Conformal Prediction for Time-Dependent PDEs

Joshua Stiller; Annika Schneider; Eyke Hüllermeier

cs.LGNov 5, 2026

Conformal Prediction for Time-Dependent PDEs

Authors: Joshua Stiller, Annika Schneider, Eyke Hüllermeier

Organizations: LMU Munich, MCML · TU Munich, Helmholtz AI, relAI, MCML · LMU Munich, MCML, DFKI

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.

Figures & tables

Figure 1: Examples of different types of non-stationarity that can arise in time-dependent systems. (a) Sudden distribution shift due to external shocks, illustrated by a stock market crash. (b) Long-term structural changes, as in climate time series. (c) Limited observation window during system development, where calibration and testing occur on different parts of the trajectory.

Figure 2: Overview of the theoretical results. The colors differentiate between function space (blue) and discretized space (yellow).

Panel of Figure 3 — Figure 3: Visualization for Proposition 4.4 and Proposition 4.7 . (a) Coverage decays exponentially with grid granularity for the baselines, while WCP grows infinite bands (dotted lines). The theoretical lower bound follows Barber et al. [3] . (b) Coverage of the analytical solution of the backward heat equation with numerical conformal bands. With finer granularity, coverage over the analytic solution converges to the target as the reconstruction error vanishes. Also here, the coverage decline effect of (a) would start dominating at higher granularity. More in Appendix H.4 .

Figure 4: Coverages for varying

a

and fixed

b=0,c=0.5

on the recalibration task. The 90% target is marked by a black dashed line. For WCP Gauss and WCP Learned, the fraction of samples with infinite bandwidth

n_{\infty}

is indicated by green and pink dotted lines, respectively. Coverage holds for some parameters, but small variations induce undercoverage at later time steps. Only WCP Gauss meets the target throughout.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

Figure E.1 : CP bands of the backward heat equation calibrated at

t=0

. Each panel shows one time point of the solution

u(x,t)

and CP bands over the spatial domain

x

. While at time

t=\delta

, the solution is properly covered, coverage is lost at later time steps.

Figure H.1 : Exemplary coverage plots for the methods (from left to right) Supr., LSCI, WCP Learned, and WCP Gauss (Ours) for a sample at time step

10

with parameters

a=-0.0075,b=0,c=0.5

Figure H.2 : Ablation study for method stability. We use data from the synthetic task (no recalibration) with parameters

a=-0.0075,b=0,c=0

. From top to bottom, the rows correspond to 5000, 4000, 3000, 2000, and 1000 calibration samples. Each plot reports coverage (left) and the fraction of infinite conformal bands (right) over time, where color corresponds to grid granularity. We see that higher granularity leads to earlier onset of infinite bands, which aligns with Proposition 4.4 . When the number of samples is insufficient relative to the dimensionality, covariance estimation degrades and undercoverage can occur.

Figure H.3 : TV distance of Theorem 4.5 at

|t_{1}-t_{2}|=0.5

over the grid granularity. For the constant drift

{\bm{v}}=\mathbf{1}

under white noise and the covariance of the granularity experiment in Figure 3(a) ,

\mathbf{1}^{\top}{\bm{\Sigma}}_{0}^{-1}\mathbf{1}=p

and both curves coincide. For the smooth Gaussian random field

{\mathcal{N}}(0,625(-\partial_{x}^{2}+25{\bm{I}})^{-2})

(0,2\pi)

, evaluated on the grid, the TV distance levels off below

1

for the constant drift, but tends to

1

for a step drift (

v_{j}=1

on the left half of the domain,

0

otherwise).

Figure I.1 : Exemplary frames from the real-world pulsed-thermography dataset. The left image shows a raw data frame from the beginning of the cooling period. The middle image shows how we cropped the frames to remove the background. The right image shows how the data looks after $100$ time steps.

Figure I.2 : Coverages for the pulsed-thermography dataset for the recalibration task (left) and the general task (right). The 90% coverage target and, for WCP Gauss (resp. Learned), the fraction of samples with infinite bandwidth

n_{\infty}

are marked as a black dashed line and a green (resp. pink) dashed line, respectively. We see that all methods achieve target coverage at first, but only WCP Gauss can preserve coverage over time.

