SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference

Ruichen Xu; Siyao Wang; Fang Wan; Jiacheng Qiu; Wenhan Gao; Jiaxing Zhang; Linsey Pang; Ravid Shwartz-Ziv

cs.LGSep 29, 2026

SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference

Authors: Ruichen Xu, Siyao Wang, Fang Wan, Jiacheng Qiu, Wenhan Gao, Jiaxing Zhang, Linsey Pang, Ravid Shwartz-Ziv, +3 more

Organizations: Stony Brook University · University of California, Davis · Independent Research · PayPal · Northeastern University · New York University

Abstract

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.

Figures & tables

Figure 2: SCOPE training and inference. (a) Context prediction: the observation-conditioned pathway reconstructs the paired fields. (b) JEPA target: the EMA teacher supplies $\bar{z}$ for comparison with $z_{M}$ through $\mathcal{L}_{J}$ ; its parameters follow the online encoder. (c) Full-view grounding: the shared online encoder and decoder support $\mathcal{L}_{G}$ , while $z_{F}$ receives variance regularization. (d) One-pass inference: only the context pathway and inverse normalization are used. Physical field losses include inverse normalization. SP replaces $D_{\omega}$ with $D_{\omega^{\prime}}$ before inference. Field maps are schematic.

Literature-Reported: Deterministic Methods
Method	Darcy		Poisson		Helmholtz		NS		NS (BCs) †
	Fwd. $\downarrow$	Inv. (BER) $\downarrow$	Fwd. $\downarrow$	Inv. $\downarrow$	Fwd. $\downarrow$	Inv. $\downarrow$	Fwd. $\downarrow$	Inv. $\downarrow$	Fwd. $\downarrow$	Inv. $\downarrow$
FNO ( Li et al., 2021 )	28.20	49.30	100.90	232.70	98.20	218.20	101.40	96.00	82.80	69.60
PINO ( Li et al., 2024 )	35.20	49.20	107.10	231.90	106.50	216.90	101.40	96.00	81.10	69.50
DeepONet ( Lu et al., 2021 )	38.30	41.10	155.50	105.80	123.10	132.80	103.20	97.20	97.70	91.90
PINN ( Raissi et al., 2019 )	48.80	59.70	128.10	130.00	142.30	160.00	142.70	146.80	100.10	105.50

Table 1: Sparse recovery across PDE settings. Forward evaluates

u

; inverse evaluates

a

. Values are percentages: Darcy inverse BER, physical relative

L_{2}

otherwise. Bold / underline mark the best/second-best unrounded results within the main common-observation group. Literature rows retain their source protocols (Appendix F.3 ); training budgets and model sizes differ (Table 2 ).

Figure 3: Grounding and readout comparisons. Bars show mean errors with per-record SD on logarithmic axes. Forward evaluates

u

; inverse evaluates

a

, using Darcy BER. Auxiliary operators use their own protocols and metric units (Appendix I.2 ); additional supervision is compared within each family. SCOPE denotes Full; all SP heads use its frozen backbone.

Appendix figures & tables18 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	SCOPE backbones	Main-table operators	Auxiliary operators
Models	Full, FO, FJV	FNO, DeepONet, CNO, Transolver	FNO/DeepONet, each with and without $G$
Inputs / outputs	Four masked-field/mask inputs; two fields	Six inputs including coordinates; two fields	Mask-aware inputs; one hidden-field head
Data exposure and observation contexts
Pretraining	10 epochs; none for FO	None	None
Main training	500 epochs	100 epochs	100 epochs
Batch size	32	125	32

Appendix

Table 2: Training settings by experiment group. The main-table operators share SCOPE’s observation/task protocol and physical field objective. The auxiliary operator pairs are a separate experiment.

