SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information

Jaehyeon Park; Inhyeok Jeong; Noseong Park

cs.LGOct 9, 2026

SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information

Authors: Jaehyeon Park, Inhyeok Jeong, Noseong Park

Organizations: SolverX · KAIST

Abstract

Partial differential equation (PDE) surrogate modeling often relies on either dense observations or access to governing equations/physics constraints. In practice, observations can be sparse and partial, and the underlying physics may be partially specified or unavailable. We introduce SBNO, an equation-free paired-data diffusion bridge matching framework that learns conditional operators for both forward and inverse PDE problems under partial observations. SBNO uses diffusion Schrödinger bridge matching to learn a pair of time-dependent drifts from a mask-aware endpoint coupling that preserves empirical input-solution pairing during training and supports both PDE-task directions within a single framework. At inference time, SBNO produces a deterministic point reconstruction by integrating the probability-flow ODE constructed from the learned drift pair, without inference-time observation or PDE guidance-weight tuning. On five PDE benchmarks with 5% observations, SBNO achieves superior or comparable accuracy to baselines while demonstrating lower test-instance error dispersion. Furthermore, SBNO is at least $9.7 \times$ faster and reduces GPU memory by at least $4.3 \times$ compared to guided-diffusion baselines.

Figures & tables

Panel of Figure 1 — Figure 1: Reconstruction error visualization . We compare our method against FunDPS, a guided-diffusion baseline, and GNOT, a deterministic baseline. All error maps are visualized using a shared color scale for fair comparison, where lower values indicate better reconstruction accuracy.

Figure 2: Overview of SBNO. (a) Training. From paired PDE samples $(\mathbf{a},\mathbf{u})$ and a random observation mask $\mathbf{m}_{\mathbf{s}}$ , SBNO lifts each partially observed field by concatenating masked values and the binary mask. It constructs a unified endpoint coupling that mixes forward and inverse pairs while preserving empirical input-solution pairing. Forward/backward drift networks are trained with diffusion Schrödinger bridge matching losses. (b) Inference. Given partial observations and their mask, SBNO produces a deterministic reconstruction of the target field using the probability-flow ODE constructed from the learned drift pair, with the same trained model supporting both PDE tasks without observation or PDE guidance-weight tuning.

Capability	SBNO (ours)	FunDPS & DiffusionPDE	FNOPE	GNOT	FNO
Observation conditioning	mask-aware	only observation	only observation	only observation	mask-aware
Unified PDE-task training	yes	yes	no	no	no
Inference-time hyperparameters	not required	required	not required	not required	not required
PDE information	equation-free	PDE guidance	equation-free	equation-free	equation-free
Inference type	both possible	stochastic	stochastic	deterministic	deterministic

Table 1: Qualitative comparison of SBNO and the baseline configurations evaluated in this study.

Dataset	Problem	SBNO (Ours)	FunDPS	DiffusionPDE	FNOPE	GNOT	FNO
Darcy flow	Forward	0.0359 $\pm$ 0.0133	0.0461 $\pm$ 0.0207	0.0552 $\pm$ 0.0849	0.2219 $\pm$ 0.0893	0.0510 $\pm$ 0.0148	0.0582 $\pm$ 0.0135
Darcy flow	Inverse	0.0269 $\pm$ 0.0125	0.0835 $\pm$ 0.0380	0.0499 $\pm$ 0.0338	0.2241 $\pm$ 0.1110	0.0779 $\pm$ 0.0367	0.0946 $\pm$ 0.0409
Poisson	Forward	0.0424 $\pm$ 0.0219	0.0775 $\pm$ 0.0473	0.0986 $\pm$ 0.1696	0.7849 $\pm$ 0.4061	0.0929 $\pm$ 0.0428	0.1458 $\pm$ 0.0518
Poisson	Inverse	0.2167 $\pm$ 0.0345	0.3664 $\pm$ 0.1232	0.2015 $\pm$ 0.0792	0.8423 $\pm$ 0.2244	0.3585 $\pm$ 0.0910	0.3912 $\pm$ 0.0991
Helmholtz	Forward	0.0519 $\pm$ 0.0315	0.0563 $\pm$ 0.0310	0.0902 $\pm$ 0.1412	0.7388 $\pm$ 0.3684	0.0926 $\pm$ 0.0393	0.2191 $\pm$ 0.0854
Helmholtz	Inverse	0.2310 $\pm$ 0.0501	0.2550 $\pm$ 0.0740	0.1893 $\pm$ 0.2557	0.7111 $\pm$ 0.2000	0.3728 $\pm$ 0.0865	0.4178 $\pm$ 0.0947

