Learning Field Reconstruction from Incomplete Data by Globally Correcting Local Estimates

Renhao Zhong; Zihan Zhou; Chiyuan Ma; Tianshu Yu

cs.AIOct 4, 2026

Learning Field Reconstruction from Incomplete Data by Globally Correcting Local Estimates

Authors: Renhao Zhong, Zihan Zhou, Chiyuan Ma, Tianshu Yu

Organizations: The Chinese University of Hong Kong, Shenzhen · Shanghai Artificial Intelligence Laboratory

Abstract

Reconstructing physical fields from training samples that are always incomplete requires learning spatial structure from fragmented observations. Existing context--query work establishes how held-out observations provide valid training targets, but this does not make the complete-field distribution identifiable when every training field is incomplete. With finite data, weak evidence of sharp transitions and localized variations can further favor averaged predictions that attenuate local detail. A structural prior is therefore needed to favor plausible completions; local spatial relationships offer one grounded in the observations. We propose a locally constructed, globally revisable estimator that explicitly learns local field estimates and subsequently corrects them using full-domain observations. A shared coordinate-conditioned predictor learns from incomplete patches, allowing relatively well-observed neighborhoods to provide direct supervision of local structure. Its overlapping predictions are reconciled into an observation-conditioned consensus field. A full-domain estimator retains the original observations and learns a residual correction around this frozen field estimate, allowing locally constructed structure to be revised by broader evidence. The local estimate serves as both an explicit input, accompanied by its discrepancies with the observations, and a prediction starting point that the global model can revise. On three real-world ocean datasets with authentic observation gaps, our estimator achieves the lowest MSE and highest PSNR on withheld source-supported values, reducing MSE by 28.9%--34.5% against the strongest external baseline.

Figures & tables

Panel of Figure 1 — Figure 1: Fine-scale detail in full-mask Baltic Sea NANO reconstruction at $64\times 64$ . OAMP (PM) is the parameter-matched OAMP control, denoted Full Direct. The yellow box marks the held-out region enlarged in the side panels. Gray denotes outside-domain locations; white in Reference denotes source-missing values.

Figure 2: Local prior construction and full-domain correction for one context. Overlapping patch predictions form a consensus prior that conditions the corrector and is added to its learned residual. The patch predictor is frozen during global training; query values enter only the training losses.

Method	Black Sea CHL		Baltic Sea NANO		Global Ocean SSS
	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
DINEOF	$21.457$	$16.89$	$37.398$	$14.37$	$44.451$	$13.68$
AmbientDiff	$12.204\,{\scriptstyle\pm 0.156}$	$19.17\,{\scriptstyle\pm 0.03}$	$28.304\,{\scriptstyle\pm 0.206}$	$15.72\,{\scriptstyle\pm 0.03}$	$47.458\,{\scriptstyle\pm 1.718}$	$14.07\,{\scriptstyle\pm 0.12}$
DINDiff	$43.830\,{\scriptstyle\pm 1.515}$	$14.77\,{\scriptstyle\pm 0.09}$	$53.153\,{\scriptstyle\pm 1.775}$	$13.22\,{\scriptstyle\pm 0.04}$	$61.082\,{\scriptstyle\pm 0.960}$	$12.73\,{\scriptstyle\pm 0.07}$
MissDiff	$25.636\,{\scriptstyle\pm 0.300}$	$15.92\,{\scriptstyle\pm 0.08}$	$47.818\,{\scriptstyle\pm 1.704}$	$13.29\,{\scriptstyle\pm 0.16}$	$41.935\,{\scriptstyle\pm 0.173}$	$14.01\,{\scriptstyle\pm 0.03}$
MSM	$15.077\,{\scriptstyle\pm 0.184}$	$18.27\,{\scriptstyle\pm 0.04}$	$30.206\,{\scriptstyle\pm 1.170}$	$15.37\,{\scriptstyle\pm 0.19}$	$40.901\,{\scriptstyle\pm 0.277}$	$14.13\,{\scriptstyle\pm 0.03}$

Table 1: Ocean reconstruction at

64\times 64

on withheld source-supported values. Learned-model results are means

\pm

standard deviations across independent training seeds; DINEOF results are point estimates. Bold marks the best result in each column.

