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.
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
zobs
rPDE
SBC
zobs
rPDE
SBC
zobs
rPDE
SBC
zobs
rPDE
SBC
zobs
rPDE
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∣
zobs
rPDE
SSRg
Without reference augmentation
11.65×
1.05
0.62
1.53
τ -scaled labels
11.84×
2.07
0.97
0.75
Augmented reference only
–
0.99
0.96
0.54
Tolerance 0.05
4.52×
0.98
0.35
0.88
Tolerance 0.15
5.20×
1.00
2.10
1.08
PhysDEM (full)
2.18✓
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.
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.
Hao Zhou, Rui Zhang, Han Wan +1
Gaoling School of Artificial Intelligence, Renmin University of China
Inferring the evolution of high-dimensional and multi-modal (e.g., spatio-temporal) physical fields from irregular sparse measurements in real time is a fundamental challenge in science and engineering. Existing approaches, including diffusion-based generative models and functional tensor methods, typically operate in offline settings, depend on full temporal observations, or incur substantial inference cost. We propose StreamPhy, an end-to-end framework that enables efficient and accurate streaming inference of full-field physical dynamics from incoming irregular sparse measurements. The framework integrates a data-adaptive observation encoder that is robust to arbitrary observation patterns, a structured state-space model that supports memory-efficient online updates across irregular time intervals, and an expressive Functional Tensor Feature-wise Linear Modulation (FT-FiLM) decoder for continuous-field generation. We prove that FT-FiLM is more expressive than the functional Tucker model, admitting a richer function class for handling complex dynamics. Experiments on three representative physical systems under challenging sampling patterns show that StreamPhy consistently outperforms state-of-the-art baselines, with at least 48% improvement in accuracy and up to 20--100X faster inference than diffusion-based methods.
Panqi Chen, Yifan Sun, Shikai Fang +2
College of Information Science and Electronic Engineering, Zhejiang University · School of EECS,Oregon State University
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.
Ruichen Xu, Siyao Wang, Fang Wan +8
Stony Brook University · University of California, Davis · Independent Research +3