Calibrated Physics-Informed Uncertainty Quantification

2025-02-06

AI Summary

Neural PDE solvers lack reliable uncertainty quantification (UQ), hindering their adoption in safety-critical scientific applications. Method: This paper proposes a model-agnostic, label-free physics-informed conformal prediction framework. Its core innovation is the first use of PDE residuals as nonconformity scores, enabling both marginal and joint statistical coverage guarantees. Crucially, convolutional layers are explicitly interpreted as finite-difference operators, ensuring UQ is fully driven by physical constraints—no training data or ground-truth labels are required. Results: The method is validated on plasma modeling and fusion reactor target design tasks. It delivers statistically rigorous uncertainty intervals for diverse complex PDEs, with empirical coverage consistently matching theoretical guarantees across benchmarks.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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Abstract

Neural PDEs offer efficient alternatives to computationally expensive numerical PDE solvers for simulating complex physical systems. However, their lack of robust uncertainty quantification (UQ) limits deployment in critical applications. We introduce a model-agnostic, physics-informed conformal prediction (CP) framework that provides guaranteed uncertainty estimates without requiring labelled data. By utilising a physics-based approach, we are able to quantify and calibrate the model's inconsistencies with the PDE rather than the uncertainty arising from the data. Our approach uses convolutional layers as finite-difference stencils and leverages physics residual errors as nonconformity scores, enabling data-free UQ with marginal and joint coverage guarantees across prediction domains for a range of complex PDEs. We further validate the efficacy of our method on neural PDE models for plasma modelling and shot design in fusion reactors.

Problem

• Uncertainty quantification in neural PDEs
• Physics-informed conformal prediction framework
• Data-free UQ with coverage guarantees

Innovation

• Physics-informed conformal prediction framework
• Convolutional layers as finite-difference stencils
• Data-free uncertainty quantification with coverage guarantees