The paper presents a method for merging distributed observations into a continuous, differentiable spline field without sharing raw data or iterative synchronization. By leveraging fixed-basis ridge regression, each data holder computes local Gram matrices and moment vectors, and the merged solution matches centralized fitting exactly. The authors validate the pipeline on four PDEs and real NOAA sea‑surface temperature data, achieving sub‑percent accuracy for linear cases and 5% for the nonlinear Burgers equation, with no degradation from distributed merging.
By Naveen Mysore
arXiv:2607. 26414v1 Announce Type: cross Abstract: Latent low-dimensional structure in datasets of natural and engineered systems enables their sparse sensing, or full-state reconstruction from historical data and very few carefully chosen localized measurements.
By Andrei A. Klishin, J. Nathan Kutz, Krithika Manohar
arXiv:2606. 24989v1 Announce Type: new Abstract: Urban flow and air-quality simulations generate high-dimensional datasets describing velocity and pollutant transport across multiple spatial, temporal, and physical-variable dimensions.
By Arindam Sengupta, Paul Jeanney, Ricardo Vinuesa, Jose Miguel Perez, Soledad Le Clainche
The paper introduces Neptune, a method that uses independent coordinate neural networks to infer parameter fields in multi-physics PDEs from sparse measurements. Neptune can accurately estimate parameters with nonlinear, spatiotemporal variations, outperforming existing techniques by reducing estimation errors by up to two orders of magnitude and improving dynamic response predictions by a factor of ten. It also demonstrates strong physical extrapolation, enabling reliable predictions beyond the training data.
By Xuyang Li, Mahdi Masmoudi, Rami Gharbi, Nizar Lajnef, Vishnu Naresh Boddeti
arXiv:2506.12045v2 Announce Type: replace-cross
Abstract: Accurate reconstruction of latent environmental fields from sparse, indirect observations is a fundamental challenge across scientific domain...
By Kazuma Kobayashi, Tapas Tripura, Jay Phil Yoo, Diab Abueidda, Seid Koric, Souvik Chakraborty, Syed Bahauddin Alam
StationPDE is a station-oriented surface PDE learning model designed for multi-station multivariate weather forecasting. It builds a terrain-aware continuous surface field from discrete station observations and separates its physical evolution into surface wind transport and upper-air inference, the latter approximating missing upper-air effects via learnable horizontal diffusion. The model also includes a data-driven diffusion branch and an adaptive router to combine forecasts, achieving a 9.6% average MSE reduction over state-of-the-art baselines on Weather2K and MeteoNet datasets.
By Xiao Wang, Changjian Chen, Rongwen Li, Hongwu Liu, Kun Fang, Zhuo Tang
arXiv:2606. 03936v1 Announce Type: new Abstract: Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters.
By Niccol\`o Perrone, Fanny Lehmann, Stefania Fresca, Filippo Gatti
arXiv:2606. 17070v1 Announce Type: cross Abstract: Accurate oceanic forecasting is critical for climate monitoring and disaster early warning.
By Qinghui Chen, Zekai Zhang, Hailong Liu, Jinglin Zhang, Cong Bai
arXiv:2605. 27527v2 Announce Type: replace-cross Abstract: Astrophysical observations from Earth are subject to weather, environmental, and scientific constraints that lead to sparse, irregular light curves.
By Siddharth Chaini, Federica B. Bianco, Ashish Mahabal
The paper introduces an online framework for functional principal component analysis (FPCA) tailored to multidimensional functional data streams. It models functional principal components with tensor product splines, enforcing smoothness and orthonormality via a penalized approach on a Stiefel manifold. The authors present efficient Riemannian stochastic gradient descent and AdaGrad algorithms, along with a dynamic smoothing parameter tuning strategy based on rolling block validation, and provide asymptotic normality results and confidence intervals for the estimators.
By Muye Nanshan, Nan Zhang, Jiguo Cao
The paper compares different Ensemble Kalman methods for calibrating climate model parameters by minimizing the misfit between modeled and observed climate statistics. It conducts systematic numerical experiments on Lorenz-type models, including neural network parameterizations, to evaluate computational efficiency and accuracy of each method. The study examines how prior information and dimensionality affect the cost of these methods.
By Rebecca Gjini, Matthias Morzfeld, Oliver R. A. Dunbar, Tapio Schneider
Neural operator surrogates (NO) approximate PDE solutions orders of magnitude faster than numerical solvers, but suffer from spectral bias: high-frequency content is systematically attenuated, limiting reliability where fine-scale structure matters. Sparse sensor measurements of the field are often available too, offering pointwise accuracy without spectral distortion but covering only a small fraction of the domain.