Table 2: The evaluation results are reported on five benchmark datasets. Each performance metric is expressed as the “mean

\pm

std” of the relative

L_{2}

error across test samples. For clarity, the best values in each column are highlighted in bold , while the second-best values are marked with underline .

Method	Time per Sample (sec)	Peak Mem (MB)
DiffusionPDE	46.65	3,846.7
FunDPS	2.71	5,428.8
SBNO (ours)	0.28	886.0

Table 3: Cost comparison between diffusion-based models. We report wall-clock time per sample and peak GPU memory. The best values are highlighted in bold .

Appendix figures & tables22 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 3: Visualizations of samples with data quality issue.

Guidance Weight $\zeta$	Darcy Flow		Poisson		Helmholtz		Nonbounded Navier-Stokes		Gray-Scott
Guidance Weight $\zeta$	Forward	Inverse	Forward	Inverse	Forward	Inverse	Forward	Inverse	Forward	Inverse
500	0.0480	0.2567	0.1649	0.4540	0.1484	0.2249	0.0624	0.1102	0.1032	0.0947
1,000	0.0374	0.1913	0.0931	0.3827	0.0725	0.2017	0.0427	0.1043	0.0920	0.0837
5,000	0.0371	0.1083	0.0775	0.2724	0.0519	0.2598	0.0358	0.2395	0.0718	0.0640
10,000	0.0449	0.0895	0.0993	0.2416	0.0499	0.4516	0.0373	0.5372	0.0662	0.0599
15,000	0.0557	0.0821	0.1324	0.2272	0.0495	0.7019	0.0426	0.4976	0.0677	0.0616

Appendix

Table 4: Sensitivity of FunDPS to the inference-time guidance weight

\zeta

under sparse observations. We report the relative

\ell_{2}

error for both forward and inverse tasks for five PDE benchmarks. The symbol “div” indicates that the inference procedure fails to converge.

Panel of Figure 4 — Figure 4: FunDPS sensitivity to guidance weight $\zeta$ in the forward and inverse problem. Divergent cases are marked with $\times$ .

Dataset	Problem	SBNO (Ours)	FunDPS fixed	FunDPS	DiffusionPDE	FNOPE	GNOT	FNO
Darcy flow	Forward	0.0359 $\pm$ 0.0133	0.0442 $\pm$ 0.0248	0.0461 $\pm$ 0.0207	0.0552 $\pm$ 0.0849	0.2219 $\pm$ 0.0893	0.0510 $\pm$ 0.0148	0.0582 $\pm$ 0.0135
Darcy flow	Inverse	0.0269 $\pm$ 0.0125	0.0742 $\pm$ 0.0348	0.0835 $\pm$ 0.0380	0.0499 $\pm$ 0.0338	0.2241 $\pm$ 0.1110	0.0779 $\pm$ 0.0367	0.0946 $\pm$ 0.0409
Poisson	Forward	0.0424 $\pm$ 0.0219	0.0775 $\pm$ 0.0420	0.0775 $\pm$ 0.0473	0.0986 $\pm$ 0.1696	0.7849 $\pm$ 0.4061	0.0929 $\pm$ 0.0428	0.1458 $\pm$ 0.0518
Poisson	Inverse	0.2167 $\pm$ 0.0345	0.2193 $\pm$ 0.0609	0.3664 $\pm$ 0.1232	0.2015 $\pm$ 0.0792	0.8423 $\pm$ 0.2244	0.3585 $\pm$ 0.0910	0.3912 $\pm$ 0.0991
Helmholtz	Forward	0.0519 $\pm$ 0.0315	0.0495 $\pm$ 0.0266	0.0563 $\pm$ 0.0310	0.0902 $\pm$ 0.1412	0.7388 $\pm$ 0.3684	0.0926 $\pm$ 0.0393	0.2191 $\pm$ 0.0854
Helmholtz	Inverse	0.2310 $\pm$ 0.0501	0.2017 $\pm$ 0.0532	0.2550 $\pm$ 0.0740	0.1893 $\pm$ 0.2557	0.7111 $\pm$ 0.2000	0.3728 $\pm$ 0.0865	0.4178 $\pm$ 0.0947