PDE	Train pairs	Test pairs	Main draws	Adaptation draws
Darcy	50,000	10,000	25,000,000	5,000,000
Poisson	50,000	1,024	25,000,000	5,000,000
Helmholtz	50,000	10,000	25,000,000	5,000,000
NS	50,000	1,000	25,000,000	5,000,000
NS (BCs)	14,000	1,000	7,000,000	1,400,000

Appendix

Table 3: Data and training exposure. An epoch is one training-set-equivalent number of draws. Main training and decoder adaptation use 500 and 100 epochs, respectively. Full and FJV additionally use complete-view pretraining.

PDE	$\mu_{a}$	$\mu_{u}$	$\sigma_{a}$	$\sigma_{u}$
Darcy	7.5	0.00569201936	4.5	0.00379030361
Poisson	0	$9.67226952\!\times\!10^{-6}$	0.2919494	0.00417478
Helmholtz	0	$1.05050595\!\times\!10^{-5}$	0.2844538	0.00428004
NS	0	0	0.26211293	0.2561266
NS (BCs)	1.80715032	2.87310427	1.00997056	1.72194188

Appendix

Table 4: Normalization constants. Physical values use the factor of two in Eq. ( 14 ).

Family	Construction
Uniform	Uniform subset without replacement and with exact cardinality.
Regular grid	Cartesian original pixels with random phase and axis swap; no interpolation. Shapes are $20\!\times\!25$ , $32\!\times\!32$ , $32\!\times\!64$ , $64\!\times\!64$ , $64\!\times\!128$ , $96\!\times\!128$ , and $128\!\times\!128$ .
Cluster	2–4 equally weighted Gaussian components, center separation at least 0.30 domain units, and widths 0.12–0.20; weighted sampling without replacement.
Random lines	Whole rows or columns in random order, with at most one partial final line to meet the exact count.
Block	Rectangle-ordered pixels about a center in $[0.2,0.8]^{2}$ , aspect ratio in $[2/3,3/2]$ , and random tie breaking.

Appendix

Table 5: Backbone observation families. Budgets count unique original-grid pixels before the cylinder union.

Component	Parameters
Online encoder	38,157,952
Predictor	1,711,488
Mask conditioner	384
Native decoder	171,010
Native online model	40,040,834
EMA teacher, training-only	38,157,952

Appendix

Table 6: Parameter accounting. Deployment excludes the EMA teacher. Decoder adaptation updates only the replacement head.

Method	Darcy		Poisson		Helmholtz		NS		NS (BCs) †
	Fwd.	Inv. (BER)	Fwd.	Inv.	Fwd.	Inv.	Fwd.	Inv.	Fwd.	Inv.
FO	2.1625	1.5832	1.8322	11.2462	1.8306	10.7070	2.6269	7.3287	1.9829	0.7076
FJV	2.1788	1.6502	2.0647	11.2334	2.1258	11.6476	2.7015	7.5940	2.3897	0.8209
Full	2.1737	1.5564	1.6804	9.5024	1.6361	8.7023	2.5813	6.9732	2.1148	0.6491

Appendix

Table 7: Backbone training recipes before decoder adaptation. All rows use their own 171,010-parameter decoder at main epoch 500, the same online architecture, training data, mask schedule, and test conditions. Full is compared with field-only training (FO) and full-target prediction without main-stage grounding (FJV).

Method	Darcy		Poisson		Helmholtz		NS		NS (BCs) †
	Fwd.	Inv. (BER)	Fwd.	Inv.	Fwd.	Inv.	Fwd.	Inv.	Fwd.	Inv.
Full / original head	2.1737	1.5564	1.6804	9.5024	1.6361	8.7023	2.5813	6.9732	2.1148	0.6491
Full / DP-5M	2.2066	1.6425	1.6760	9.5222	1.6320	8.7066	2.5749	6.9797	2.0027	0.4789
Full / SP-5M	2.1687	1.5509	1.6678	9.4277	1.6285	8.6436	2.5733	6.9402	1.9958	0.4773
Full / SP-10M	2.1684	1.5503	1.6674	9.4260	1.6279	8.6425	2.5729	6.9384	1.9728	0.4524

Appendix

Table 8: Readout adaptation at a fixed Full backbone. DP fits a new head on complete-view encoder latents; SP fits it on sparse predictor latents. Both use 100 additional epochs. Every row is tested through the sparse predictor path. Four decimal places expose small changes hidden by main-table rounding.

