaranteed prediction sets for functional surrogate models. We use zonotopes as basis of the set construction, due to their well studied set-propagation and verification properties. The method is model agnostic and can thus be applied to complex Sci-ML models, including Neural Operators, but also in simpler settings. We also elicit a technique to capture the truncation error of the SVD, ensuring the guarantees of the method.

Problem

• Partial Differential Equations
• Machine Learning
• Prediction Accuracy

Innovation

• Machine Learning
• Zonotopes
• Prediction Intervals