Operator Learning for Reconstructing Flow Fields from Sparse Measurements: a Language Model Approach

Reconstructing high-fidelity flow fields from low-fidelity observations is a central problem in scientific machine learning, yet recent diffusion and flow-matching models typically rely on iterative sampling, making them costly for latency-sensitive workflows such as ensemble forecasting, real-time visualization, and simulation-in-the-loop inference. We study whether a high-fidelity flow-matching generative model can be compressed into a compact one-step model for fast scientific flow reconstruction. Our approach distills an optimal-transport flow-matching teacher into a one-step consistency model. Low-fidelity observations are incorporated at inference by initializing the generative trajectory from a noised observation along the transport path, allowing an unconditional high-fidelity flow model to perform conditional reconstruction without retraining the teacher. We evaluate this distillation strategy on three fluid benchmarks, Smoke Buoyancy, Turbulent Channel Flow, and Kolmogorov Flow, using coarse-to-fine reconstruction as a controlled testbed at field sizes up to $256 \times 256$. Across these settings, the distilled student retains similar performance of the teacher's model on spectrum metrics, while using roughly half as many parameters and achieving a $12\times$ inference speedup over the flow-matching teacher. Under the same training budget, the distilled student also outperforms a one-step consistency model trained directly from scratch by $23.1\%$ in SSIM, showing that teacher distillation improves training efficiency rather than merely accelerating sampling. These results suggest a promising route for turning future high-capacity scientific generative models into compact reconstruction models that are faster to train, cheaper to run, and easier to deploy.

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.

Reconstructing continuous flow fields from sparse surface-mounted sensors is central to aerodynamic design, flow control, and digital-twin instrumentation. Existing neural methods for this task typically encode sensor readings into implicit latent codes with little spatial interpretability and limited formal guidance on how representational capacity should scale with observation count. Inspired by 3D Gaussian Splatting, we introduce FLUIDSPLAT, a sensor-conditioned model that predicts K anisotropic Gaussian primitives forming a partition-of-unity scaffold, a spatially explicit and interpretable intermediate representation of the flow. For an idealized Gaussian primitive estimator, we prove an $O(K^{-s/d})$ approximation rate for fields with Sobolev smoothness $s$; incorporating $N$ noisy observations yields a squared-risk decomposition with bias $O(K^{-2s/d})$ and variance $O(σ^{2}K/N)$.Balancing the two yields $K^{*}\!\sim\!(N/σ^{2})^{d/(2s+d)}$: primitive count cannot grow freely under sparse sensing, revealing a variance bottleneck that motivates complementing the scaffold with a state-conditioned residual decoder. Across four benchmarks spanning 2D and 3D, FLUIDSPLAT achieves 11-28% error reduction over several strong baselines on cylinder flow, AirfRANS, FlowBench LDC-3D, and PhySense-Car 3D benchmarks.