Figure I.3 : Coverages for the Lake Maggiore nighttime cooling dataset for the recalibration task (left) and the general task (right). The 90% target is marked by a black dashed line. For WCP Gauss (resp. Learned), the fraction of samples with infinite bandwidth

n_{\infty}

is shown as a green (resp. pink) dotted line. WCP Gauss and LSCI maintain the target coverage on both tasks. Deviations at single time steps are statistically expected given the test size of

14

Figure I.4 : ( Top ) Coverages for the FRAP dataset for the recalibration task (left) and the general task (right). The 90% target is marked by a black dashed line. For WCP Gauss (resp. Learned), the fraction of samples with infinite bandwidth $n_{\infty}$ is shown as a green (resp. pink) dotted line. ( Bottom ) Recovery field snapshots from a real FRAP experiment on $32$ wt% sucrose solution, showing the normalized fluorescence change $\Delta F/F_{0}$ just after photobleaching ( $t=0$ ) and after partial recovery ( $t=5$ ). The dashed circle marks the bleach center. WCP Gauss is the only method that provides non-trivial target coverage for every time step on both tasks, where WCP Learned immediately provides trivial bands.

Figure J.1 : Multi-seed stability. (a) Coverages for the general task of the synthetic PDE (

a=-0.0075

b=0

) for

c\in\{-0.5,0,0.5\}

, mean

\pm

one standard deviation over 10 reshuffled calibration/test splits. (b) Coverages for the pulsed-thermography dataset for the recalibration task (left) and the general task (right), mean

\pm

one standard deviation over 10 plate-level reshuffles with a retrained surrogate per seed. The 90% target is marked by a black dashed line. For WCP Gauss (resp. Learned), the fraction of samples with infinite bandwidth

n_{\infty}

is shown as a green (resp. pink) dotted line.

Figure J.2 : Nonlinear ablation for

u_{t}=au_{xx}+cu-\gamma u^{3}

(

a=-0.0075

b=0

). (a) Coverages, mean

\pm

one standard deviation over 10 reshuffled calibration/test splits, for

c=0.5

(top row) and

c=3

(bottom row). The 90% target is marked by a black dashed line. For WCP Gauss (resp. Learned), the fraction of samples with infinite bandwidth

n_{\infty}

is shown as a green (resp. pink) dotted line. (b) Marginal distribution of the middle grid point at

t=1

for

c=3

with a Gaussian fit.

Figure J.3 : Bandwidth of WCP Gauss per test sample against the test log likelihood-ratio

\log r(x_{\mathrm{test}})

for the synthetic PDE (

a=-0.0075

b=0

c=0.5

, general task) at the first time steps with infinite bands. Infinite bands are marked by triangles at the top of each panel.

Figure K.1 : Coverages for the recalibration task for varying

a,c

and fixed

b=-0.5

across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

Figure K.2 : Coverages for the recalibration task for varying

a,c

and fixed

b=0

across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

Figure K.3 : Coverages for the recalibration task for varying

a,c

and fixed

b=0.5

across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

			$c=-0.5$					$c=0.0$					$c=0.5$
		T:	1	5	10	15	20	1	5	10	15	20	1	5	10	15	20
$a=-0.005$	Supr.	Cov.	.91	.92	.91	.90	.90	.91	.91	.90	.91	.89	.91	.89	.88	.89	.88
	Supr.	Bw.	.0125	.0109	.0089	.0073	.0065	.0133	.0134	.0135	.0145	.0161	.0142	.0171	.0243	.0419	.0793
	LSCI	Cov.	.96	.96	.92	.91	.90	.96	.95	.91	.94	.86	.97	.96	.97	.97	.96
	LSCI	Bw.	.0121	.0069	.0052	.0052	.0056	.0126	.0108	.0101	.0148	.0152	.0133	.0167	.0238	.0508	.0988
	DtACI	Cov.	.91	.97	.96	.96	.94	.91	.93	.93	.93	.86	.91	.88	.89	.91	.91

Appendix

Table K.1 : Coverage and mean bandwidth for the recalibration task for varying

a,c

and fixed

b=-0.5

. We also report the fraction of samples where infinite bands were reported

n_{\infty}

. The mean bandwidth is taken over finite bands only. The gray font is chosen for better readability.