Appendix

Table 5: The evaluation results are reported on five benchmark datasets. Each performance metric is expressed as the “mean

\pm

standard deviation” of the relative

L_{2}

error across test set samples. For clarity, the best mean and best standard deviation in each column are highlighted in bold , while the second-best values are marked with underline .

Problem	Forward problem		Inverse problem
Problem	mean	std	mean	std
Darcy Flow	0.0359 $\pm$ 0.000708	0.0133 $\pm$ 0.000627	0.0269 $\pm$ 0.000297	0.0125 $\pm$ 0.000185
Poisson	0.0424 $\pm$ 0.000595	0.0219 $\pm$ 0.000712	0.2167 $\pm$ 0.002005	0.0345 $\pm$ 0.001032
Helmholtz	0.0519 $\pm$ 0.001751	0.0315 $\pm$ 0.002282	0.2310 $\pm$ 0.003499	0.0501 $\pm$ 0.002147
Nonbounded Navier-Stokes	0.0368 $\pm$ 0.000786	0.0086 $\pm$ 0.000206	0.1097 $\pm$ 0.000403	0.0167 $\pm$ 0.000080
Gray-Scott	0.0393 $\pm$ 0.001163	0.0193 $\pm$ 0.000897	0.0382 $\pm$ 0.001187	0.0167 $\pm$ 0.000884

Appendix

Table 6: Sensitivity of SBNO to inference randomness over five different seeds. For each seed, we compute the test-set mean and standard deviation of the relative

L_{2}

error. Each entry reports the average and standard deviation of the corresponding metric across seeds.

Method	Darcy forward	Darcy inverse	Helmholtz forward	Helmholtz inverse
DiffusionPDE	2.1465 $\pm$ 2.4595	1.2648 $\pm$ 0.1078	8.9174 $\pm$ 11.6318	0.4139 $\pm$ 0.2540
FunDPS	11.1638 $\pm$ 1.7811	1.3643 $\pm$ 0.0712	4.2907 $\pm$ 1.7299	0.5651 $\pm$ 0.1460
SBNO	1.6347 $\pm$ 0.1468	1.1543 $\pm$ 0.0621	1.9629 $\pm$ 1.1171	0.3278 $\pm$ 0.1533

Appendix

Table 7: High-frequency relative spectral error under 5% observations. Values are mean

\pm

standard deviation across test instances; lower is better.

Method	Darcy forward	Darcy inverse	Helmholtz forward	Helmholtz inverse
DiffusionPDE	10.6676 $\pm$ 25.2757	0.9310 $\pm$ 0.3236	213.7049 $\pm$ 530.5397	0.9117 $\pm$ 0.2235
FunDPS	127.4999 $\pm$ 41.0276	1.1889 $\pm$ 0.1975	22.2814 $\pm$ 22.8030	0.9577 $\pm$ 0.2244
SBNO	2.6015 $\pm$ 0.5581	0.7429 $\pm$ 0.1215	6.2776 $\pm$ 8.0286	0.9932 $\pm$ 0.5402

Appendix

Table 8: High-frequency energy ratios under 5% observations. Values are mean

\pm

standard deviation across test instances. A ratio of one indicates matching energy when the normalization floor is inactive; the ratios should be interpreted together with spectral error.

