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.