			$c=-0.5$					$c=0.0$					$c=0.5$
		T:	1	5	10	15	20	1	5	10	15	20	1	5	10	15	20
$a=-0.005$	Supr.	Cov.	.91	.92	.92	.92	.91	.91	.91	.91	.91	.89	.91	.90	.90	.89	.88
	Supr.	Bw.	.0125	.0108	.0089	.0073	.0063	.0132	.0133	.0136	.0144	.0157	.0141	.0172	.0249	.0417	.0798
	LSCI	Cov.	.96	.97	.93	.90	.88	.96	.95	.93	.89	.87	.97	.97	.98	.97	.96
	LSCI	Bw.	.0120	.0073	.0049	.0048	.0051	.0124	.0126	.0098	.0110	.0145	.0131	.0187	.0235	.0495	.0972
	DtACI	Cov.	.91	.97	.96	.97	.95	.91	.94	.94	.95	.91	.91	.89	.91	.91	.90

Appendix

Table K.2 : Coverage and mean bandwidth for the recalibration task for varying

a,c

and fixed

b=0.0

. We also report the fraction of samples where infinite bands were reported

n_{\infty}

. The mean bandwidth is taken over finite bands only. The gray font is chosen for better readability.

			$c=-0.5$					$c=0.0$					$c=0.5$
		T:	1	5	10	15	20	1	5	10	15	20	1	5	10	15	20
$a=-0.005$	Supr.	Cov.	.91	.92	.92	.91	.91	.91	.91	.91	.91	.89	.91	.89	.89	.89	.89
	Supr.	Bw.	.0127	.0109	.0090	.0075	.0067	.0134	.0135	.0139	.0147	.0164	.0144	.0172	.0248	.0426	.0809
	LSCI	Cov.	.96	.95	.93	.92	.90	.97	.94	.91	.89	.86	.96	.95	.97	.97	.97
	LSCI	Bw.	.0100	.0067	.0053	.0053	.0056	.0102	.0096	.0095	.0127	.0150	.0108	.0137	.0238	.0505	.0986
	DtACI	Cov.	.91	.97	.96	.96	.95	.91	.94	.93	.93	.89	.91	.88	.90	.91	.90

Appendix

Table K.3 : Coverage and mean bandwidth for the recalibration task for varying

a,c

and fixed

b=0.5

. We also report the fraction of samples where infinite bands were reported

n_{\infty}

. The mean bandwidth is taken over finite bands only. The gray font is chosen for better readability.

Figure K.4 : Mean coverages for the general task (no recalibration) for varying

a,c

and fixed

b=-0.5

across an increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

Figure K.5 : Mean coverages for the general task (no recalibration) for varying

a,c

and fixed

b=0

across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

Figure K.6 : Mean coverages for the general task (no recalibration) for varying

a,c

and fixed

b=0.5

across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

			$c=-0.5$					$c=0.0$					$c=0.5$
		T:	1	5	10	15	20	1	5	10	15	20	1	5	10	15	20
$a=-0.005$	Supr.	Cov.	.91	.93	.95	.97	.98	.91	.91	.91	.90	.90	.91	.89	.85	.78	.69
	Supr.	Bw.	.0129	.0129	.0129	.0129	.0129	.0132	.0132	.0132	.0132	.0132	.0136	.0136	.0136	.0136	.0136
	LSCI	Cov.	.96	.98	.99	.99	.99	.96	.95	.92	.87	.72	.96	.90	.72	.42	.11
	LSCI	Bw.	.0122	.0122	.0123	.0123	.0123	.0124	.0124	.0124	.0124	.0124	.0127	.0127	.0126	.0125	.0124
	WCP Learned	Cov.	.90	.90	.90	.91	.91	.90	.90	.89	.89	.87	.90	.89	.87	.84	.82

Appendix

Table K.4 : Coverage and mean bandwidth for the general task (no recalibration) for varying

a,c

and fixed

b=-0.5

. We also report the fraction of samples where infinite bands were reported

n_{\infty}

. The mean bandwidth is taken over finite bands only. A dash (–) marks cells where every band was infinite. The gray font is chosen for better readability.

			$c=-0.5$					$c=0.0$					$c=0.5$
		T:	1	5	10	15	20	1	5	10	15	20	1	5	10	15	20
$a=-0.005$	Supr.	Cov.	.91	.93	.95	.97	.98	.91	.91	.91	.91	.90	.91	.89	.83	.79	.70
	Supr.	Bw.	.0128	.0128	.0128	.0128	.0128	.0132	.0132	.0132	.0132	.0132	.0136	.0136	.0136	.0136	.0136
	LSCI	Cov.	.96	.98	.98	.99	1.00	.95	.94	.91	.86	.75	.96	.90	.73	.45	.13
	LSCI	Bw.	.0121	.0121	.0122	.0122	.0122	.0123	.0123	.0123	.0123	.0123	.0127	.0126	.0125	.0124	.0123
	WCP Learned	Cov.	.90	.89	.90	.91	.91	.90	.90	.89	.89	.88	.90	.89	.88	.85	.82

Appendix

Table K.5 : Coverage and mean bandwidth for the general task (no recalibration) for varying

a,c

and fixed

b=0.0

. We also report the fraction of samples where infinite bands were reported

n_{\infty}

. The mean bandwidth is taken over finite bands only. A dash (–) marks cells where every band was infinite. The gray font is chosen for better readability.