Problem	Mask-aware forward	No-mask forward	Mask-aware inverse	No-mask inverse
Darcy Flow	0.0359 $\pm$ 0.0133	0.0579 $\pm$ 0.0172	0.0269 $\pm$ 0.0125	0.0297 $\pm$ 0.0138
Poisson	0.0424 $\pm$ 0.0219	0.3339 $\pm$ 0.1829	0.2167 $\pm$ 0.0345	0.3950 $\pm$ 0.1254
Helmholtz	0.0519 $\pm$ 0.0315	0.2411 $\pm$ 0.1428	0.2310 $\pm$ 0.0501	0.3334 $\pm$ 0.1149
Nonbounded Navier-Stokes	0.0368 $\pm$ 0.0086	0.1471 $\pm$ 0.0238	0.1097 $\pm$ 0.0167	0.2267 $\pm$ 0.0243
Gray-Scott	0.0393 $\pm$ 0.0193	0.0448 $\pm$ 0.0248	0.0382 $\pm$ 0.0167	0.0385 $\pm$ 0.0210

Appendix

Table 9: Mask-channel ablation on all five benchmark datasets under 5% observations. Values are mean

\pm

standard deviation of relative

L_{2}

error across test instances. For Gray–Scott, entries average the corresponding statistics of the two reconstructed channels.

Training coupling	Forward error	Inverse error
Unified mixture	0.0359 $\pm$ 0.0133	0.0269 $\pm$ 0.0125
Forward-task coupling only	0.0315 $\pm$ 0.0073	0.5063 $\pm$ 0.0525
Inverse-task coupling only	0.5420 $\pm$ 0.0269	0.0150 $\pm$ 0.0072

Appendix

Table 10: Mask-channel ablation on all five benchmark datasets under 5% observations. Values are mean

\pm

standard deviation of relative

L_{2}

error across test instances. For Gray–Scott, entries average the corresponding statistics of the two reconstructed channels.

Forward-drift training coupling	Inference	Forward error
Unified mixture	Forward-drift-only SDE	0.0263 $\pm$ 0.0064
Forward-task coupling only	Forward-drift-only SDE	0.0174 $\pm$ 0.0056

Appendix

Table 11: Mask-channel ablation on all five benchmark datasets under 5% observations. Values are mean

\pm

standard deviation of relative

L_{2}

error across test instances. For Gray–Scott, entries average the corresponding statistics of the two reconstructed channels.

Dataset	Problem	SBNO (Ours)	FunDPS	DiffusionPDE	FNOPE	GNOT	FNO
Korteweg –De Vries	Forward	0.2858 $\pm$ 0.0650	1.5024 $\pm$ 0.2236	0.2177 $\pm$ 0.2017	0.8498 $\pm$ 0.0446	1.1628 $\pm$ 0.0692	0.5356 $\pm$ 0.0348
Korteweg –De Vries	Inverse	0.2866 $\pm$ 0.1005	1.5401 $\pm$ 0.2491	0.1765 $\pm$ 0.0791	0.9399 $\pm$ 0.0571	1.1798 $\pm$ 0.0739	0.7483 $\pm$ 0.0202
Kuramoto– Sivashinsky	Forward	0.3236 $\pm$ 0.0295	1.2990 $\pm$ 0.0754	0.4517 $\pm$ 0.0992	1.0500 $\pm$ 0.0711	1.0707 $\pm$ 0.0579	0.6808 $\pm$ 0.0359
Kuramoto– Sivashinsky	Inverse	0.3674 $\pm$ 0.0283	1.2962 $\pm$ 0.0733	0.3973 $\pm$ 0.0584	1.0451 $\pm$ 0.0716	1.0660 $\pm$ 0.0568	0.6727 $\pm$ 0.0389
Kolmogorov Flow	Forward	0.5322 $\pm$ 0.0978	1.2445 $\pm$ 0.0560	0.5551 $\pm$ 0.1133	0.6180 $\pm$ 0.0882	0.9275 $\pm$ 0.0860	0.6211 $\pm$ 0.0764
Kolmogorov Flow	Inverse	0.4849 $\pm$ 0.0810	1.2329 $\pm$ 0.0525	0.5739 $\pm$ 0.2634	0.5687 $\pm$ 0.0632	0.9071 $\pm$ 0.0814	0.6039 $\pm$ 0.0466

