Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervision

Kevin Valencia; Xihaier Luo; Shinjae Yoo; David Keetae Park

cs.LGFeb 17, 2026

Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervision

Authors: Kevin Valencia, Xihaier Luo, Shinjae Yoo, David Keetae Park

Organizations: University of California, Los Angeles · Brookhaven National Laboratory

Abstract

Physical fields are typically observed only at sparse, time-varying sensor locations, making forecasting and reconstruction ill-posed and uncertainty-critical. We present SOLID, a mask-conditioned diffusion framework that learns spatiotemporal dynamics from sparse observations alone: training and evaluation use only observed target locations, requiring no dense fields and no pre-imputation. Unlike prior work that trains on dense reanalysis or simulations and only tests under sparsity, SOLID is trained end-to-end with sparse supervision only. SOLID conditions each denoising step on the measured values and their locations, and introduces a dual-masking objective that (i) emphasizes learning in unobserved void regions while (ii) upweights overlap pixels where inputs and targets provide the most reliable anchors. This strict sparse-conditioning pathway enables posterior sampling of full fields consistent with the measurements, achieving up to an order-of-magnitude improvement in probabilistic error and yielding calibrated uncertainty maps (\r{ho} > 0.7) under severe sparsity.

Figures & tables

Figure 1: Conditional Field Forecasting. This paper develops realistic training scheme with sparse data only which contrasts with prior works assuming fully observed spatiotemporal fields. We evaluate capacities on spatiotemporal field forecasting during the inference learned purely from sparse data.

Figure 2: Training the Proposed SOLID. A dual-masking strategy is introduced. Coupled with sparse conditioning, it realizes a simple yet effective spatiotemporal learning capable of field reconstruction robust with sparse-only training data.

Figure 3: Qualitative Comparisons on Forecasting. White boxes denote MSE between prediction and target GT, and gray-scale images show corresponding error.

Figure 4: Performance and Parameter Efficiency. Navier–Stokes baseline comparisons summarizing (a) Per parameter efficiency and (b) data efficiency (full data vs.10%).

Figure 5: AirDelhi: Performance and Uncertainty. (a) Each model is run for ten different seeds, shown as standard deviation. (b) 30-minute air pollution prediction and its uncertainty. Black contours denote available sample positions.

Figure 6: Qualitative Analyses of Uncertainty. Regions of high uncertainty co-locate with (a) large error (Navier–Stokes). Orange-colored arrows point to representative areas where error and uncertainty highly match. (c) Rollouts on a single sparse input show that uncertainty propagates over time, following closely with the elevating error.

Figure 7: Ablations on $\lambda$ and Sparsity Rate.

Figure 8: Uncertainty Calibration and Sample Coverage. (a) Per-sample uncertainty correlates with error in CRPS. (b) Uncertainty heightens with distance to the nearest observed location.

Figure 9: Sparsity Patterns on Performance. (a) Increasing sensor density (2% $\rightarrow$ 20%) sharpens SOLID’s forecasting performance while error maps fade. (b) For a similar overall coverage, having sensors aggregated into condensed regions increases error.

Strategy	CRPS
Strategy	$M_{i}$	$M_{o}\setminus M_{i}$	$U^{t_{o}}$
Pre-Interp (NN)	0.169	0.229	0.222
Pre-Interp (RBF)	0.081	0.139	0.133
SOLID	0.025	0.066	0.069

Table 1: Comparison of strategies under sparse conditioning on Navier–Stokes. We report the performance at input (conditioned) locations (

M_{i}

), target-only locations (

M_{o}\setminus M_{i}

), and on the full grid (

U^{t_{o}}

). SOLID (no pre-interpolation) attains the lowest error across all three views. ( Interp : Pre-Interpolation; NN : Nearest Neighbor; RBF : Radial Basis Function)

Pattern	Sample Density	Sensor Scenarios
Pattern	Sample Density	$S_{\text{Global}}$	$S_{\text{Instance}}$
Random	4%	1.37	1.23
	10%	0.67	0.48
	40%	0.31	0.10
Block (8 $\times$ 8)	2 blocks	1.37	2.06
	6 blocks	2.04	1.84

Table 2: CRPS for Navier–Stokes experiments under different sparsity patterns and fixing regimes.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Basis	FLOPs (GFLOPs)	# Params
SCENT	per forward pass	$20.351$	$28\,988\,739$
OmniField	per forward pass	$9.966$	$10\,272\,327$
FFNO	per forward pass	$0.771$	$535\,746$
OFormer	per forward pass	$6.682$	$2\,662\,849$
UNet	per forward pass	$0.397$	$466\,593$
StyleGAN3	per forward pass	$10\,473.000$	$76\,320\,261$

Appendix

Table 3: Compute and model size. FLOPs are reported in GFLOPs.

Model
Hyperparameter	Navier–Stokes	AirDelhi
Backbone	UNet	UNet
UNet base dim	64	64
Image size	64	$21\times 20$
Dim multipliers	(1, 2, 2, 2)	(1, 2, 2,2)
Attention resolutions	(16)	(16)

Appendix

Table 4: SOLID hyperparameters. Navier–Stokes values are from the provided config. AirDelhi values are from the provided training cell.

	Full			10% Subset
Baseline	4%	10%	40%	4%	10%	40%
SCENT	1.6638	0.8490	0.5568	1.9158	1.3971	1.2591
OmniField	1.6982	1.1019	0.4765	2.1117	1.6541	1.0651
FFNO	1.6863	1.0993	0.4545	1.8689	1.3265	0.7042
OFormer	1.6873	1.0358	0.5645	1.9677	1.2387	0.9052
UNet	1.6256	0.9981	0.4673	1.8204	1.2387	0.6722

Appendix

Table 5: Full Results on Navier–Stokes across sparsity (CRPS, corresponds to Fig. 4 ). Best per column in bold.

Baseline	AD-B
SOLID	25.713
SCENT	$28.467$
OmniField	$29.244$
FFNO	$30.156$
UNet	$32.897$
OFormer	$57.837$

Appendix

Table 6: Full Results on AirDelhi (CRPS, corresponds to Fig. 5 ).

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Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.

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