Estimator	Black Sea CHL		Baltic Sea NANO		Global Ocean SSS
	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
Local Prior	$15.227\,{\scriptstyle\pm 0.103}$	$18.15\,{\scriptstyle\pm 0.03}$	$30.576\,{\scriptstyle\pm 0.479}$	$15.17\,{\scriptstyle\pm 0.04}$	$32.053\,{\scriptstyle\pm 0.276}$	$15.13\,{\scriptstyle\pm 0.04}$
Full Direct	$9.350\,{\scriptstyle\pm 0.306}$	$20.32\,{\scriptstyle\pm 0.12}$	$23.559\,{\scriptstyle\pm 0.834}$	$16.41\,{\scriptstyle\pm 0.16}$	$32.895\,{\scriptstyle\pm 0.967}$	$14.99\,{\scriptstyle\pm 0.15}$
Low-rank2Global	$11.067\,{\scriptstyle\pm 0.243}$	$19.79\,{\scriptstyle\pm 0.11}$	$22.841\,{\scriptstyle\pm 0.171}$	$16.43\,{\scriptstyle\pm 0.04}$	$30.282\,{\scriptstyle\pm 0.649}$	$15.46\,{\scriptstyle\pm 0.08}$
Global2Global	$9.175\,{\scriptstyle\pm 0.499}$	$20.55\,{\scriptstyle\pm 0.16}$	$25.097\,{\scriptstyle\pm 1.442}$	$16.23\,{\scriptstyle\pm 0.21}$	$41.695\,{\scriptstyle\pm 0.670}$	$14.30\,{\scriptstyle\pm 0.04}$
Prior Direct	$8.546\,{\scriptstyle\pm 0.062}$	$20.82\,{\scriptstyle\pm 0.03}$	$21.445\,{\scriptstyle\pm 0.417}$	$16.89\,{\scriptstyle\pm 0.09}$	$30.151\,{\scriptstyle\pm 0.197}$	$15.40\,{\scriptstyle\pm 0.03}$

Table 2: Estimator organization and prior replacement for full-mask ocean reconstruction at

64\times 64

. Results are means

\pm

standard deviations across independent training seeds. Bold marks the best result in each column.

Panel of Figure 3 — Figure 3: Full-mask ocean reconstructions at $64\times 64$ , with supplied observations restored in Ours. Black and red outlines mark supplied observations and withheld source-supported scoring locations, respectively. Gray denotes outside-domain locations; white in Reference denotes source-missing values.

Panel of Figure 4 — Figure 4: Navier–Stokes Block 8/9 reconstruction at $64\times 64$ . IDCD (PM) is the parameter-matched IDCD control, denoted Full Direct. Red dashes mark the held-out block, black lines the acquisition partition, and orange boxes the zoomed region.

Appendix figures & tables29 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Original setting	Principal mechanism	Use in this paper
Evaluated baselines
DINEOF [ Beckers and Rixen, 2003 , Alvera-Azcárate et al., 2005 ]	Incomplete ocean fields	Low-rank EOF reconstruction	Ocean baseline; low-rank-prior control
AmbientDiff [ Daras et al., 2023 ]	Corrupted images	Further corruption; prediction supervised on available measurements	Ocean and PDE baseline
DINDiff [ Barth et al., 2024 ]	Satellite ocean chlorophyll	Noise prediction on additionally withheld observations	Ocean and PDE baseline
MissDiff [ Ouyang et al., 2023 ]	Incomplete tabular data	Masked denoising score matching	Ocean and PDE baseline
MSM [ Park et al., 2026 ]	Natural and medical imaging	Partial measurement scores; stochastic sampling	Ocean and PDE baseline

Appendix

Table 3: Original settings, reconstruction mechanisms, and experimental roles of the evaluated baselines.

