Sampling from Neural Operator Priors to Solve Partial Differential Equations

Andrew Millard; Joshua Murphy; Zheng Zhao

cs.LGOct 12, 2026

Sampling from Neural Operator Priors to Solve Partial Differential Equations

Authors: Andrew Millard, Joshua Murphy, Zheng Zhao

Organizations: Department of Computer and Information Science, Linköping University, Sweden. · Department of Political Science, Trinity College Dublin, Ireland.

Abstract

Guided sampling with sparse observations and PDE residuals has recently been combined with neural operator priors to solve partial differential equations in function space, their natural setting. We pair such a prior with pseudo-bootstrap sequential Monte Carlo (pBS SMC), a guided sampling framework that accommodates proposals without tractable densities, and introduce a new proposal by adapting the DPM++ algorithm to the guidance setting. Across five benchmark PDE systems this proposal attains the best error--time trade-off at every step count tested, and wrapping it in pBS gives the lowest error on most inverse-problem settings, with the largest gains when few sampling steps are used.

Figures & tables

Darcy Flow †
$K$	DPS		SOSaG			G-DPM++
$K$	$\mathcal{E}$	$\mathcal{E}{\times}t$	$\mathcal{E}$	$\mathcal{E}{\times}t$	+pBS	$\mathcal{E}$	$\mathcal{E}{\times}t$	+pBS
25	0.1918	1.5881	0.1850	1.3234	0.1527	0.1588	0.6706	0.1203
50	0.1252	2.1180	0.1333	1.8811	0.1071	0.1196	1.0048	0.0937
100	0.0949	3.2017	0.1026	2.8351	0.0836	0.0926	1.5665	0.0770
150	0.0812	4.0898	0.0892	3.7497	0.0735	0.0807	2.0726	0.0716

Table 1: Relative

L_{2}

error (

\mathcal{E}

) and efficiency (

\mathcal{E}{\times}t

) across all PDE benchmarks, where

t

is wall-clock time in seconds. +pBS columns give the error of each proposal wrapped in the pBS SMC framework. Best

\mathcal{E}

and best

\mathcal{E}{\times}t

per row in bold . † Darcy Flow reports error rate rather than relative

L_{2}

error.

Figure 1: Sampler error vs. wall-clock time (efficiency) across the five PDE benchmarks. Each curve traces the trade-off between accuracy and compute as the number of sampling steps is varied; lower-and-leftward is better.

Appendix figures & tables15 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 2: Contour plot of the coefficient and solution fields overlayed with the observed values (highlighted pixels) for each PDE. The top row shows the coefficient field and the bottom row shows the solution field.

PDE	Solver	25 steps	50 steps	100 steps	150 steps	200 steps	500 steps
Darcy	DPM++	0.0703	0.0409	0.0355	0.0347	0.0343	0.0330
	DPS	0.1011	0.0457	0.0346	0.0340	0.0324	0.0316
	G-DPM++ pBS	0.0425	0.0368	0.0353	0.0339	0.0344	0.0339
	SOSaG pBS	0.0665	0.0434	0.0381	0.0374	0.0354	0.0339
	SOSaG	0.0864	0.0452	0.0377	0.0354	0.0362	0.0343
Helmholtz	DPM++	0.0953	0.0605	0.0432	0.0390	0.0343	0.0300

Appendix

Table 2: Forward problems: relative

L_{2}

error. Bold indicates best performance per column within each PDE.

Solver	Forward wins	Inverse wins	Combined wins	Combined (%)
DPS	17	4	21	35.0%
DPM++	4	2	6	10.0%
G-DPM++ pBS	8	23	31	51.7%
SOSaG	0	0	0	0.0%
SOSaG pBS	1	1	2	3.3%
Total	30	30	60	100%

Appendix

Table 3: Number of step-count slots (out of 6 per PDE) where each solver achieves the lowest error. Forward: 5 PDEs

\times

6 steps

=

30 slots. Inverse: 5 PDEs

\times

6 steps

=

30 slots. Combined total: 60 slots.

