Sparse Reconstruction by Direct Measurement-to-Field Generation in Function Space
Authors: Linzheng Wang, Jason Chen, Nicolas Tricard, Zituo Chen, Xingsen Guo, Sili Deng
Organizations: Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA · Department of Civil, Environmental and Geomatic Engineering, University College London, London WC1E 6BT, UK · Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK
Understanding physical systems often requires recovering their full spatial state from measurements available only at sparse locations. This reconstruction becomes challenging when physical boundaries are unknown, high-resolution training fields are unavailable, or important quantities cannot be measured directly. Sparse observations may also be consistent with multiple plausible states, leading to significant reconstruction uncertainty. Here we introduce Direct Measurement-to-Field Generation (\DMF{}), a functional rectified flow framework that learns how sparse measurements drive physical-field generation end to end. Instead of relying on grid-based rasterization or runtime guidance, \DMF{} jointly learns a sensor--global--sensor representation with the generative transport, dynamically fusing local grounding with global context to evaluate physical fields at arbitrary query coordinates. Across benchmarks involving unobserved boundaries, missing high-resolution training fields, and unmeasured physical quantities, \DMF{} outperforms alternative baselines by significantly reducing reconstruction errors and preserving the fidelity of physical properties. Furthermore, its stochastic ensemble generation reliably quantifies uncertainty while improving continuous ranked probability scores. These results extend sparse reconstruction from fixed-setting point estimation toward measurement-conditioned generation of physical fields under incomplete and changing information.
Figures & tables
Figure 1: Direct measurement-to-field generation. a , Each observation records a coordinate, physical channel, and value. Query coordinates are specified separately. b , Sensor tokens exchange information through global latents and return context to sensor locations. Local evidence and a global readout condition the velocity at each query coordinate. c , Rectified flow transports a smooth Gaussian reference function to a conditional field. Independent draws give multiple reconstructions. d , Evaluations vary the domain Ω , output coordinates X and measurements Y .
Figure 2: Field and hidden-boundary reconstruction on irregular domains. Results include 400 elasticity and 498 airfoil test cases. a,c , Von Mises stress and derived Mach number from 16 stress and 32 pressure measurements. b,d , Signed normalized errors in Kt and CD . Labels give mean absolute normalized errors. e , Mean masked relative- L2 error across sensor counts, with interquartile bands. f , Gradient-magnitude similarity deviation (GMSD) distributions and means. g , True and inferred boundaries for selected cases. h , Thresholded geometry relative- L2 error distributions and means.
Figure 3: Reconstruction across training and output resolutions. a , Low- (L, 32×32 ), medium- (M, 64×64 ) and high-resolution (H, 128×128 ) fields and five training recipes. Bar labels give spatial-degree-of-freedom exposure relative to H-only. b , Mean H-resolution physical relative- L2 error using 512 sparse sensors. Whiskers are 95% simple case-bootstrap intervals. c , Error across 64–512 nested H-grid sensors. d , Full field, matched enlargement and local absolute error for a held-out Zero-H-M-rich state at 512 sensors. e , Wavelet-scale truth and absolute residuals for DMF-Gen and Senseiver on the same state at 256 sensors. f , Median scale-wise pattern correlation and squared-energy allocation bias over 300 held-out case–time pairs at 256 sensors.
Figure 4: Reconstruction of missing channels in turbulent combustion. a , DMF-Gen reconstructions and absolute errors under three observed-channel sets with 256 sensors per channel. Selected temperature-only baselines appear on the right. b , Mean physical relative- L2 error by field over 1,000 evaluation states. Teal marks observed fields. c , Spectra and mean log-spectral distance for unobserved YCH4 , p and U1 under temperature-only conditioning. Error bars show 95% simple snapshot-bootstrap intervals. d , Ground-truth joint PDFs pooled over 25 states and per-state base-2 Jensen–Shannon divergence (JSD) distributions over 1,000 states for T – U1 , YCH4 – U1 and p – U1 .
Figure 5: Conditional generation, ablations, and cost. All analyses use 256 T measurements and target the four unobserved combustion fields. a , Normalized pointwise continuous ranked probability score (CRPS) from 64 draws for each of 200 paired states. b , Reconstruction error after retaining states with the lowest ensemble spread, normalized by each method’s full-cohort mean. Whiskers in a and bands in b are 95% moving-block-bootstrap intervals. c , Ensemble spread versus ensemble-mean relative- L2 error for the same states. Labels give Spearman ρ . Facet axes differ. d,e , Aggregate and high-band errors and median U1 spectra from 1,000-state ablation comparisons. f , Reconstruction error, time per training update, inference latency, and GPU memory use.
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
Reconstructing continuous physical fields from sparse measurements is central to scientific monitoring, inverse modeling, and digital-twin construction. Generative reconstruction has recently emerged as a promising paradigm for this task by learning data-driven physical priors that complete plausible full fields from limited observations. However, existing methods largely assume fixed, batch conditioning, whereas real sensing systems often produce structured streams: probes scan local regions, instruments observe moving fields of view, and communication constraints may leave entire frames missing. We propose TRACE, a retrospective streaming generative reconstruction framework for physical fields under structured sensing. TRACE performs approximate Bayesian inference in a learned continuous-coordinate latent space, converting sparse off-grid measurements into generative latent evidence, fusing it with a state-space temporal prior through Kalman-style filtering, and refining under-observed past frames via retrospective smoothing. Experiments on active matter, ocean sound-speed fields, and supernova simulations show that TRACE matches or surpasses frame-wise generative reconstructors, offline spatiotemporal methods, and streaming data-assimilation baselines in reconstruction quality under temporally sparse and spatially localized sensing protocols.
Xinyu Zhang, Lihao Chen, Panqi Chen +4
College of Information Science and Electronic Engineering, Zhejiang University
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.