PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements

Zhenyu Liang; Yining Huang; Yubo Zhao; Jack C. P. Cheng

cs.CVOct 1, 2026

PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements

Authors: Zhenyu Liang, Yining Huang, Yubo Zhao, Jack C. P. Cheng

Organizations: HKUST · UT Austin

Abstract

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.

Figures & tables

Method family and representative	P	D	R	S
Data-trained GANs (e.g., DGMR ( Ravuri et al., 2021 ) )	$\times$	$\times$	$\times$	$\times$
Physics-informed GANs (e.g., PI-GAN ( Yang et al., 2020 ) )	$\checkmark$	$\checkmark$	$\times$	$\checkmark$
Simulation-based flow matching (e.g., FNOPE ( Moss et al., 2025 ) )	$\times$	$\times$	$\checkmark$	$\checkmark$
Data-trained diffusion (e.g., DPS ( Chung et al., 2023 ) )	$\times$	$\times$	$\checkmark$	$\checkmark$
Physics-loss diffusion (e.g., PIDM ( Bastek et al., 2025 ) )	$\checkmark$	$\times$	$\times$	$\times$
Physics-guided diffusion (e.g., DiffusionPDE ( Huang et al., 2024 ) )	$\checkmark$	$\times$	$\checkmark$	$\checkmark$

Table 1: Comparison of spatiotemporal physical generation methods. P: explicit use of prescribed PDE constraints in generative modeling; D: training without requiring a full-field dataset obtained from measurements or forward simulations; R: reuse across different numbers of measurements without retraining; S: full-field inference from spatially sparse measurements.

Figure 1: Overview of our proposed PhysDEM, a physics-defined diffusion framework for spatiotemporal field generation under scarce measurements.

	Darcy			Adv.–diff.			Fisher–KPP			SPE10			Bemidji
Method	SBC	$z_{\mathrm{obs}}$	$r_{\mathrm{PDE}}$	SBC	$z_{\mathrm{obs}}$	$r_{\mathrm{PDE}}$	SBC	$z_{\mathrm{obs}}$	$r_{\mathrm{PDE}}$	SBC	$z_{\mathrm{obs}}$	$r_{\mathrm{PDE}}$	SBC	$z_{\mathrm{obs}}$	$r_{\mathrm{PDE}}$
DiffusionPDE	3/5	3.99	1.11	5/7	8.11	1.02	4/5	4.40	1.02	1/5	13.47	0.89	4/7	9.51	1.39
DPS	3/5	14.79	2.53	1/7	30.03	1.07	3/5	16.56	1.49	3/5	42.85	0.77	3/7	38.19	1.95
DAPS	3/5	1.59	1.34	5/7	4.79	0.79	1/5	1.65	1.32	3/5	5.22	0.70	2/7	5.41	1.20
PnP-DM	2/5	3.25	3.28	4/7	3.81	1.49	1/5	3.82	1.95	1/5	5.37	0.88	5/7	6.04	1.68
FunDPS	4/5	0.21	1.39	3/7	0.24	3.31	5/5	0.54	4.92	3/5	1.87	0.79	3/7	0.01	1.64

Table 2: Baseline diagnostics at

K=4

with 64-member ensembles. Bold marks the best values by the stated criteria: largest SBC counts and values closest to one elsewhere.

Figure 2: Darcy baseline comparison of fields and observation consistency at $K=4$ .

Figure 3: Reuse across measurement counts without retraining.

Variant	SBC $\max\|z\|$	$z_{\mathrm{obs}}$	$r_{\mathrm{PDE}}$	$\mathrm{SSR}_{g}$
Without reference augmentation	$11.65\,\times$	1.05	0.62	1.53
$\tau$ -scaled labels	$11.84\,\times$	2.07	0.97	0.75
Augmented reference only	–	0.99	0.96	0.54
Tolerance $0.05$	$4.52\,\times$	0.98	0.35	0.88
Tolerance $0.15$	$5.20\,\times$	1.00	2.10	1.08
PhysDEM (full)	$\mathbf{2.18}\,\checkmark$	1.00	1.01	0.89

Table 3: Component ablations and tolerance sensitivity on Darcy (

K=4

, 64-member ensembles).

Figure 4: Site-informed Bemidji fields and descriptive comparison with 147 USGS records.

Figure 5: Amortized sampling cost and observation misfit on Darcy at

K=4

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Adaptation used in the comparison
DiffusionPDE	Joint diffusion with observation and PDE guidance ( Huang et al., 2024 ) .
DPS	A learned joint diffusion prior with likelihood-gradient guidance ( Chung et al., 2023 ) .
DAPS	Decoupled annealing with diffusion denoising and Langevin likelihood updates ( Zhang et al., 2025 ) .
PnP-DM	Alternating diffusion-prior and Gaussian-likelihood sampling ( Wu et al., 2024 ) .
FunDPS	Correlated-noise diffusion with observation guidance; PDE-guidance strength is validation-selected ( Yao et al., 2025 ) .
PIDM	Physics-informed diffusion training with DPS observation guidance at inference ( Bastek et al., 2025 ) .

Appendix

Table 4: Implemented baseline variants.

Figure 7: Additional Darcy baseline comparisons at $K=4$ with 64-member ensembles. Reference fields, observations, sample selection, and plotting conventions match Figure 2 ; rows contain the five remaining methods. FNOPE pressure fields are completed by its deterministic forward solver.

Figure 8: Measurement reuse with SPE10 static-field uncertainty. Panels (a–d) repeat Figure 3 ; panel (e) shows the ensemble standard deviation of log permeability with a common logarithmic color scale. Circles mark measured wells, and labels give spatial means.

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PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics

Hao Zhou, Rui Zhang, Han Wan, Hao Sun

Gaoling School of Artificial Intelligence, Renmin University of China

Rectified Flow Spatiotemporal Fields Partial Differential Equations Spatiotemporal Uncertainty Quantification

Open paper AskPDF arXiv

Reconstructing PDE-governed fields from sparse and irregular measurements is challenging due to their ill-posed nature. Deterministic surrogates are trained on dense fields that struggle with limited measurements and uncertainty quantification. Generative models, by learning distributions over spatiotemporal fields, can better handle sparsity and uncertainty. However, existing generative approaches enforce data consistency and PDE constraints simultaneously via sampling-time gradient guidance, resulting in slow and unstable inference. To this end, we propose PerFlow, a Physics-embedded rectified Flow for efficient sparse reconstruction and uncertainty quantification of spatiotemporal dynamics. PerFlow decouples observation conditioning from physics enforcement, performing guidance-free conditioning by feeding observations into rectified-flow dynamics while embedding hard physics via a constraint-preserving projection (e.g., incompressibility or conservation). Theoretically, we establish invariance guarantees to ensure that trajectories remain on the physics-consistent manifold throughout sampling. Experiments on various PDE systems demonstrate competitive reconstruction accuracy with sound physics consistency, while enabling efficient conditional sampling (e.g., 50 steps) and up to 320x faster inference than 2000-step guided diffusion baselines.

Abstract

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