Method	Original training setting	Construction and revision	Distinction from our estimator
GLCIC [ Iizuka et al., 2017 ]	Artificially masked complete images	A completion network trained with global and local discriminators	Local–global consistency is imposed through training losses, rather than a local-value consensus followed by a learned full-domain corrector.
Contextual Attention [ Yu et al., 2018 ]	Artificially masked complete images	Coarse completion followed by contextual-attention refinement	Uses a coarse-to-fine image-completion pipeline rather than a reference learned from incomplete spatial patches.
EdgeConnect [ Nazeri et al., 2019 ]	Images and derived edge targets	Predicted edges condition subsequent image completion	The intermediate representation is an edge map, not a field-valued estimate used as a residual anchor.
Patch Diffusion [ Wang et al., 2023 ]	Clean images with coordinate-conditioned, multiscale training	Learns scores from patch and full-image training views	Patch learning serves diffusion-model training rather than held-out-query prediction followed by a separate full-domain corrector.
PaDIS [ Hu et al., 2024 ]	Clean image patches for prior learning	Patch scores define a prior used by an inverse-problem solver	Global measurement constraints enter the solver; there is no separately trained full-domain residual corrector around an observed-query-trained local consensus.
MultiDiffusion [ Bar-Tal et al., 2023 ]	A pretrained text-to-image diffusion model	Reconciles regional diffusion updates during sampling	Fusion operates on sampling updates, rather than constructing a frozen clean-value reference for a learned corrector.

Appendix

Table 4: Comparison with related local–global and staged reconstruction methods in their original training settings.

Setting	Value
Anchor retention $\rho$	0.7
Guidance scale $w_{g}$	80 ; 32 for the first two updates
Sampling steps $S$	20
Reverse solver / schedule	Guided BFN ODE; uniform time grid $t:10^{-3}\to 1$
Sampling seed	42 ; run-level random-number state
Empty-support handling	Initial proposal plus at most 8 retries; post-loop minimum-size repair

Appendix

Table 5: Guided BFN sampling configuration for Black Sea CHL and Baltic Sea NANO.

	Black Sea CHL	Baltic Sea NANO	Global Ocean SSS
Product ID	OCEANCOLOUR_BLK_BGC_L3_MY_009_153	OCEANCOLOUR_BAL_BGC_L3_MY_009_133	MULTIOBS_GLO_PHY_SSS_L3_MYNRT_015_014
DOI	10.48670/moi-00303	10.48670/moi-00296	10.48670/mds-00368
Variable	CHL	NANO	Sea_Surface_Salinity
Spatial subset	Full Black Sea domain; approximately $26.5^{\circ}$ – $42^{\circ}$ E, $40^{\circ}$ – $48^{\circ}$ N	$15^{\circ}$ – $25^{\circ}$ E, $54^{\circ}$ – $59^{\circ}$ N	$180^{\circ}$ W– $179.8^{\circ}$ E, $83.6^{\circ}$ S– $83.6^{\circ}$ N
Native grid	Regional grid; approximately 1-km spacing	Regional grid; approximately 1-km spacing	Global cylindrical EASE-Grid 2.0

Appendix

Table 6: Ocean data products and geographic subsets.

Setting	Available training support	Training context rule	Training query	Validation/test input	Scoring support
Black Sea CHL	All source-supported values	Guided-mask intersection within the current support	Remaining available values	Cached support from a same-split authentic-mask overlay	Withheld source-supported values, $\bm{M}_{\mathrm{raw}}\odot(1-\bm{M}_{\mathrm{in}})$
Baltic Sea NANO	All source-supported values	Guided-mask intersection within the current support	Remaining available values	Cached support from a same-split authentic-mask overlay	Withheld source-supported values, $\bm{M}_{\mathrm{raw}}\odot(1-\bm{M}_{\mathrm{in}})$
Global Ocean SSS	All source-supported values	Empirical-track intersection within the current support	Remaining available values	Cached support from a same-split authentic-mask overlay	Withheld source-supported values, $\bm{M}_{\mathrm{raw}}\odot(1-\bm{M}_{\mathrm{in}})$
Pixel $30\%$	Fixed $q^{\mathrm{pix}}_{0.3}$ mask; approximately $30\%$	Retention probability $0.7$ ; approximately $21\%$ of the field	Approximately $9\%$ of the field	Fixed $q^{\mathrm{pix}}_{0.3}$ mask; approximately $30\%$	$V\setminus\bm{M}_{\mathrm{in}}$
Block $3/9$	Three fixed blocks	Two available blocks	One available block	Three blocks	Remaining six blocks
Block $8/9$	Eight fixed blocks	Seven available blocks	One available block	Eight blocks	Remaining one block

