ICML 2025

Calibrated Physics-Informed Uncertainty Quantification
Vignesh Gopakumar 1 2 Ander Gray 3 Lorenzo Zanisi 2 Timothy Nunn 2 Daniel Giles 1 Matt J. Kusner 4 5
Stanislas Pamela 2 Marc Peter Deisenroth 1
Abstract
Simulating complex physical systems is cru-
cial for understanding and predicting phenomena
across diverse fields, such as fluid dynamics and
heat transfer, as well as plasma physics and struc-
tural mechanics. Traditional approaches rely on
solving partial differential equations (PDEs) using
numerical methods, which are computationally ex-
pensive and often prohibitively slow for real-time
applications or large-scale simulations. Neural
PDEs have emerged as efficient alternatives to
these costly numerical solvers, offering significant
computational speed-ups. However, their lack of
robust uncertainty quantification (UQ) limits de-
ployment in critical applications. We introduce a
model-agnostic, physics-informed conformal pre-
diction (CP) framework that provides guaranteed
uncertainty estimates without requiring labelled
data. By utilising a physics-based approach, we
can quantify and calibrate the model’s inconsisten-
cies with the physics rather than the uncertainty
arising from the data. Our approach utilises con-
volutional layers as finite-difference stencils and
leverages physics residual errors as nonconfor-
mity 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.
1. Introduction
Numerical PDE solvers are essential tools in scientific and
engineering simulations (Danabasoglu et al., 2020; Giudi-
1Centre for Artificial Intelligence, University College Lon-
don
2Computing Division,
UK Atomic Energy Authority
3Heudiasyc Laboratory 4Polytechnique Montr´eal 5Mila - Que-
bec AI Institute.
Correspondence to:
Vignesh Gopakumar
<v.gopakumar@ucl.ac.uk>.
Proceedings...