			$c=-0.5$					$c=0.0$					$c=0.5$
		T:	1	5	10	15	20	1	5	10	15	20	1	5	10	15	20
$a=-0.005$	Supr.	Cov.	.91	.93	.95	.97	.98	.91	.91	.91	.91	.90	.91	.89	.84	.77	.68
	Supr.	Bw.	.0129	.0129	.0129	.0129	.0129	.0133	.0133	.0133	.0133	.0133	.0137	.0137	.0137	.0137	.0137
	LSCI	Cov.	.96	.98	.98	1.00	1.00	.97	.95	.91	.84	.74	.96	.90	.73	.41	.15
	LSCI	Bw.	.0122	.0122	.0122	.0123	.0123	.0124	.0124	.0124	.0124	.0124	.0127	.0126	.0125	.0124	.0123
	WCP Learned	Cov.	.90	.90	.90	.90	.91	.90	.89	.88	.88	.87	.90	.89	.87	.85	.82

Appendix

Table K.6 : Coverage and mean bandwidth for the general task (no recalibration) for varying

a,c

and fixed

b=0.5

. We also report the fraction of samples where infinite bands were reported

n_{\infty}

. The mean bandwidth is taken over finite bands only. A dash (–) marks cells where every band was infinite. The gray font is chosen for better readability.

Figure K.7 : Coverages for the general task for varying

a,c

and fixed

b=0

with Laplace initial distribution across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

Figure K.8 : Coverages for the general task for varying

a,c

and fixed

b=0

with Logistic initial distribution across increasing prediction horizon. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

for WCP Learned and WCP Gauss as dotted line in their respective color.

Figure K.9 : Coverages of WCP with Cauchy likelihood-ratio weights for the general task for varying

a,c

and fixed

b=0

with heavy-tailed elliptical Cauchy initial distribution, mean

\pm

one standard deviation over 10 reshuffled calibration/test splits. The 90% coverage target is marked as black dashed line and the fraction of samples with infinite bandwidth

n_{\infty}

as dotted line. The Cauchy likelihood ratios stay bounded, so no infinite bands occur. The heavy tails increase the variance of the coverage across splits, and for

c=-0.5

the mean coverage falls slightly below the target at late time steps.

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Conformal Prediction for Neural Operators: Distribution-Free Uncertainty Quantification in Physics Simulation

Michael Chin

Independent Researcher

Uncertainty Quantification PDE Surrogate Modeling Conformal Prediction Neural Operators Regression Prediction Intervals

Open paper AskPDF arXiv

Neural operators such as the Fourier Neural Operator (FNO) have emerged as powerful surrogates for solving partial differential equations (PDEs), achieving speedups of several orders of magnitude over traditional numerical solvers. However, deploying these models in safety-critical engineering applications -- such as thermal management of electronic components and battery systems -- requires not only accurate point predictions but also rigorous uncertainty guarantees. Existing uncertainty quantification (UQ) methods for neural operators, including Monte Carlo Dropout and Deep Ensembles, provide only relative uncertainty estimates without formal coverage guarantees. In this work, we propose the first application of split conformal prediction to neural operator-based physics simulation, providing distribution-free prediction intervals with finite-sample coverage guarantees. We further introduce a normalized conformal prediction scheme that leverages MC Dropout uncertainty to produce adaptive-width intervals, yielding tighter intervals in regions of low uncertainty and wider intervals where the model is less certain. Full-scale experiments (33.7M parameters, 800 training samples, 5 ensemble members, NVIDIA V100) on steady-state heat conduction benchmarks demonstrate that our method achieves 89.1% empirical coverage at the target level of alpha=0.1, while producing spatially adaptive prediction intervals that reflect the underlying physical uncertainty structure. We also provide an uncertainty decomposition framework that separates epistemic uncertainty (68% of total) from aleatoric uncertainty (32% of total), offering actionable guidance for data collection and model improvement. Our method is implemented in an open-source platform with REST API endpoints and interactive 3D visualization.

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