Spatiotemporal Fields

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6 papers in the last four weeks, against 1 the four weeks before. 0.1% of all new papers.

Jul 6Week of Sep 21

Latest papers 36

Oct 1, 2026cs.CV

PhysDEM: Physics-Defined Energy-Matching Diffusion for Spatiotemporal Field Generation under Scarce Measurements

Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.

Zhenyu Liang, Yining Huang, Yubo Zhao +1

HKUST · UT Austin

Spatiotemporal Fields Diffusion Dynamics Gaussian Primitives Demographics

Sep 29, 2026cs.LG

SCOPE: Observation-Conditioned Full-Target Prediction for Sparse PDE Inference

Recovering complete physical fields from sparse observations is challenging because the measurements may not uniquely determine the underlying state. Diffusion-based PDE solvers address this problem through iterative sampling whereas neural operators provide deterministic one-pass predictions. We propose SCOPE (Sparse-Context Observability-aware Predictive Embeddings) to recover complete PDE fields from sparse observations by coupling full-field latent prediction with physical reconstruction. A shared decoder reconstructs fields from both predicted and complete-view representations so that representation learning is guided by both physical recovery and latent matching. We derive a quadratic risk decomposition at fixed teacher-decoder pairs showing why optimal latent prediction need not yield optimal field reconstruction. We also establish sufficient conditions for decoder improvements on complete inputs to transfer to recovery from partial observations. Experiments across five PDE settings show that SCOPE outperforms mask-aware neural operators on all ten forward and inverse tasks and achieves lower errors than those reported for diffusion-based solvers including DiffusionPDE and FunDPS. Decoder-only adaptation further improves recovery without retraining the backbone while retaining deterministic single-pass inference.

Ruichen Xu, Siyao Wang, Fang Wan +8

Stony Brook University · University of California, Davis · Independent Research +3

Neural Partial Differential Equation Solvers Partial Differential Equations Spatiotemporal Fields Joint-Embedding Predictive Architectures

Sep 29, 2026cs.LG

PDE-OBS: Controlled Evaluation Across Observation Patterns

Physical-field reconstruction and forecasting depend on both measurement density and spatial layout, yet evaluation under a single observation pattern does not characterize performance when that pattern changes. We introduce PDE-OBS, an integrated benchmarking platform spanning numerical data generation, model training, and inference and evaluation under varying observation conditions. It combines 560,000 fields and trajectories from seven partial differential equation families with configurable observation operators and seven adapted baseline methods for stationary reconstruction and short-horizon forecasting. Separating observation construction from physical records allows users to specify parameterized patterns and deterministic mixtures for training and testing while preserving prediction targets and data splits. The evaluation protocol uses references trained for each test pattern to compare models on identical test observations and targets, alongside equal-count groups for spatial-layout comparisons. On a 14,000-record subset, we evaluate 441 trained models under nine test patterns, yielding 3,969 evaluations. Mean cross-pattern error exceeds mean matched-pattern error in all 49 PDE-method pairs, and this finding persists in a configuration-matched subset of 117 models. Denser test observations do not consistently reduce error for a fixed model. Mixed-pattern training on five completed pairs reduces large single-pattern transfer errors, although destination-trained references usually remain more accurate. Together, the benchmark and findings support systematic evaluation of observation-pattern sensitivity and provide a reusable workflow for developing methods under changing measurement conditions. Code: https://github.com/ru1ch3n/PDE-OBS.

Ruichen Xu, Siyao Wang, Fang Wan +7

Stony Brook University · University of California, Davis · Independent Research +3

Spatiotemporal Fields Short-Horizon Forecasting Model Training

Sep 28, 2026cs.AI

Calibrated Uncertainty for Informative Path Planning in Aquatic Environmental Monitoring

Informative Path Planning for scalar field reconstruction uses predictive uncertainty to direct sensing vehicles toward maximally informative locations. Gaussian Processes provide this signal but their stationary isotropic kernels are misspecified for non-homogeneous phenomena such as oil spills, producing miscalibrated estimates that degrade planning. We investigate whether replacing the Gaussian Process with a well-calibrated Deep Ensemble improves path planning outcomes, and whether uncertainty quality interacts with the choice of planning algorithm. Five strategies (

ε

-Greedy, Value Greedy, Uncertainty Greedy, Monte Carlo Tree Search, and Receding Horizon Orienteering) share a common Deep Ensemble backbone trained on physics-based oil spill simulations. On held-out stochastic spill scenarios, the Deep Ensemble reduces normalised reconstruction error by

83\%

relative to the Gaussian Process baseline. Crucially, well-calibrated uncertainty amplifies the importance of the planning strategy: the performance gap between algorithms is negligible under miscalibrated models but becomes substantial under the ensemble, where multi-step lookahead planners outperform greedy selection by up to

32\%

in reconstruction error and achieve IoU above

0.85

. Monte Carlo Tree Search is the recommended planner, matching Orienteering in reconstruction quality at an order-of-magnitude lower computational cost.

Samuel Yanes Luis, Alejandro Casado Pérez, Alejandro Mendoza Barrionuevo +3

Department of Electronic Technology, University of Sevilla, Sevilla, Spain · Department of Electronic Engineering, University of Sevilla, Sevilla, Spain

Calibrated Uncertainty Path Planning Gaussian Process Coral Spatiotemporal Fields Path+2

Sep 24, 2026cs.CV

Efficient Continuous DEM Reconstruction under Limited Target-Resolution Supervision

High-resolution digital elevation models (DEMs) support Earth observation applications, but paired training references are often available only at coarser output resolutions. Reconstructing finer terrain grids therefore requires both effective transfer beyond the supervised scale and control of dense-query computation. To address this problem, SCOPE learns a continuous terrain representation from coarser-resolution pairs. It predicts a latent coefficient field on the low-resolution grid and reuses local Fourier residual functions through basis evaluation and geometry-guided ensemble fusion. This separates high-dimensional coefficient prediction from output-grid construction. Experiments on geographically distributed land--ocean samples assess supervised reconstruction, unseen-scale inference, cross-domain generalization, and theoretical computation. SCOPE leads the compared methods across six metrics in the main supervised-scale evaluation. At an unseen factor three times the training factor, land reconstruction reduces RMSE and MAE by approximately 12% relative to bicubic interpolation, with errors close to target-scale fine-tuning. Ninefold output density increases counted multiply--accumulate operations by only about 2%. Frozen-model validation on held-out external marine regions reduces RMSE relative to the DEM-specific implicit baseline EBCF-CDEM by approximately 19% under self-downsampling and 2% with cross-product inputs, while also yielding lower RMSE than LIIF-MS in both settings. These results demonstrate the value of reusable coefficient fields for accurate reconstruction beyond the supervised resolution with low incremental arithmetic cost.