PDE	Model / head	Forward		Inverse		Darcy aux.
		$a$ ( $L_{2}$ )	$u$ ( $L_{2}$ )	$a^{\star}$	$u$ ( $L_{2}$ )	Inv. $a$ ( $L_{2}$ )
Darcy	FO	15.6697	2.1625	1.5832	0.4097	10.7508
Darcy	FJV	15.7062	2.1788	1.6502	0.4502	10.9896
Darcy	Full	15.6970	2.1737	1.5564	0.4213	10.6789
Darcy	Full / SP-5M	15.6806	2.1687	1.5509	0.4008	10.6478
Poisson	FO	8.5021	1.8322	11.2462	0.7304	—

Appendix

Table 9: Both reconstructed channels at uniform/500. All entries are held-out percentages, not training losses. Inverse

a^{\star}

uses BER for Darcy and relative

L_{2}

elsewhere; all other channel errors use relative

L_{2}

. The final column retains Darcy inverse relative

L_{2}

only as an auxiliary diagnostic. Cylinder

u

means use 997 defined targets.

PDE	Model / head	Forward		Inverse		Darcy aux.
		$a$ ( $L_{2}$ )	$u$ ( $L_{2}$ )	$a^{\star}$	$u$ ( $L_{2}$ )	Inv. $a$ ( $L_{2}$ )
Darcy	DP-5M	18.3474	2.2066	1.6425	0.4874	12.4596
Darcy	DP-10M	18.4497	2.2021	1.6468	0.4857	12.4984
Darcy	DP-15M	18.1494	2.2016	1.6316	0.4837	12.3332
Darcy	SP-5M	15.6806	2.1687	1.5509	0.4008	10.6478
Darcy	SP-10M	15.6781	2.1684	1.5503	0.3998	10.6458

Appendix

Table 10: Decoder size and training-view sensitivity on the Full backbone. All heads are newly initialized and trained for 100 epochs. Inverse

a^{\star}

uses BER for Darcy and relative

L_{2}

elsewhere; the final column retains Darcy inverse relative

L_{2}

as auxiliary only. All values are percentages. A dash denotes an unreported channel-specific measurement.

Head	Forward $a$	Forward $u$	Inverse $a$	Inverse $u$
SP-5M	1.0278	2.5322	0.7664	2.0122
SP-10M	1.0115	2.5174	0.7494	1.9952
SP-15M	1.0044	2.5137	0.7402	1.9895

Appendix

Table 11: A separate cylinder observation condition. SP is trained only on uniform/500. These held-out rows use known-disk conditioning plus 1% Bernoulli observations of the fluid region; “100” in the saved condition name denotes basis points, not 100 fixed sensors. Errors are relative

L_{2}

(%).

Operator references: 100 epochs; 3% Bernoulli masks
Method	Poisson	Helmholtz	Darcy	NS (periodic)	NS (cylinder) ‡
DeepONet + G	$3.4783\,\pm\,1.0679$	$3.0379\,\pm\,1.1545$	$2.5830\,\pm\,0.8954$	$4.3536\,\pm\,0.8093$	$13.9307\,\pm\,11.0177$
DeepONet	$3.4458\,\pm\,1.2496$	$2.3259\,\pm\,1.0340$	$2.6641\,\pm\,0.8977$	$4.4076\,\pm\,0.8204$	$13.0824\,\pm\,9.6002$
FNO + G	$3.1172\,\pm\,1.4950$	$3.1115\,\pm\,1.5640$	$2.8842\,\pm\,1.1010$	$3.4010\,\pm\,0.8183$	$15.1163\,\pm\,19.4803$
FNO	$3.6179\,\pm\,1.6686$	$4.1611\,\pm\,1.6845$	$2.7522\,\pm\,1.0165$	$3.0042\,\pm\,0.7315$	$14.3843\,\pm\,17.5732$
SCOPE backbone recipes: 500 main epochs; uniform/500

