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This paper addresses a critical theoretical gap in scientific machine learning by analyzing the breakdown of conformal prediction guarantees in time-dependent partial differential equation systems. The authors demonstrate that distribution shifts over time can lead to systematic undercoverage and propose weighted conformal prediction methods to restore finite-sample coverage for evolving physical simulations. This work is essential for deploying reliable AI models in safety-critical engineering applications like flood forecasting and thermal control.
Conformal Prediction for Time-Dependent PDEs
arXiv:2611.08409v1 Announce Type: new Abstract: Uncertainty quantification is crucial in scientific machine learning, where models inform safety-critical tasks such as flood forecasting, financial risk management, and thermal control in machines. Conformal prediction provides distribution-free coverage guarantees, but in time-dependent settings common to physics and engineering, these guarantees can break down, leading to systematic undercoverage. We study this problem in the context of surrogate models for time-dependent physical systems described by partial differential equations. We prove that in a function space setting, distributions at arbitrarily close times can be mutually singular, making exact coverage guarantees impossible. We further show that in discretized settings, the total variation distance between distributions often saturates exponentially fast with the grid resolution, mandating restrictions on the spatial discretization to ensure coverage. Finally, for certain settings, we show how to use principled weighted conformal prediction to obtain finite-sample coverage guarantees over growing time horizons.
Abstract:Uncertainty quantification is crucial in scientific machine learning, where models inform safety-critical tasks such as flood forecasting, financial risk management, and thermal control in machines. Conformal prediction provides distribution-free coverage guarantees, but in time-dependent settings common to physics and engineering, these guarantees can break down, leading to systematic undercoverage. We study this problem in the context of surrogate models for time-dependent physical systems described by partial differential equations. We prove that in a function space setting, distributions at arbitrarily close times can be mutually singular, making exact coverage guarantees impossible. We further show that in discretized settings, the total variation distance between distributions often saturates exponentially fast with the grid resolution, mandating restrictions on the spatial discretization to ensure coverage. Finally, for certain settings, we show how to use principled weighted conformal prediction to obtain finite-sample coverage guarantees over growing time horizons.
Machine Learning (cs.LG); Machine Learning (stat.ML)
arXiv:2611.08409 [cs.LG]
(or arXiv:2611.08409v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2611.08409 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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