Appendix

Table 7: Available training observations, training context–query partitions, validation/test inputs, and scoring supports by dataset and protocol.

Method	Shallow Water Pixel $30\%$		Shallow Water Block $3/9$		Navier–Stokes Pixel $30\%$
	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
AmbientDiff	$0.903\,{\scriptstyle\pm 0.018}$	$37.168\,{\scriptstyle\pm 0.092}$	$13.777\,{\scriptstyle\pm 0.025}$	$26.516\,{\scriptstyle\pm 0.035}$	$6.338\,{\scriptstyle\pm 0.448}$	$28.146\,{\scriptstyle\pm 0.311}$
DINDiff	$0.866\,{\scriptstyle\pm 0.033}$	$37.233\,{\scriptstyle\pm 0.005}$	$17.656\,{\scriptstyle\pm 0.106}$	$26.170\,{\scriptstyle\pm 0.017}$	$7.995\,{\scriptstyle\pm 0.017}$	$27.074\,{\scriptstyle\pm 0.009}$
MissDiff	$1.177\,{\scriptstyle\pm 0.089}$	$35.779\,{\scriptstyle\pm 0.388}$	$16.839\,{\scriptstyle\pm 0.164}$	$25.798\,{\scriptstyle\pm 0.106}$	$8.740\,{\scriptstyle\pm 0.467}$	$26.728\,{\scriptstyle\pm 0.260}$
MSM	$4.025\,{\scriptstyle\pm 0.094}$	$30.352\,{\scriptstyle\pm 0.056}$	$18.660\,{\scriptstyle\pm 0.023}$	$25.017\,{\scriptstyle\pm 0.003}$	$17.756\,{\scriptstyle\pm 0.017}$	$23.613\,{\scriptstyle\pm 0.005}$
IDCD	$0.880\,{\scriptstyle\pm 0.111}$	$37.223\,{\scriptstyle\pm 0.275}$	$8.001\,{\scriptstyle\pm 0.055}$	$29.279\,{\scriptstyle\pm 0.241}$	$7.545\,{\scriptstyle\pm 0.085}$	$27.359\,{\scriptstyle\pm 0.056}$

Appendix

Table 8: PDE reconstruction at

64\times 64

, evaluated on valid unsupplied coordinates. Results are means

\pm

standard deviations across independent training seeds. Bold marks the best result in each column.

Method	Shallow Water Block $8/9$		Navier–Stokes Block $8/9$
	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
AmbientDiff	$8.608$	$38.859$	$33.725$	$21.260$
DINDiff	$5.530$	$42.014$	$34.068$	$21.360$
MissDiff	$16.510$	$35.989$	$53.201$	$19.207$
MSM	$16.099$	$31.509$	$45.947$	$20.013$
IDCD	$4.751$	$41.694$	$28.444$	$22.054$

Appendix

Table 9: PDE reconstruction under Block 8/9 observations at

64\times 64

. Entries are point estimates. Bold marks the best result in each column.

Aspect	MSM [ Park et al., 2026 ]	Ours
Training observations	Partial measurements, including noisy or subsampled measurement settings	Incomplete physical-field observations
Supervised coordinates	Denoising targets lie on the measurement support supplied in noisy form	Query values are withheld from the network input and supervise local and full-domain predictions
Learned object	Partial-measurement denoisers and associated scores	Local clean-value predictions and a full-domain residual corrector
Aggregation object	Lifted partial scores or corresponding denoising estimates	Overlapping spatial-patch predictions reconciled into a field-valued consensus
Role of full-domain information	Propagated through sampling and observation-consistency updates	Supplied to a separately trained corrector together with the local reference and discrepancy features
Reconstruction procedure	Iterative stochastic sampling for generation or conditional reconstruction	A local-to-global reconstruction path per selected context at near-zero diffusion time; final fields are averaged when context ensembling is used

Appendix

Table 10: Mechanistic comparison of MSM and our estimator.