			Forward Problem					Inverse Problem
PDE	Steps	Solver	TC 1	TC 50	TC 100	TC 500	TC 1000	TC 1	TC 50	TC 100	TC 500	TC 1000
Darcy Flow	25 steps	DPS	0.1011	–	–	–	–	0.1918	–	–	–	–
		SOSaG-pBS	0.0665	0.0495	0.0513	0.0501	0.0466	0.1527	0.1236	0.1223	0.1208	0.1195
		G-DPM++-pBS	0.0425	0.0424	0.0429	0.0408	0.0423	0.1203	0.1130	0.1121	0.1124	0.1122
	50 steps	DPS	0.0457	–	–	–	–	0.1252	–	–	–	–
		SOSaG-pBS	0.0434	0.0384	0.0380	0.0367	0.0385	0.1071	0.0955	0.0891	0.0919	0.0899

Appendix

Table 4: Temper constant and steps ablation for pBS SMC methods (relative

L_{2}

error), including DPS at temperature 1 (DPS has no defined temper constant, so it is reported only in the TC 1 column; ‘–’ marks cells without data). Bold indicates the best result among DPS, SOSaG-pBS, and G-DPM++-pBS across all temper constants, for each Steps block, reported separately for the forward and inverse problems.

Solver	Forward wins	Inverse wins	Combined wins	Combined (%)
DPS	12	1	13	26.0%
SOSaG pBS	1	5	6	12.0%
G-DPM++ pBS	12	19	31	62.0%
Total	25	25	50	100%

Appendix

Table 5: Number of (PDE, Steps) slots (out of 25 per problem type) where each solver achieves the lowest error, best result taken across all temper constants for that PDE and step count. Forward: 5 PDEs

\times

5 steps

=

25 slots. Inverse: 5 PDEs

\times

5 steps

=

25 slots. Combined total: 50 slots.

Panel of Figure 3 — Figure 3: Temper constant ablation across step counts for the forward PDE problems. Generally we find that the tempering parameter value is not correlated with performance.

Panel of Figure 4 — Figure 4: Temper constant ablation across step counts for the inverse PDE problems. We find in the inverse problems that the correlation between the tempering parameter value and relative error is a lot stronger than the forward problems. Darcy Flow and Poisson in particular show strong correlations between better relative error and a higher tempering parameter. We also notice that the increased performance is more pronounced at higher step counts.

Panel of Figure 5 — Figure 5: Inverse problem coefficient reconstruction $a$ . 25 sampling steps.

Panel of Figure 6 — Figure 6: Inverse problem solution field reconstruction $u$ . 25 sampling steps.

Panel of Figure 7 — Figure 7: Inverse problem coefficient reconstruction $a$ . 50 sampling steps.

Panel of Figure 8 — Figure 8: Inverse problem solution field reconstruction $u$ . 50 sampling steps.

Panel of Figure 9 — Figure 9: Inverse problem coefficient reconstruction $a$ . 100 sampling steps.

Panel of Figure 10 — Figure 10: Inverse problem solution field reconstruction $u$ . 100 sampling steps.

Panel of Figure 11 — Figure 11: Inverse problem coefficient reconstruction $a$ . 500 sampling steps.

Panel of Figure 12 — Figure 12: Inverse problem solution field reconstruction $u$ . 500 sampling steps.

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Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems

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Recovering Partial Differential Equation (PDE) coefficient fields from extremely sparse observations is a severely ill-posed inverse problem for which generative machine learning methods (e.g., diffusion models) have become a leading way to encode the prior. Recent state-of-the-art diffusion solvers lift these priors to function spaces, finding a physics-consistent reconstruction in the output space of the diffusion denoiser. We prove that, in a discontinuous PDE setting, output space methods can result in failure to appropriately minimize the unobserved error with the correct coefficient field. Consequently, we propose Function space Backward-Consistent Sampling (FunBCS), an input space optimization approach for solving PDE problems which aims to find the best input such that the denoiser reconstruction is physics-consistent. We then prove that FunBCS appropriately minimizes the unobserved error, unlike output space optimization methods. Per our theoretical analysis, we also provide insights on how to dynamically allocate the number of input space optimization steps used throughout the sampling process. Our evaluations, across four PDE inverse problems (including the discontinuous Darcy flow), demonstrate that FunBCS reduces the reconstruction error by

27

64\%

while running

1.4

2.1\times

faster when compared to the current state-of-the-art.

Abstract

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