Appendix

Table 12: Forward recovery of the hidden response

u

. Mean

\pm

standard deviation of per-record relative

L_{2}

errors (%). SCOPE uses uniform/500 and physical units. Auxiliary operators use their own Bernoulli-

3\%

protocol and the metric units specified below. These are not fully matched cross-family comparisons.

Operator references: 100 epochs; 3% Bernoulli masks
Method	Poisson	Helmholtz	Darcy (BER)	NS (periodic)	NS (cylinder) ‡
DeepONet + G	$12.7519\,\pm\,3.1407$	$12.0210\,\pm\,2.7818$	$5.8548\,\pm\,2.1429$	$10.3008\,\pm\,1.7708$	$13.4073\,\pm\,7.0904$
DeepONet	$12.0413\,\pm\,3.0069$	$11.6675\,\pm\,2.7691$	$5.6359\,\pm\,2.0694$	$10.3639\,\pm\,1.8348$	$13.3365\,\pm\,7.0118$
FNO + G	$15.0845\,\pm\,3.8692$	$14.8467\,\pm\,3.7329$	$3.4798\,\pm\,1.6474$	$9.3371\,\pm\,1.6888$	$11.6073\,\pm\,6.9658$
FNO	$15.0947\,\pm\,3.9014$	$14.4706\,\pm\,3.5408$	$3.6251\,\pm\,1.7206$	$7.5056\,\pm\,1.4283$	$10.9079\,\pm\,6.6674$
SCOPE backbone recipes: 500 main epochs; uniform/500

Appendix

Table 13: Inverse recovery of the hidden input

a

. Mean

\pm

standard deviation of per-record target errors (%): BER for Darcy, relative

L_{2}

for the other PDEs. Operators use their own Bernoulli 3% protocol; SCOPE uses exactly 500 uniform points. These are not fully matched cross-family comparisons.

Operator references: own observation protocol
Method	BER (%) $\downarrow$	Pixel accuracy (%) $\uparrow$
DeepONet + G	$5.8548\,\pm\,2.1429$	94.1452
DeepONet	$5.6359\,\pm\,2.0694$	94.3641
FNO + G	$3.4798\,\pm\,1.6474$	96.5202
FNO	$3.6251\,\pm\,1.7206$	96.3749
SCOPE backbone recipes

Appendix

Table 14: Darcy inverse classification. Mean

\pm

per-record SD of binary error rate and mean pixel accuracy. SCOPE uses uniform/500 on 10,000 fields; operators use their separate 3% protocol on 1,000 fields with five mask draws.

PDE	$n$	$\Delta$ (pp)	Paired SD	$t$	First better (%)
SP-5M vs. Full
Poisson	1,024	$-0.0126$	0.0473	$-8.5$	59.9
Helmholtz	10,000	$-0.0077$	0.0278	$-27.6$	61.9
Darcy	10,000	$-0.0051$	0.0212	$-23.9$	60.1
NS (periodic)	1,000	$-0.0080$	0.0143	$-17.6$	71.8
SP-10M vs. Full

Appendix

Table 15: Forward

u

: paired differences at uniform/500.

\Delta

and paired SD are relative-

L_{2}

percentage points;

\Delta<0

favors the first method named in each group. Statistics are computed on paired records for the fitted models.