Appendix

Table 12: Evaluation results on three additional APEBench datasets. Each performance metric is reported as the “mean

\pm

standard deviation” of the relative

L_{2}

error across test samples. For clarity, the best mean and best standard deviation in each row are highlighted in bold , while the second-best values are marked with underline .

Category	Metric	Darcy Flow		Poisson		Helmholtz		Nonbounded Navier-Stokes		Gray-Scott
Category	Metric	Forward	Inverse	Forward	Inverse	Forward	Inverse	Forward	Inverse	Forward	Inverse
Sample mean reconstruction	Rel $L_{2}$ error mean	0.0263	0.1296	0.0519	0.1271	0.0539	0.1362	0.0717	0.1342	0.0254	0.0272
	Rel $L_{2}$ error std	0.0064	0.0334	0.0210	0.0361	0.0297	0.0362	0.0072	0.0182	0.0119	0.0162
	Abs $L_{1}$ error mean	0.0001	0.2217	0.0001	0.0257	0.0002	0.0260	0.0142	0.0267	0.0089	0.0037
Uncertainty magnitude	Global mean std	0.0002	0.2016	0.0001	0.0144	0.0001	0.0131	0.0128	0.0172	0.0116	0.0060
Uncertainty magnitude	Global mean var	0.0000	0.6258	0.0000	0.0002	0.0000	0.0002	0.0002	0.0003	0.0002	0.0001

Appendix

Table 13: Uncertainty-quantification results of SBNO using stochastic posterior sampling. For each dataset and for both forward and inverse problems, we draw 60 stochastic bridge samples per test case and report metrics for sample-mean reconstruction, uncertainty magnitude, calibration and sharpness, and the relation between predictive error and estimated uncertainty.

Figure 5: Representative uncertainty-quantification example of SBNO under partial observations using stochastic posterior sampling. We draw 60 stochastic bridge samples and visualize the ground truth, sample mean, absolute error of the sample mean, sample standard deviation, 90% posterior interval width, and the corresponding 90% empirical coverage mask. The sample mean summarizes the reconstruction, while the posterior spread highlights spatial regions associated with larger predictive uncertainty.

Panel of Figure 6 — Figure 6: Comparison of ODE and SDE inference on representative Darcy-flow and nonbounded Navier–Stokes forward cases under 5% observations. For each case, we show the ground truth, deterministic ODE reconstruction, one stochastic SDE sample, and the mean of 60 SDE samples. All predictions use the same observation mask, and panels within each case share a color scale. The individual SDE sample is shown separately from the sample-mean reconstruction.

Noise std	Problem	SBNO (Ours)	FunDPS	DiffusionPDE	GNOT
0.1	Forward	0.0359	0.0461	0.0371	0.0510
0.1	Inverse	0.0269	0.0834	0.0743	0.1031
0.5	Forward	0.0359	0.0461	0.0372	0.1709
0.5	Inverse	0.0274	0.0837	0.0749	0.4092
1.0	Forward	0.0358	0.0461	0.0373	0.2657
1.0	Inverse	0.0291	0.0838	0.0766	0.4764

Appendix

Table 14: Evaluation results with Gaussian noise added to partial observations. Noise std denotes the standard deviation of the Gaussian noise. For clarity, the best mean in each column is highlighted in bold , while the second-best values are marked with underline .