Method	Black Sea CHL		Baltic Sea NANO		Global Ocean SSS
	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
DINEOF	$21.806$	$17.17$	$38.067$	$14.57$	$51.323$	$13.02$
AmbientDiff	$12.594$	$19.37$	$30.786$	$15.45$	$55.249$	$13.52$
DINDiff	$43.359$	$15.22$	$42.796$	$14.27$	$72.747$	$12.06$
MissDiff	$31.779$	$15.27$	$51.621$	$12.94$	$50.593$	$13.19$
MSM	$30.211$	$16.06$	$34.102$	$15.07$	$45.435$	$13.71$

Appendix

Table 11: Ocean reconstruction at

128\times 128

on withheld source-supported values. Entries are point estimates. Bold marks the best result in each column.

A: Local coordinate conditioning
Variant	Black Sea CHL		Baltic Sea NANO		Global Ocean SSS
	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
With local coordinates	$\mathbf{7.164}$	$\mathbf{21.79}$	$\mathbf{19.345}$	$\mathbf{17.37}$	$\mathbf{26.798}$	$\mathbf{15.93}$
No local coordinates	$7.286$	$21.66$	$19.910$	$17.17$	$30.614$	$15.33$
B: Explicit prior–evidence representation
Full state	$\mathbf{7.164}$	$\mathbf{21.79}$	$\mathbf{19.345}$	$\mathbf{17.37}$	$\mathbf{26.798}$	$\mathbf{15.93}$

Appendix

Table 12: Local-coordinate and corrector-input ablations for final full-mask reconstruction on

64\times 64

ocean fields. Bold marks the best result in each column within each panel.

A: identical overlap views; fusion weights vary
View family / fusion	Black Sea CHL	Baltic Sea NANO	Global Ocean SSS
	MSE $\downarrow$ ( $\times 10^{-3}$ )	MSE $\downarrow$ ( $\times 10^{-3}$ )	MSE $\downarrow$ ( $\times 10^{-3}$ )
Confidence-weighted mean	$15.525$	$31.106$	$32.185$
Gaussian-weighted mean	$15.569$	$31.177$	$32.203$
Uncertainty-weighted fusion	$15.481$	$30.922$	$32.102$
Uniform mean (ours)	$\mathbf{15.227}$	$\mathbf{30.576}$	$\mathbf{32.053}$

Appendix

Table 13: Local-prior reconstruction before global correction using full supplied masks on

64\times 64

ocean fields. Bold marks the lowest MSE within each dataset and panel.

Patch / stride	Views per field	Black Sea CHL		Baltic Sea NANO		Global Ocean SSS
		MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)
$32\times 32$ / 16	9	$\mathbf{7.164}$	$\mathbf{21.79}$	$\mathbf{19.345}$	$\mathbf{17.37}$	$28.297$	$15.76$
$16\times 16$ / 8	49	$7.542$	$21.45$	$21.405$	$16.84$	$\mathbf{26.798}$	$\mathbf{15.93}$

Appendix

Table 14: Patch-scale comparison for final full-mask residual reconstruction on

64\times 64

ocean fields. Bold marks the better result in each column.

Item	Setting
Resolution	$64\times 64$
Black Sea / Baltic bank	$W_{32}$ : $32\times 32$ , stride 32; four windows per field
Global SSS bank	$W_{16}$ : $16\times 16$ , stride 16; 16 windows per field
Eligibility	At least eight scored coordinates, $\|B(W)\|\geq 8$
Black Sea / Baltic local views	$32\times 32$ patches, stride 16; nine views
Global SSS local views	$16\times 16$ patches, stride 8; 49 views

Appendix

Table 15: Diagnostic settings for remote-evidence interventions on full-mask ocean reconstructions.

Dataset	$\max d_{\bm{\Pi}}^{\mathrm{far}}$	Mean $d_{\widehat{\bm{x}}}^{\mathrm{far}}$ [95% CI]	Median $d_{\widehat{\bm{x}}}^{\mathrm{far}}$	Max $d_{\widehat{\bm{x}}}^{\mathrm{far}}$
		$(\times 10^{-5})$	$(\times 10^{-6})$	$(\times 10^{-2})$
Baltic Sea NANO	$0$	$6.46$ [-1pt] $[4.71,\,8.61]$	$3.35$	$1.617$
Black Sea CHL	$0$	$5.80$ [-1pt] $[4.71,\,7.09]$	$2.67$	$1.156$
Global Ocean SSS	$0$	$62.30$ [-1pt] $[56.96,\,68.02]$	$28.00$	$6.348$

Appendix

Table 16: Local-prior and final-reconstruction displacement after deleting the complete Remote Far tercile from the full supplied mask.

	$\Delta$ block MSE $(\times 10^{-5})$
Dataset	Remote Near	Remote Mid	Remote Far	Local Near
Baltic Sea NANO	$0.95$ [-1pt] $[-3.32,\,5.23]$	$\mathbf{4.66}$ [-1pt] $[1.08,\,8.56]$	$\mathbf{5.25}$ [-1pt] $[1.14,\,9.56]$	$\mathbf{50.11}$ [-1pt] $[32.63,\,69.56]$
Black Sea CHL	$0.53$ [-1pt] $[-1.57,\,2.77]$	$\mathbf{2.53}$ [-1pt] $[0.43,\,4.84]$	$\mathbf{3.72}$ [-1pt] $[0.65,\,6.75]$	$5.87$ [-1pt] $[-1.73,\,13.10]$
Global Ocean SSS	$\mathbf{6.76}$ [-1pt] $[2.99,\,10.54]$	$0.98$ [-1pt] $[-1.53,\,3.49]$	$\mathbf{-12.85}$ [-1pt] $[-18.64,\,-7.09]$	$\mathbf{64.12}$ [-1pt] $[55.49,\,73.16]$

Appendix

Table 17: Final-reconstruction block-MSE changes under matched-dose deletion from the full supplied mask. Entries are means with 95% confidence intervals; positive values indicate increased error after deletion. Bold marks means whose intervals exclude zero.

A: Black Sea CHL
Model	Inference	MSE $\downarrow$ ( $\times 10^{-3}$ )	PSNR $\uparrow$ (dB)	Median latency (ms/field)	P95 latency (ms/field)
Full Direct	Full-mask	$9.350\,{\scriptstyle\pm 0.306}$	$20.32\,{\scriptstyle\pm 0.12}$	$69.46\,{\scriptstyle\pm 0.20}$	$69.78\,{\scriptstyle\pm 0.31}$
Full Direct	Context $K=1$	$13.922\,{\scriptstyle\pm 0.055}$	$18.61\,{\scriptstyle\pm 0.07}$	$69.41\,{\scriptstyle\pm 0.10}$	$69.76\,{\scriptstyle\pm 0.12}$
Full Direct	Context $K=2$	$13.823\,{\scriptstyle\pm 0.065}$	$18.64\,{\scriptstyle\pm 0.07}$	$80.05\,{\scriptstyle\pm 0.19}$	$80.29\,{\scriptstyle\pm 0.26}$
Full Direct	Context $K=4$	$13.762\,{\scriptstyle\pm 0.072}$	$18.66\,{\scriptstyle\pm 0.08}$	$105.30\,{\scriptstyle\pm 0.20}$	$105.51\,{\scriptstyle\pm 0.15}$
Full Direct	Context $K=8$	$13.721\,{\scriptstyle\pm 0.083}$	$18.66\,{\scriptstyle\pm 0.08}$	$210.30\,{\scriptstyle\pm 0.41}$	$210.60\,{\scriptstyle\pm 0.50}$

Appendix

Table 18: Accuracy and request latency under full-mask and context-ensemble inference at

64\times 64

. Entries are means

\pm

standard deviations across independent training seeds. For latency, each seed contributes its median or 95th-percentile request time. Bold marks the best mean accuracy within each dataset.