PDE	$n$	$\Delta$ (pp)	Paired SD	$t$	First better (%)
SP-5M vs. Full
Poisson	1,024	$-0.0747$	0.0306	$-78.0$	99.6
Helmholtz	10,000	$-0.0587$	0.0331	$-177.5$	97.3
NS (periodic)	1,000	$-0.0330$	0.0143	$-73.0$	98.8
SP-10M vs. Full
Poisson	1,024	$-0.0764$	0.0326	$-75.1$	99.3

Appendix

Table 16: Inverse

a

: paired differences at uniform/500.

\Delta

and paired SD are relative-

L_{2}

percentage points;

\Delta<0

favors the first method named in each group. Statistics are computed on paired records for the fitted models. Darcy is omitted here because its primary inverse metric is BER, for which paired summaries are unavailable.

Comparison	$n$	$\Delta$ (pp)	Paired SD	$t$	First better (%)
SP-5M vs. Full	10,000	$-0.0311$	0.0417	$-74.6$	77.8
SP-10M vs. Full	10,000	$-0.0331$	0.0443	$-74.7$	77.8
SP-15M vs. Full	10,000	$-0.0346$	0.0461	$-75.0$	78.2
Full vs. FO	10,000	$-0.0719$	0.3327	$-21.6$	58.6
Full vs. FJV	10,000	$-0.3106$	0.3270	$-95.0$	83.7

Appendix

Table 17: Auxiliary Darcy inverse continuous-error diagnostic (not BER). These retained paired relative-

L_{2}

statistics are not the classification endpoint used in the main comparisons. Both

\Delta

and paired SD are relative-

L_{2}

percentage points. No BER significance or improved-record rate is inferred from them.

Model	Parameters	Main configuration
FNO	5,029,946	Width 40; 14 Fourier modes; 4 layers; projection width 128; padding 9.
DeepONet	4,977,354	CNN branch channels 32/64/128/192; branch hidden width 360; 256 basis functions; trunk width 128, depth 2.
CNO	4,774,776	3 levels; 2 residual blocks; 4 bottleneck blocks; channel multiplier 52.
Transolver	4,792,482	8 layers; width 160; 8 heads; 32 slices; MLP ratio 2.

Appendix

Table 18: Main-comparison operator configurations. These are the SCOPE-protocol models in Table 1 , not the auxiliary models in Figure 3 . Counts are real trainable degrees of freedom, including both parts of Fourier weights.

Architecture	Parameter elements	Real degrees of freedom
FNO / PINO	4,803,521	9,522,113
DeepONet	13,239,987	13,239,987
U-Net	4,740,101	4,740,101
CNO	5,281,721	5,281,721

Appendix

Table 19: Auxiliary operator parameter accounting. These architectures are not parameter-matched and differ from Table 18 . Each complex Fourier parameter contributes two real degrees of freedom.

Explore similar work

Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements
Jan 28, 2026Ofek Aloni, Barak FishbainNeural Partial Differential Equation Solvers Partial Differential Equations
Department of Mathematics, Technion, Israel · Department of Civil and Environmental Engineering, Technion, Israel
Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations
May 13, 2026Zhonghao Li, Chaoyu Liu, Qian ZhangNeural Partial Differential Equation Solvers Bayesian Inverse Problems
School of Science, Harbin Institute of Technology, Shenzhen, China · Department of Theoretical Physics and Applied Mathematics, University of Cambridge, Cambridge, United Kingdom
Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity
Apr 26, 2026Valerie Tsao, Nathaniel Chaney, Manolis VeveakisLatent Diffusion Model Spatiotemporal Fields
Department of Civil & Environmental Engineering, Duke University, NC, USA.

Jan 28, 2026, stat.MLJ/K move · Enter open · S save

Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements

Ofek Aloni, Barak Fishbain

Department of Mathematics, Technion, Israel · Department of Civil and Environmental Engineering, Technion, Israel

Neural Partial Differential Equation Solvers Partial Differential Equations Neural Reconstruction Spatiotemporal Fields Scalar Field Radial Basis Function

Open paper AskPDF arXiv

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

Abstract

Explore similar work

Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements

Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations

Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity