UQ-SHRED: uncertainty quantification of shallow recurrent decoder networks for sparse sensing via engression

Reconstructing high-dimensional spatiotemporal fields from sparse sensor measurements is critical in a wide range of scientific applications. The SHallow REcurrent Decoder (SHRED) architecture is a recent state-of-the-art architecture that reconstructs high-quality spatial domain from hyper-sparse sensor measurement streams. An important limitation of SHRED is that in complex, data-scarce, high-frequency, or stochastic systems, portions of the spatiotemporal field must be modeled with valid uncertainty estimation. We introduce UQ-SHRED, a distributional learning framework for sparse sensing problems that provides uncertainty quantification through a neural network-based distributional regression called engression. UQ-SHRED models the uncertainty by learning the predictive distribution of the spatial state conditioned on the sensor history. By injecting stochastic noise into sensor inputs and training with an energy score loss, UQ-SHRED produces predictive distributions with minimal computational overhead, requiring only noise injection at the input and resampling through a single architecture without retraining or additional network structures. On complicated synthetic and real-life datasets including turbulent flow, atmospheric dynamics, neuroscience and astrophysics, UQ-SHRED provides a distributional approximation with well-calibrated confidence intervals. We further conduct ablation studies to understand how each model setting affects the quality of the UQ-SHRED performance, and its validity on uncertainty quantification over a set of different experimental setups.

SENDAI: A Hierarchical Sparse-measurement, EfficieNt Data AssImilation Framework

Bridging the gap between data-rich training regimes and observation-sparse deployment conditions remains a central challenge in spatiotemporal field reconstruction, particularly when target domains exhibit distributional shifts, heterogeneous structure, and multi-scale dynamics absent from available training data. We present SENDAI, a hierarchical Sparse-measurement, EfficieNt Data AssImilation Framework that reconstructs full spatial states from hyper sparse sensor observations by combining simulation-derived priors with learned discrepancy corrections. We demonstrate the performance on satellite remote sensing, reconstructing MODIS (Moderate Resolution Imaging Spectroradiometer) derived vegetation index fields across six globally distributed sites. Using seasonal periods as a proxy for domain shift, the framework consistently outperforms established baselines that require substantially denser observations -- SENDAI achieves a maximum SSIM improvement of 185% over traditional baselines and a 36% improvement over recent high-frequency-based methods. These gains are particularly pronounced for landscapes with sharp boundaries and sub-seasonal dynamics; more importantly, the framework effectively preserves diagnostically relevant structures -- such as field topologies, land cover discontinuities, and spatial gradients. By yielding corrections that are more structurally and spectrally separable, the reconstructed fields are better suited for downstream inference of indirectly observed variables. The results therefore highlight a lightweight and operationally viable framework for sparse-measurement reconstruction that is applicable to physically grounded inference, resource-limited deployment, and real-time monitor and control.

Mesh-free sparse identification of nonlinear dynamics

Identifying the governing equations of a dynamical system is one of the most important tasks for scientific modeling. However, this procedure often requires high-quality spatio-temporal data uniformly sampled on structured grids. In this paper, we propose mesh-free SINDy, a novel algorithm which leverages the power of neural network approximation as well as auto-differentiation to identify governing equations from arbitrary sensor placements and non-uniform temporal data sampling. We show that mesh-free SINDy is robust to high noise levels and limited data while remaining computationally efficient. In our implementation, the training procedure is straight-forward and nearly free of hyperparameter tuning, making mesh-free SINDy widely applicable to many scientific and engineering problems. In the experiments, we demonstrate its effectiveness on a series of PDEs including the Burgers' equation, the heat equation, the Korteweg-De Vries equation and the 2D advection-diffusion equation. We conduct detailed numerical experiments on all datasets, varying the noise levels and number of samples, and we also compare our approach to previous state-of-the-art methods. It is noteworthy that, even in high-noise and low-data scenarios, mesh-free SINDy demonstrates robust PDE discovery, achieving successful identification with up to 75% noise for the Burgers' equation using 5,000 samples and with as few as 100 samples and 1% noise. All of this is achieved within a training time of under one minute.