Zekai Shi, Meng Zhang, Haokun Zhang +1

School of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an, 710049, China · School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University, Xi’an, 710072, China

Digital Elevation Models Spatiotemporal Fields Ocean

Sep 15, 2026cs.LG

Walking the Score Manifold: Continuous-time Generative Dynamics on Learned Data Manifolds

Generative modeling of time-dependent data is typically formulated on a discrete temporal grid, restricting supervision to the observed timestamps in the training data. We instead frame generation as continuous-time evolution on a learned data manifold. To this end, we leverage pretrained score-based models as geometric priors and learn a vector field that evolves data along score-induced interpolation paths. Because these dynamics follow transitions that respect the geometry learned by the score model, they support generation at arbitrary timestamps and temporal super-resolution beyond the discretization of the training data. Moreover, this geometric formulation allows us to train the vector field simulation-free through a regression objective. To improve long-horizon rollout robustness, we introduce an objective that promotes path-relative transverse exponential stability. While motivated by stability theory, it admits a practical interpretation as denoising score matching transverse to the interpolation path. Further, we extend the framework to a probabilistic setting that models a distribution over plausible future trajectories. We demonstrate the method on natural video and scientific dynamical data, including temporal super-resolution, PDE-based spatiotemporal fields, and molecular dynamics. Our results show that score-based priors provide a strong foundation for learning stochastic continuous-time generative dynamics.

Jan Tauberschmidt, Brian B. Moser, Stanislav Frolov +3

DFKI Kaiserslautern · RPTU Kaiserslautern · Imperial College London

Generative Models Score-Based Diffusion Model Spatiotemporal Fields Interpolation Geometric Priors

Sep 15, 2026cs.RO

Gaussian Processes for Modelling Spatial Fields with Robot Swarms

Robot swarms, by virtue of their decentralised architecture, are a natural tool for scalable, robust modelling of spatial fields, such as water temperature, wind velocity, or terrain elevation. However, existing methods rely on external positioning systems that allow each robot to determine its own position in space. Here, we introduce location-unaware Gaussian process regression (LU-GPR) as a solution to the modelling of spatial fields in the absence of such positioning systems. LU-GPR allows each robot to infer the posterior mean and variance of the field in space, while simultaneously agreeing on a common frame of reference with its peers, using only local sensing and communication. We propose an online algorithm that allows each robot to consistently infer local estimates as its local frame of reference converges to the common one. By means of a product of experts model, each robot also combines the estimates of its peers with its own to obtain a global model. Our results show that LU-GPR scales well with the number of robots and is robust to limited communication ranges. We also demonstrate how it can be used in real-world monitoring scenarios to estimate the flow of an evacuating crowd.

Guillermo Legarda Herranz, Gianpiero Francesca, Mauro Birattari

IRIDIA, Universit´e libre de Bruxelles, 1050 Brussels, Belgium · Toyota Motor Europe

Swarms Spatiotemporal Fields Gaussian Process Robot Systems

Sep 15, 2026cs.LG

Recovering Governing Dynamics from Distributed Observations via Exact Spline Merging

Scientific observations are frequently distributed across locations, time periods, and institutions. Combining such observations into a continuous, differentiable field enables recovering governing physical parameters from its derivatives. This paper makes two contributions in this setting. First, the established additive structure of fixed-basis ridge-regression statistics is applied to tensor-product spline fields: each data holder computes a local Gram matrix and moment vector, and the merged solution is mathematically identical to centralized fitting, with no raw data shared and no iterative synchronization. This property is specific to the fixed-feature squared-error setting; the present derivation does not establish an analogous guarantee for general jointly trained multilayer networks. Second, a complete pipeline connects distributed observations to physical parameter inference through field reconstruction, derivative extraction, and linear regression. The pipeline is validated on four PDEs: diffusion, wave, heat-with-source, and the nonlinear viscous Burgers equation, recovering governing parameters to sub-percent accuracy in the linear cases and 5% for Burgers. In all cases, distributed merging introduces zero degradation relative to centralized fitting. Application to 41 years of NOAA sea-surface temperature data confirms the result on real spatiotemporal observations. Source code to reproduce all experiments is available at https://github.com/NAVEENMN/gramfield.

Naveen Mysore

University of California, Santa Barbara · Dyssonance AI

Spatiotemporal Fields Splines Sea-Surface Temperature Data Implicit Differentiation Kernel Ridge Regression

Sep 9, 2026eess.IV

Scale-Aware 3D Deep Learning for Robust Brain Metastasis Detection in Multimodal MRI

Detecting brain metastases in magnetic resonance imaging (MRI) remains challenging because lesions vary widely in size and appearance, with very small metastases occupying only a minute fraction of a three-dimensional input. We investigate whether combining different spatial fields of view (FOVs) improves lesion detection in multimodal MRI and present a scale-aware 3D deep-learning framework. The method uses independently trained

96^3

and

64^3

3D U-Nets whose whole-volume probability maps are combined by weighted late fusion. This design allows us to study the effect of spatial context separately from image resolution and modality choice. On a 97-patient development cohort, cross-FOV fusion improved lesion-level precision and F1 while substantially reducing false positives relative to the individual models. A same-FOV ensemble control showed that these gains were not explained solely by averaging independently trained networks, supporting a contribution from complementary spatial context. An exploratory cross-FOV agreement filter reduced false positives but did not improve overall F1. These results support cross-FOV probability fusion as a simple and computationally practical strategy for improving the precision-false-positive trade-off in 3D brain-metastasis detection.

Sylvain Jaume, Hongming Wang, Simon K. Warfield

Massachusetts Institute of Technology · Harvard University · Boston Children’s Hospital, Harvard Medical School

Brain Tumor Segmentation Magnetic Resonance Imaging Cross-View Fusion Lesion 3D Generation Spatiotemporal Fields

Sep 1, 2026cs.LG

HarmoCore: Functional Latent Diffusion for Sparse Reconstruction of Oscillatory Wave Fields

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.

Lihao Chen, Xinyu Zhang, Panqi Chen +4

College of Information Science and Electronic Engineering, Zhejiang University

Complex Wavefield Signal Reconstruction Bayesian Inverse Problems Spatiotemporal Fields Identity-Conditioned Latent Diffusion Models Oscillatory Dynamics+1

Aug 9, 2026stat.ML

Physics-Informed Learning for Robust Acoustic Localization with Calibrated Uncertainty

Recent advances in Passive Acoustic Monitoring (PAM) offer an opportunity to obtain ecological spatial point-process data at unprecedented scale. However, realizing this opportunity necessitates the development of accurate and scalable localization methods. In real-world outdoor soundscapes, however, the assumptions underlying classical localization methods such as hyperbolic and score-based localization are routinely violated by multipath dominance, near-field effects, and complex propagation. Under these conditions, classical localization methods become brittle, with extreme errors possible even in small detection arrays. Rather than statistically replacing the underlying physics, we propose a method to refine it and increase robustness outside of ideal operating conditions: a learned model operating on physics-informed acoustic features corrects a fast hyperbolic solver where it produces implausible solutions, substantially reducing catastrophic worst-case errors while matching its median accuracy on field data. We further provide calibrated, geometry-aware uncertainty estimates suitable for propagation into downstream spatial models. Evaluating on distributed microphone arrays in real and simulated outdoor environments, we demonstrate that the proposed method yields robust, uncertainty-aware localization, providing a step toward scalable automated wildlife monitoring in complex acoustic environments.

Jennifer N. Kampe, Changwoo J. Lee, Xin Shen +5

Department of Biological and Environmental Science, University of Jyväskylä, Jyväskylä, Finland · Department of Statistical Science, Duke University, Durham, NC, USA · 3Digital Services, University of Jyväskylä, Jyväskylä, Finland +1

Sound Source Localization Indoor Localization Calibrated Uncertainty Physics-Informed Learning Spatiotemporal Fields Uncertainty+1

Jul 31, 2026cs.RO

Localization in Spatiotemporal Fields via Environmental PDEs

This paper proposes a localization framework that uses spatiotemporal fields governed by partial differential equations (PDEs) as localization signatures. Two PDE classes are considered: the shallow water equations, which describe free-surface flows in coastal and riverine environments, and the advection-diffusion equation, which models the transport and mixing of scalar quantities such as temperature, salinity, and dissolved oxygen. A numerical PDE solver provides predicted fields over the domain, and multiple field channels are fused as multimodal measurements to improve localization accuracy. We formulate the problem within a Rao-Blackwellized particle filter (RBPF) that partitions the vehicle state into a nonlinear component sampled by particles and a linear sensor bias component tracked analytically via per-particle Kalman filters. This factorization reduces the required number of particles compared to a standard particle filter while accounting for realistic sensor drift. Simulation studies on both PDE scenarios show that the RBPF consistently outperforms a standard particle filter in terms of final position error and Root Mean Square Error (RMSE) across varying particle counts. Field experiments with an autonomous surface vehicle measuring salinity, temperature, and dissolved oxygen validate that PDE-governed environmental fields provide sufficient spatial variability for practical localization. Related experimental videos are available at https://localization-environmental-pdes.github.io/.

Jose Fuentes, Abdullah Al Redwan Newaz, Ana Cavalcanti +1

Faculty of Computer Engineering and Information Sciences, Florida International University, Miami, FL 33174, USA · University of New Orleans, New Orleans, LA 70148, USA

Particle Filters Spatiotemporal Fields Indoor Localization Extended Kalman Filter One-Dimensional Viscous Burgers Equation Reynolds-Averaged Navier-Stokes+3

Jul 29, 2026cs.LG

Existence-Field Diffusion Model for Spatial Point Processes with Variable Cardinality

We study generative modeling of spatial point processes (SPP), where both the number of points and their spatial configuration are governed by a joint distribution. While diffusion models have achieved strong performance in modeling complex distributions, extending them to variable-cardinality SPP remains challenging. Existing approaches either decouple the modeling of cardinality and spatial structure, or rely on discrete trans-dimensional operations to modify the number of points, resulting in inflexible and asymmetric generative dynamics. We propose the existence-field diffusion model (EFDM) for spatial point processes modeling, where each potential point is associated with an existence variable representing its degree of presence. This enables a unified diffusion process that jointly models both spatial locations and cardinality without requiring explicit discrete transitions. We demonstrate that our approach provides a flexible and general framework for generative modeling of spatial point processes, achieving improved modeling capability on datasets with varying cardinality.

Xiaoyin Pan, Christian R. Shelton, Rakshith Mahishi +1

University of California, Riverside · New York University · Zhongguancun Academy

Determinantal Point Process Spatiotemporal Fields Cardinality Generative Models Diffusion Models

Jul 28, 2026stat.ML

Can Deep Generative Models Reproduce Non-Stationary Gaussian Random Fields?

Deep generative models (DGMs) are widely used for complex high-dimensional data and increasingly applied to spatial and spatio-temporal modeling. Their generated samples implicitly represent the learned data distribution and associated uncertainty. However, for real-world data, assessing whether DGMs have learned the underlying process is difficult because the ground truth is unknown and evaluation often relies on observations alone. We evaluate representative DGMs, flow matching (FM), DDPM, score-SDE, and VAE, on a known non-stationary Gaussian random field. This paper provides comprehensive metrics to assess recovery of the ground-truth mean and covariance structures, with oracle samples and a stationary control as references. All four models recover the mean surface, while their covariance recovery differs across model families: DDPM and score-SDE recover the covariance structure reasonably well, FM exhibits mildly attenuated non-stationarity and slight variance under-dispersion, and VAE has difficulty recovering the covariance structure. An experiment on ERA5 temperature anomalies further demonstrates how the framework can support the validation and development of DGMs for complex real-world spatio-temporal data.

Daniel Kua, Yan Song

Department of Statistics, University of British Columbia

Spatiotemporal Fields Generative Models Gaussian Process Covariance

Jul 14, 2026cs.CV

Semantic-Edge Response Decoding of SAM3 for Zero-Shot Crack Segmentation

Crack segmentation is essential for infrastructure inspection and structural health assessment, but existing high-performance methods typically require task-specific pixel-level annotations and training. Text-promptable vision foundation models enable zero-shot deployment, yet their final mask proposals are poorly suited to thin, fragmented, and low-contrast cracks, whose evidence may be suppressed, truncated, or over-expanded during mask generation. We find that language-conditioned semantic responses within the SAM3 decoder preserve more continuous and complete crack evidence than its final masks. Based on this observation, we propose Semantic-Edge Response Decoding (SERD), which interprets internal responses as a dense crack-likelihood field, calibrates them with a lightweight edge prior, and generates crack masks using a unified global threshold, without annotation or fine-tuning. Experiments on six public datasets show that SERD consistently improves over native SAM3 and outperforms the compared zero-shot and open-vocabulary segmentation methods, achieving an average Crack IoU of 61.14%, 4.63 points higher than SAM3. Further analyses show that most gains arise from directly decoding internal semantic responses, while edge calibration improves structural recovery and false-positive control without increasing end-to-end inference overhead. These results suggest that, for thin and non-compact targets, internal continuous responses can provide a more transferable interface than the final masks of foundation models. Code is available at: https://github.com/xauat-liushipeng/SERD

Shipeng Liu, Zhanping Song, Liang Zhao +1

Xi’an University of Architecture and Technology

Segment Anything Model Pixel-Level Annotation Spatiotemporal Fields

Jul 13, 2026cs.CV

FoundationGeo: Learning Spatial Pixel-Wise Fields for Monocular Metric Geometry

We present FoundationGeo, a two-stage framework that explicitly bridges relative and metric prediction via spatial calibration and principled data design. Stage 1 learns a high-fidelity, affine-invariant geometry model by initializing with DINOv3 and training on a curated 10.2M-sample multi-domain corpus with complementary local-detail supervision, yielding sharp boundaries and strong cross-domain generalization. Stage 2 moves beyond global scaling by introducing lightweight pixel-wise calibration fields for metric estimation: a scale field for spatially varying metric alignment and a ray-direction correction field that mitigates directional bias in point-map geometry, together producing metrically consistent 3D point maps. Beyond model design, we identify camera intrinsic coverage, especially focal length distribution mismatch between training and test data, as a key bottleneck for zero-shot metric generalization: performance drops sharply when test intrinsics fall outside the training distribution. To address this, we synthesize additional training data across diverse focal lengths using a Blender-based data engine, repairing under-covered focal regimes and improving robustness under intrinsic shift. Extensive zero-shot evaluations across seven benchmarks show that FoundationGeo significantly strengthens cross-domain robustness, staying near the top across diverse domains while avoiding the sharp cross-domain performance drops observed in other methods. This consistency translates into the best overall performance, surpassing heavier baselines by over 5.2% on average.

Muxin Liu, Xiaoyang Lyu, Tianhe Ren +7

The University of Hong Kong · Voyager Research, DiDi Chuxing

3D Foundation Models Geometric Metrics Geogs-Slam Spatiotemporal Fields

Jul 11, 2026stat.ML

TSCoNet: A Two-Stage Copula CNN-LSTM for Uncertainty-Aware Spatio-Temporal Forecasting

Reliable forecasting of several interrelated environmental variables - such as regional precipitation and temperature, or other correlated geophysical fields - across many locations calls for accurate predictions accompanied by trustworthy statements of their uncertainty. Modern deep-learning models forecast such variables accurately but usually report no uncertainty, and forcing them to output uncertainty through maximum likelihood tends to degrade their accuracy, especially when the variables are strongly correlated. Motivated by this tension, we develop TSCoNet, a two-stage convolutional-recurrent model coupled with a Gaussian copula that jointly forecasts multiple variables over space and time while quantifying predictive uncertainty. The method first learns accurate mean forecasts and then, holding the mean fixed, refines a shared representation to estimate the predictive variance, yielding calibrated prediction intervals after a standard recalibration, so that uncertainty is added without sacrificing point accuracy. We study the approach on simulated non-stationary spatial fields on the sphere and on a real dataset of monthly precipitation and temperature for fifty cities over 2000-2020. The model matches the accuracy of a strong deterministic forecaster while supplying calibrated prediction intervals that the deterministic model cannot, giving a single tool that provides both accurate point forecasts and reliable uncertainty for multivariate spatio-temporal data.

Jongwook Kim, Jong-Min Kim

Statistics Discipline, Division of Science and Mathematics, University of Minnesota-Morris

Spatio-Temporal Forecasting Predictive Uncertainty Gaussian Copula Spatiotemporal Fields

Jul 5, 2026physics.geo-ph

Generative wave propagator

Seismic wavefield simulation is fundamental to seismology, but conventional finite-difference (FD) methods remain limited by numerical dispersion and stability constraints, which often require dense spatial grids and small time steps and thereby severely limit the effectiveness of iterative inversion workflows. We introduce a conditional diffusion-based wavefield propagator that advances seismic wavefields recursively from one time step to the next. Instead of learning an unconditional data distribution of wavefield evolution, the model is conditioned by a short history of recent wavefield time steps (snapshots), the velocity model, and the wavefield time step index, allowing it to represent the conditional transition between adjacent physical states. By training the network to directly predict the clean next wavefield snapshot, this strong physical conditioning makes it possible to replace the iterative reverse diffusion process with a single network evaluation for each predicted snapshot. To improve stability over long recursive rollouts, we further introduce a causal time-weighted loss, in which adaptive weights, accumulated as exponential moving averages of per-snapshot training errors, emphasize training directions that are consistent with the forward propagation sequence and reduce the amplification of one-step prediction errors. Because the learned propagator is tied to the temporal spacing of the training snapshots rather than to the FD stability limit, it can advance the wavefield using a physical time step ten times larger than that required by the underlying solver. Experiments on the Overthrust, SEG/EAGE, and Marmousi models show that the proposed method accurately reproduces wavefield snapshots and shot gathers and achieves an end-to-end speedup of 2.17 x over a GPU-accelerated tenth-order staggered-grid FD implementation under matched hardware conditions.

Shijun Cheng, Tariq Alkhalifah

Division of Physical Science and Engineering King Abdullah University of Science and Technology Thuwal 23955-6900, Saudi Arabia

Complex Wavefield Full Waveform Inversion Time Discretization Spatiotemporal Fields

Jun 30, 2026stat.ML

Dynamic Gaussian Processes and the Vanilla-SPDE Exchange

Gaussian process inference is often limited by cubic computational costs, a challenge that becomes more pronounced in spatio-temporal settings where posterior inference is required over dense grids. While state-space SPDE formulations enable linear complexity in time, exact inference remains cubic in space and deteriorates further when observation locations are disjoint from the prediction locations, which inflates the number of considered spatial points. To address this, we propose the Vanilla-SPDE Exchange, which exploits an equivalence between the standard and SPDE formulations of GP inference to construct a hybrid scheme with improved computational cost. We demonstrate these gains through complexity analysis and numerical experiments.

Rui-Yang Zhang, Lachlan Astfalck, Edward Cripps +2

School of Mathematical Sciences, Lancaster University. · School of Mathematics and Statistics, The University of New South Wales. · School of Physics, Mathematics and Computing, the University of Western Australia.

Gaussian Process Spatiotemporal Fields Computational Cost

Jun 23, 2026cs.LG

Low-Cost High-Order Singular Value Decomposition for Tensor-Based Reconstruction from Sparse Sensor Measurements: Urban Flow and Air-Quality Applications

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.

Arindam Sengupta, Paul Jeanney, Ricardo Vinuesa +2

ETSI Aeronáutica y del Espacio, Universidad Politécnica de Madrid, Plaza Cardenal Cisneros, 3, Madrid, 28040, Spain · Ove Arup, C. de Alfonso XI, 12, Retiro, Madrid, 28014, Spain · Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, United States

Tensor Decomposition Singular Value Decomposition Spatiotemporal Fields Air Pollution Velocity Field

Jun 11, 2026math.NA

Approximating Gaussian Whittle-Matern Fields over Well-Centered Triangulations of Riemannian Manifolds

Markovian Whittle-Matérn fields have been convergently approximated by discrete Gauss Markov Random Fields (GMRFs) with sparse precision matrices using a Finite Element approximation of the two-parameter family,

(κ^2 - Δ)^{α/2} u = \mathcal{W}, \;\; κ\in \mathbb{R}, \; α\in \mathbb{N}.

of SPDEs. Using recent developements in the analysis of Discrete Exterior Calculus (DEC), we present a different, yet closely related, convergent GMRF approximation to these Matérn fields over complete, boundaryless Riemannian manifolds discretized as well-centered simplicial complexes. This convergent method (i) is agnostic to

α, κ

and thus allows a universal approximation scheme for the precision and covariance matrices of the entire

(α, κ)

-family of GMRFs, so they may be inferred rather than guessed. (ii) inherently models pointwise and piecewise-smoothed measurements of a random field and approximates both equally well (iii) is computationally independent of the interpolants used - it suffers no overhead if one convergent interpolant were replaced with another suitable interpolant over the same mesh. Furthermore, we show that, on discretizations that are well-connected in a precise sense, and volume-concentrated, the precision matrices are spectral functions of a graph-laplacian. We provide a low rank approximator to the family of such Matérn GMRFs and mention a use case: reducing the number of measurements needed to model the GMRF by compressed-sensing.

Srinivas Nambirajan

Riemannian Manifolds Spatiotemporal Fields Triangulations Matrix

Jun 1, 2026physics.app-ph

Uncertainty-Aware Graph Neural Reconstruction of Urban Temperature Fields from Sparse Sensors under Deployment Constraints

Reconstructing spatially continuous daily temperature fields from sparse observations is important for urban climate monitoring and heat-risk analysis, but practical deployments are limited by sensor budgets and spacing constraints. This study proposes an uncertainty-aware graph neural network (GNN) framework for reconstructing daily maximum temperature fields from sparse sensors while supporting distance-constrained sensor placement and probabilistic exceedance mapping. The model predicts both the temperature field and a spatially varying predictive uncertainty field using a graph-attention-based mean-residual architecture trained with a Gaussian negative log-likelihood. Sensor placement is addressed using a Proper Orthogonal Decomposition with QR factorization (POD-QR) strategy with a 4 km minimum inter-sensor distance constraint and is compared with random feasible placement and farthest-point sampling. The framework is evaluated over a Montreal-area polygon using Daymet v4.1 daily temperature data (1 km resolution) under a strict temporal hold-out protocol (training: 2020-2023; testing: 2024). Across sensor budgets (10-40 sensors), the proposed GNN consistently outperforms inverse distance weighting and ordinary kriging in RMSE and MAE on unobserved nodes. Sensor-placement effects are most pronounced at low budgets and diminish at higher budgets, with a practical saturation regime emerging around 30 sensors under the imposed spacing constraint. Probabilistic evaluation further shows improved uncertainty calibration with increasing sensor density and a better sharpness-calibration trade-off than kriging. These results support the proposed framework as an effective tool for uncertainty-aware temperature field reconstruction and decision-oriented heat-risk mapping.

Reda Snaiki, Abdelatif Merabtine

Department of Construction Engineering, École de Technologie Supérieure, Université du Québec, Montréal, Québec, Canada

Optimal Sensor Placement Temporal Graph Neural Networks Spatiotemporal Fields Graph Neural Networks

May 31, 2026cs.LG

Data Enrichment for Symbolic Regression Using Diffusion Models

Symbolic regression (SR) offers a route to scientific discovery by converting observations into interpretable governing equations. However, despite its promise, its reliability degrades sharply when spatiotemporal measurements are sparse, noisy, or physically incomplete, as commonly occurring in practice. Data enrichment (DE) has been shown to be able to mitigate this limitation, yet additional samples can mislead equation discovery unless they preserve the physical structure of the target system. Such implication of DE requires narrow domain expertise as well as technical fluidity, highly limiting its practical usefulness. In this study, we introduce a physics-guided latent diffusion framework for DE for down the line SR models. The proposed framework combines a variational autoencoder, a conditional latent diffusion model, and a physics-informed residual corrector to complete sparse observations with synthetic fields constrained by governing relations. We evaluate the approach on heat conduction, incompressible Navier-Stokes flow, and a moving single-mass Newtonian gravitational potential, using GPLearn, DEAP, and PySR as downstream SR backends. Our results reveal that physics-corrected enrichment consistently improves recovery in sparse regimes across physical dynamics and SR models. These results show that generative enrichment can strengthen equation discovery without additional domain expertise.

Simon De Reuver, Tamas Kristof Toth, Teddy Lazebnik

Department of Computing, Jönköping University, Jönköping, Sweden · Department of Information Science, University of Haifa, Haifa, Israel

Symbolic Regression Spatiotemporal Fields Enrichment Analysis Diffusion Models

May 29, 2026cs.LG

Scalable Bayesian Inference for Nonlinear Conservation Laws

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Tim Weiland, Philipp Hennig

T"{u}bingen AI Center, University of T"{u}bingen, T"{u}bingen, Germany

Bayesian Inference Uncertainty Quantification Bayesian Inverse Problems Classical Numerical Solver Spatiotemporal Fields

May 25, 2026cs.LG

PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems

Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space transformations. While existing benchmarks of machine learning methods for PDEs primarily focus on the forward problem, there are no similar comprehensive studies and benchmark datasets on PDE inverse problems, i.e., mapping solution fields to underlying physical parameters. We fill this gap by introducing PDEInvBench, a comprehensive benchmark dataset consisting of numerical simulations for both time-dependent and time-independent PDEs across a wide range of physical behaviors and parameters. Our dataset includes evaluation splits that assess performance in both in-distribution and various out-of-distribution settings. Using our benchmark dataset, we comprehensively explore the design space of neural networks for PDE inverse problems along three key dimensions: (1) optimization procedures, analyzing the role of supervised, self-supervised, and test-time training objectives on performance, (2) problem representations, where we study the value of architectural choices with different inductive biases and various conditioning strategies, and (3) scaling, which we perform with respect to both model and data size. Our experiments reveal several practical insights: 1) neural networks perform best with a two-stage training procedure: initial supervision with PDE parameters followed by test-time fine-tuning using the PDE residual, 2) incorporating PDE derivatives as input features consistently improves accuracy, and 3) increasing the diversity of initial conditions in the training data yields greater performance gains than expanding the range of PDE parameters. We make our dataset and codebase publicly available.

Divyam Goel, Nithin Chalapathi, Sanjeev Raja +1

Department of Computer Science, UC Berkeley · Departments of Computer Science and Chemical EngineeringMay UC Berkeley; LBNL

Neural Partial Differential Equation Solvers Bayesian Inverse Problems Neural Network Benchmark Datasets Spatiotemporal Fields Inductive Bias

May 23, 2026cond-mat.mtrl-sci

SPLIT-PINN: Separable Probability Learning Technique via Physics-Informed Neural Networks for High-Dimensional Probabilistic Modeling

We present a probabilistic modeling framework for incorporating small-scale spatial heterogeneity into macroscopic descriptions of material behavior for polycrystalline metallic materials. Spatially heterogeneous material state fields are represented using probability density functions (PDFs), providing a principled statistical description of microstructural variability and state evolution across different computational polycrystalline realizations. The framework is built on the inverse identification of a probabilistic transport model, formulated as a Liouville equation with an unknown drift term. To enable accurate, stable, and interpretable inference of this drift field in high-dimensional, transport-dominated settings, we develop a Separable Probability Learning Technique via Physics-Informed Neural Networks (SPLIT-PINN). This method incorporates a marginal-correction drift decomposition, orthogonality constraints, and residual-based adaptive training to enhance well-posedness, numerical stability, and physical consistency without imposing restrictive parametric assumptions. Using SPLIT-PINN, the drift field governing the temporal evolution of joint state PDFs is inferred directly from data. After benchmark validation, the framework is applied to physical computational datasets describing the evolution of polycrystalline microstructural states, including von Mises stress, dislocation density, and equivalent plastic strain rate. The learned Liouville model, trained on a single dataset, is subsequently used in forward predictions of the temporal evolution of joint and marginal PDFs for multiple unseen polycrystal realizations. Quantitative comparisons with reference PDFs demonstrate that the proposed framework yields accurate and robust probabilistic predictions and generalizes effectively across datasets.

Pouria Behnoudfar, Deekshith Naidu Ponnana, Noah J. Schmelzer +6

Department of Mechanical Engineering, University of Wisconsin-Madison, Madison, WI, USA · Department of Civil Engineering, Johns Hopkins University, Baltimore, MD, USA · Materials Physics and Applications Division, Los Alamos National Laboratory, Los Alamos, NM, USA +2

Parametric Physics-Informed Neural Network Porous Media Spatiotemporal Fields

May 10, 2026cs.LG

DiffATS: Diffusion in Aligned Tensor Space

Direct diffusion modeling of high-resolution spatiotemporal fields is computationally challenging. Parameter-efficient primitives address this by representing high-dimensional data with a compact set of parameters. In this paper, we construct data-dependent tensor primitives without pretrained compression autoencoders. Our construction starts from Tucker decomposition, which captures low-rank multilinear structure through a core tensor and mode-wise factors. However, Tucker factors are non-unique: the same tensor can be represented by different rotated factors, which complicates generative modeling. We address this issue with orthogonal Procrustes (OP) alignment. Specifically, we select medoid anchor matrices from the data and align the factor matrices to resolve the gauge ambiguity. This yields matrix Grassmannian primitives and tensor Grassmannian primitives that are compact, data-adaptive, and directly decodable by explicit multilinear reconstruction. Theoretically, we prove that the proposed primitive maps are homeomorphisms between low-rank tensors and their corresponding primitive spaces, certifying that the representations are non-degenerate and topologically faithful. Building on these primitives, we propose Diffusion in Aligned Tensor Space (DiffATS), a generative framework that trains diffusion models directly on aligned tensor primitives. Across images, videos, and PDE solutions, DiffATS achieves strong unconditional and conditional generation performance while compressing original data by

3.9\times

to

210\times

, without relying on any pretrained deep compression autoencoders.

Jinhua Lyu, Tianmin Yu, Brian Kim +3

Northwestern University

Diffusion Alignment Diffusion Models Autoencoder Architectures Spatiotemporal Fields

May 8, 2026eess.SP

PropSplat: Map-Free RF Field Reconstruction via 3D Gaussian Propagation Splatting

Building a site-specific propagation model typically requires either ray-tracing over detailed 3D maps or dense measurement campaigns. Both approaches are expensive and often infeasible for rapid deployments where geographic data is unavailable or outdated. We present PropSplat, a map-free propagation modeling method that reconstructs radio frequency (RF) fields using 3D anisotropic Gaussian primitives. Each Gaussian encodes a scalar path loss offset relative to an explicit baseline path loss model with a learnable path loss exponent. Gaussians are initialized along observed transmitter--receiver paths and optimized end-to-end to learn the propagation environment without external information like floor plans, terrain databases, or clutter data. We evaluate PropSplat against wireless radiance field methods NeRF

^2

, GSRF, and WRF-GS+ on two real-world datasets. On large-scale outdoor drive-tests spanning multiple topographical regions at six sub-6 GHz frequencies, PropSplat achieves 5.38 dB RMSE when training measurements are spaced 300m apart and outperforms WRF-GS+ (5.87 dB), GSRF (7.46 dB), and NeRF

^2

(14.76 dB). On indoor Bluetooth Low Energy measurements, PropSplat achieves 0.19m mean localization error, an order of magnitude better than NeRF

^2

(1.84m), while achieving near-identical received signal strength prediction accuracy. These results show that accurate site-specific propagation reconstruction is achievable from sparse RF-native measurements. The need for geographic data as a prerequisite for scalable RF environment modeling is reduced.

William Bjorndahl, Maninder Pal Singh, Farhad Nouri +1

Department of Electrical and Computer Engineering, Southern Methodist University, Dallas, TX, USA · Department of Engineering Technology, University of Houston, Houston, TX, USA

Radio Map Estimation Wireless Radiance Field Propagation Gaussian Splatting Spatiotemporal Fields 3D Gaussian+1

May 8, 2026cs.LG

StreamPhy: Streaming Inference of High-Dimensional Physical Dynamics via State Space Models

Inferring the evolution of high-dimensional and multi-modal (e.g., spatio-temporal) physical fields from irregular sparse measurements in real time is a fundamental challenge in science and engineering. Existing approaches, including diffusion-based generative models and functional tensor methods, typically operate in offline settings, depend on full temporal observations, or incur substantial inference cost. We propose StreamPhy, an end-to-end framework that enables efficient and accurate streaming inference of full-field physical dynamics from incoming irregular sparse measurements. The framework integrates a data-adaptive observation encoder that is robust to arbitrary observation patterns, a structured state-space model that supports memory-efficient online updates across irregular time intervals, and an expressive Functional Tensor Feature-wise Linear Modulation (FT-FiLM) decoder for continuous-field generation. We prove that FT-FiLM is more expressive than the functional Tucker model, admitting a richer function class for handling complex dynamics. Experiments on three representative physical systems under challenging sampling patterns show that StreamPhy consistently outperforms state-of-the-art baselines, with at least 48% improvement in accuracy and up to 20--100X faster inference than diffusion-based methods.

Panqi Chen, Yifan Sun, Shikai Fang +2

College of Information Science and Electronic Engineering, Zhejiang University · School of EECS,Oregon State University

Streaming Inference Spatiotemporal Fields Inference Dynamics State Space Models Streaming Nonlinear Dynamics

May 5, 2026cs.LG

PerFlow: Physics-Embedded Rectified Flow for Efficient Reconstruction and Uncertainty Quantification of Spatiotemporal Dynamics

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.

Hao Zhou, Rui Zhang, Han Wan +1

Gaoling School of Artificial Intelligence, Renmin University of China

Rectified Flow Spatiotemporal Fields Partial Differential Equations Spatiotemporal Uncertainty Quantification

May 5, 2026cs.LG

GRIFDIR: Graph Resolution-Invariant FEM Diffusion Models in Function Spaces over Irregular Domains

Score-based diffusion models in infinite-dimensional function spaces provide a mathematically principled framework for modelling function-valued data, offering key advantages such as resolution invariance and the ability to handle irregular discretisations. However, practical implementations have struggled to fully realise these benefits. Existing backbones like Fourier neural operators are often biased towards regular grids and fail to generalise to complex domain topologies. We propose a novel architecture for function-space diffusion models that represents generalised graph convolutional kernels as finite element functions, enabling the model to naturally handle unstructured meshes and complex geometries. We demonstrate the efficacy of our network architecture through a series of unconditional and conditional sampling experiments across diverse geometries, including non-convex and multiply-connected domains. Our results show that the proposed method maintains resolution invariance and achieves high fidelity in capturing functional distributions on non-trivial geometries.

James Rowbottom, Elizabeth L. Baker, Nick Huang +3

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK · Department of Applied Mathematics and Computer Science, Technical University of Denmark, Denmark · Department of Mathematics, Simon Fraser University, Canada +1

Diffusion Models Discretization Mesh Generation Graph Foundation Model Spatiotemporal Fields Fourier Neural Operators

May 5, 2026cs.LG

PODiff: Latent Diffusion in Proper Orthogonal Decomposition Space for Scientific Super-Resolution

Probabilistic super-resolution of high-dimensional spatial fields using diffusion models is often computationally prohibitive due to the cost of operating directly in pixel space. We propose PODiff, a structured conditional generative framework that performs diffusion in a fixed, variance-ordered Proper Orthogonal Decomposition (POD) coefficient space, exploiting the orthogonality of POD modes to impose an interpretable, variance-ordered latent geometry. This design enables efficient ensemble generation, preserves dominant spatial structure, and yields spatially interpretable, well-calibrated uncertainty at substantially lower computational cost. We evaluate PODiff on sea surface temperature downscaling over the West Australian coast and on a controlled advection-diffusion benchmark. PODiff achieves reconstruction accuracy comparable to pixel-space diffusion while requiring significantly less memory and producing more reliable uncertainty estimates than deterministic and Monte Carlo Dropout baselines.