Mask statistic	Black Sea CHL	SW Pixel 30%
Full-mask observations / valid domain	50.72%	29.96%
Subcontext observations / valid domain	27.70%	20.97%
Supplied observations retained per subcontext	51.23%	70.00%
Supplied observations covered by the first 4 contexts	72.93%	99.19%
Supplied observations covered by the first 8 contexts	78.05%	99.994%
Mean nearest-observation distance: full-mask	3.09 pixels	1.15 pixels

Appendix

Table 19: Test-time observation retention and spatial coverage on Black Sea CHL and Shallow Water Pixel 30%. Entries are equally weighted means over test samples.

Panel of Figure 5 — Figure 5: Local-to-global decomposition for full-mask ocean reconstruction at $64\times 64$ . Model predictions are shown before restoring supplied observations, preserving the pointwise relation $\text{Ours}=\text{Local Prior}+\text{Learned Residual}$ .

Panel of Figure 6 — Figure 6: Full-mask reconstructions on Black Sea CHL at $64\times 64$ .

Panel of Figure 7 — Figure 7: Full-mask reconstructions on Baltic Sea NANO at $64\times 64$ .

Panel of Figure 8 — Figure 8: Full-mask reconstructions on Global Ocean SSS at $64\times 64$ .

Panel of Figure 9 — Figure 9: Full-mask reconstructions on Black Sea CHL at $128\times 128$ .

Panel of Figure 10 — Figure 10: Full-mask reconstructions on Baltic Sea NANO at $128\times 128$ .

Panel of Figure 11 — Figure 11: Full-mask reconstructions on Global Ocean SSS at $128\times 128$ .

Panel of Figure 12 — Figure 12: Navier–Stokes Pixel 30% reconstructions at $64\times 64$ , comparing Ours with AmbientDiff.

Panel of Figure 13 — Figure 13: Shallow Water Pixel 30% reconstructions at $64\times 64$ , comparing Ours with DINDiff.

Panel of Figure 14 — Figure 14: Shallow Water Block 3/9 reconstructions at $64\times 64$ , comparing Ours with AmbientDiff.

Panel of Figure 15 — Figure 15: Shallow Water Block 8/9 reconstructions at $64\times 64$ , comparing Ours with DINDiff.

Panel of Figure 16 — Figure 16: Navier–Stokes Block 8/9 reconstructions at $64\times 64$ , comparing Ours with AmbientDiff.

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Sep 15, 2026cs.LG

Recovering Governing Dynamics from Distributed Observations via Exact Spline Merging

Scientific observations are frequently distributed across locations, time periods, and institutions. Combining such observations into a continuous, differentiable field enables recovering governing physical parameters from its derivatives. This paper makes two contributions in this setting. First, the established additive structure of fixed-basis ridge-regression statistics is applied to tensor-product spline fields: each data holder computes a local Gram matrix and moment vector, and the merged solution is mathematically identical to centralized fitting, with no raw data shared and no iterative synchronization. This property is specific to the fixed-feature squared-error setting; the present derivation does not establish an analogous guarantee for general jointly trained multilayer networks. Second, a complete pipeline connects distributed observations to physical parameter inference through field reconstruction, derivative extraction, and linear regression. The pipeline is validated on four PDEs: diffusion, wave, heat-with-source, and the nonlinear viscous Burgers equation, recovering governing parameters to sub-percent accuracy in the linear cases and 5% for Burgers. In all cases, distributed merging introduces zero degradation relative to centralized fitting. Application to 41 years of NOAA sea-surface temperature data confirms the result on real spatiotemporal observations. Source code to reproduce all experiments is available at https://github.com/NAVEENMN/gramfield.

Naveen Mysore

University of California, Santa Barbara · Dyssonance AI

Spatiotemporal Fields Splines Sea-Surface Temperature Data Implicit Differentiation Kernel Ridge Regression

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