Reconstructing oscillatory wave fields from scattered sensors is a severely underdetermined inverse problem. Beyond the challenges of general physical-field reconstruction, wave responses are complex-valued, frequency-sensitive, and highly oscillatory, while costly simulation and sensing often leave only extreme-sparse observations. Existing low-rank, operator, and diffusion approaches are largely designed for real-valued, smoother fields; dense pixel-space diffusion is particularly inefficient for oscillatory complex fields and difficult to scale to 3D. We propose HarmoCore, which places a generative prior in a compact, continuous, and structured wave-field latent. HarmoCore represents joint real--imaginary channels with Functional Tucker cores over shared continuous spatial bases, learns a frequency-conditioned core diffusion prior, and performs Diffusion Posterior Sampling directly in core space. At fixed sensor coordinates, the multilinear decoder induces an explicit likelihood guidance operator, avoiding dense pixel-space correction. Optional target-equation residual guidance further promotes physical consistency. Experiments on 2D Helmholtz, 2D synthetic wave fields, and 3D Helmholtz show substantial gains under 1%--2% sensing while remaining practical in three dimensions.
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.
Arindam Sengupta, Paul Jeanney, Ricardo Vinuesa +2cs.LG
Urban flow and air-quality simulations generate high-dimensional datasets describing velocity and pollutant transport across multiple spatial, temporal, and physical-variable dimensions. Reconstructing these fields from sparse sensor measurements is a fundamental challenge in environmental monitoring, digital twins, forecasting, and data assimilation. Existing low-cost reconstruction approaches are commonly based on matrix decompositions, which require multidimensional datasets to be flattened into two-dimensional snapshot matrices, thereby discarding important structural information. This work introduces the low-cost High-Order Singular Value Decomposition (lcHOSVD), a novel tensor-based sparse-sensing reconstruction framework for high-dimensional environmental fields. To the authors' knowledge, this is the first methodology that combines sparse sensing and HOSVD for field reconstruction. Unlike matrix-based approaches, lcHOSVD preserves the natural tensor structure of the data, enabling the exploitation of correlations across spatial, temporal, and physical-variable dimensions while substantially reducing the computational requirements of conventional HOSVD. The methodology is applied to urban flow and air-quality datasets, where three-dimensional velocity and pollutant concentration fields are reconstructed using only 1-4% of the available spatial locations. While lcSVD provides larger computational speed-ups, lcHOSVD consistently achieves lower reconstruction errors in configurations characterized by strong multidimensional coupling and heterogeneous dynamics across dimensions. Additional sensor-anisotropy analyses demonstrate that the tensor formulation is significantly more robust to uneven sensor distributions, a common situation in practical environmental monitoring networks.