Step number	32	64	96	128	dopri
Poisson Inverse	0.2167	0.2074	0.2072	0.2073	0.2166
Helmholtz Inverse	0.2310	0.2220	0.2243	0.2249	0.2310
Time per Sample(sec)	0.28	0.56	0.83	1.10	0.57

Appendix

Table 15: Reconstruction performance under different ODE solver configurations on the challenging inverse tasks where SBNO did not outperform DiffusionPDE in the main text, together with the corresponding per-sample inference time. We increase the number of RK4 integration steps and additionally evaluate an adaptive DOPRI solver to assess whether improved numerical integration can close the remaining performance gap.

Observation ratio	5%	4%	3%	2%	1%	0.5%
Forward	0.0359	0.0402	0.0749	0.1226	0.2212	0.2727
Inverse	0.0269	0.0498	0.0865	0.2107	0.2924	0.3362

Appendix

Table 16: Evaluation results with varying ratios of partial observations. The observation ratio denotes the proportion of points in the full grid that are provided as partial observations for evaluation.

Observation ratio	10%	8%	5%	4%	3%	2%	1%	0.5%
Forward	0.0664	0.0703	0.0836	0.0880	0.0954	0.1112	0.1435	0.1790
Inverse	0.0442	0.0437	0.0474	0.0511	0.0595	0.0773	0.1240	0.1710

Appendix

Table 17: Evaluation results with varying ratios of partial observations. The observation ratio denotes the proportion of points in the full grid that are provided as partial observations for evaluation.

Dataset	Problem	SBNO	SBNO (loop)
Darcy flow	Forward	0.0359 $\pm$ 0.0133	0.0369 $\pm$ 0.0132
Darcy flow	Inverse	0.0269 $\pm$ 0.0128	0.0531 $\pm$ 0.0216
Helmholtz	Forward	0.0519 $\pm$ 0.0315	0.1886 $\pm$ 0.1069
Helmholtz	Inverse	0.2310 $\pm$ 0.0501	0.2506 $\pm$ 0.0722

Appendix

Table 18: Evaluation results for the closed-loop experiment, where inference is run again in the opposite direction using the model prediction from the first pass as the new partial observation. For forward inference, the prediction obtained from backward inference is used as the partial input observation. Conversely, for backward inference, the prediction obtained from forward inference is used as the partial solution observation. The resulting performance is reported as “SBNO (loop).”

Figure 7: Representative closed-loop visualization . We show the original SBNO predictions and the reconstructions obtained after reusing those predictions as partial observations for inference in the opposite direction.

Models	Hyperparameter	Darcy Flow		Poisson		Helmholtz		Nonbounded Navier-Stokes		Gray-Scott
		Forward	Inverse	Forward	Inverse	Forward	Inverse	Forward	Inverse	Forward	Inverse
Ours	learning rate	0.0001
	ODE solver step	32
	Sigma min	0.2		0.001		0.001		0.001		0.001
	Sigma max	0.2		0.3		0.3		0.45		0.3
FunDPS multi-res	Observation guidance	1,000	25,000	5,000	10,000	10,000	5,000	5,000	1,000	5,000	10,000

Appendix

Table 19: Best hyperparameters for each model used in the main experiments.

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Open paper AskPDF arXiv

Recovering complete physical fields from sparse observations is challenging because the measurements may not uniquely determine the underlying state. Diffusion-based PDE solvers address this problem through iterative sampling whereas neural operators provide deterministic one-pass predictions. We propose SCOPE (Sparse-Context Observability-aware Predictive Embeddings) to recover complete PDE fields from sparse observations by coupling full-field latent prediction with physical reconstruction. A shared decoder reconstructs fields from both predicted and complete-view representations so that representation learning is guided by both physical recovery and latent matching. We derive a quadratic risk decomposition at fixed teacher-decoder pairs showing why optimal latent prediction need not yield optimal field reconstruction. We also establish sufficient conditions for decoder improvements on complete inputs to transfer to recovery from partial observations. Experiments across five PDE settings show that SCOPE outperforms mask-aware neural operators on all ten forward and inverse tasks and achieves lower errors than those reported for diffusion-based solvers including DiffusionPDE and FunDPS. Decoder-only adaptation further improves recovery without retraining the backbone while retaining deterministic single-pass inference.

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