Onkar Jadhav, Tim French, Matthew Rayson +1

School of Earth and Oceans, UWA Oceans Institute, University of Western Australia, Crawley, WA, Australia · School of Physics, Mathematics and Computing, Computer Science and Software Engineering, University of Western Australia, Crawley, WA, Australia.

Spatiotemporal Fields Latent Diffusion Model Dynamical Downscaling

Apr 26, 2026cs.LG

Learning Interpretable PDE Representations for Generative Reconstructions with Structured Sparsity

Scientific measurements are often bottlenecked by suboptimal conditions, whether that be noise, incomplete spatial coverage, or limited resolution, rendering accurate field reconstruction a difficult task. We introduce LatentPDE, a latent diffusion framework designed to simultaneously resolve sparse-observation reconstruction and super-resolution. While existing physics-guided diffusion models typically rely on soft loss penalties or uninterpretable representations, our approach enforces physical compliance by constructing an inherently interpretable latent space. Specifically, we parameterize the latent variables directly as the coefficients and source terms of an assumed governing PDE. In doing so, LatentPDE is able to reliably reconstruct dynamics across highly disparate and structured data gaps. Empirical results on diverse configurations demonstrate that our model achieves high-fidelity recovery at any desired resolution while also tracking the underlying predictive uncertainty.

Valerie Tsao, Nathaniel Chaney, Manolis Veveakis

Department of Civil & Environmental Engineering, Duke University, NC, USA.

Latent Diffusion Model Spatiotemporal Fields Partial Differential Equations Sparsity

Jan 28, 2026stat.ML

Leveraging Differentiable PDE Solvers for Semi-Neural Spatial Reconstruction From Sparse Measurements

Generating dense physical fields from sparse measurements is a fundamental question in sampling, signal processing, and many other applications. State-of-the-art approaches to this problem either rely on spatial statistics that ignore the governing physics, integrate the physics into a multiple-objective optimization process, or require examples of the complete, fully-resolved simulation state during training, which are frequently unavailable outside of synthetic benchmarks. Here, we present a novel alternative that leverages recent advances in the integration of numerical simulators with data-driven models. Namely, we propose a hybrid modeling pipeline that couples Radial Basis Function (RBF) reconstruction with a Neural Network (NN) correction and a Partial Differential Equation (PDE) solver, so that the numerical simulator itself is embedded directly in the training loop of the learned component. Notably, the NN is trained without assuming availability of examples of the fully-resolved simulation state. This is made possible by implementing the PDE solver so that it is end-to-end differentiable, allowing gradients to be backpropagated through the simulation step during training. This grey-box methodology is evaluated on three standard benchmarks from fluid mechanics, where it achieves superior results over statistical and machine-learning-based reconstruction methods.

Ofek Aloni, Barak Fishbain

Department of Mathematics, Technion, Israel · Department of Civil and Environmental Engineering, Technion, Israel

Neural Partial Differential Equation Solvers Partial Differential Equations Neural Reconstruction Spatiotemporal Fields Scalar Field Radial Basis Function

Jan 5, 2026cs.LG

Attention in Geometry: Scalable Spatial Modeling via Adaptive Density Fields and FAISS-Accelerated Kernels

Spatial computation in geographic systems increasingly requires query-conditioned, local, interpretable aggregation under metric constraints. Many classical approaches rely on global summation and treat approximation as an implementation concern, limiting interpretability and scalability at large scales. We propose the Adaptive Density Field (ADF), a geometric attention framework that formulates spatial aggregation as a query-conditioned, metric-induced attention operator in continuous space. Given a set of labelled spatial points with associated scalar scores, ADF defines a continuous intensity field over space. For a given query location, the field value is obtained via a local adaptive Gaussian kernel mixture centered on the query's nearest neighbors, where kernel bandwidths are modulated by point-specific scores to evaluate local aggregated influence. Additionally, approximate nearest-neighbor search is introduced, enabling scalable execution while preserving locality. The proposed ADF bridges concepts from adaptive kernel methods, classical GIS methods, and attention mechanisms by reinterpreting spatial influence as geometry-embedded attention, grounded in physical distance rather than learned latent projections. The proposed framework is formulation-level rather than algorithm-specific, allowing flexible kernel choices, score-to-bandwidth mappings, and approximation parameters. This approach provides a unifying perspective on spatial influence modeling that emphasizes structure, scalability, and geometric interpretability, with relevance to geographic information systems and spatial machine learning.

Zhaowen Fan, Yunxiang Han

School of Computer, SICHUAN UNIVERSITY, CHENGDU, CHINA, 610065

Spatiotemporal Fields Approximate Nearest-Neighbor Search

Jun 30, 2024cs.CL

LaSEr-Edit: Localized Span-level Error Editing with Energy-based Localization

As large language models (LLMs) are widely adopted in real-world applications, it has become critical to ensure LLMs satisfy safety constraints, such as non-toxicity and logical consistency, as well as task- and situation-specific constraints. Controlling the output through instructions is a simple and tempting approach; however, it remains brittle, is opaque in how it influences model behavior, and thus cannot reliably ensure constraint satisfaction. Moreover, most recent controlled text generation (CTG) methods require access to the internal components of language models--such as weights or logits--making them incompatible with popular API-based LLMs. In this work, we propose LaSEr-Edit, a constraint-satisfying text revision method that can be applied to any LLMs, black- or white-box. We first find that lightweight, task-specific energy-based models (EBMs) achieve error-localization performance competitive with or even better than that of much larger LLMs, while operating substantially faster. Based on this finding, we propose two variants of text revision methods that incorporate energy-based error localization: LaSEr-LLM Edit, which instructs an LLM to edit text given EBM-predicted error spans, and LaSEr-EBM Edit, which uses the EBM not only for localization but also for editing by reranking edit candidates. Through experiments in diverse single-constraint control tasks, we show that LaSEr-LLM Edit controls text better than plain LLM-based editing in most of the tasks. We also find that LaSEr-EBM Edit further improves the control performance of LaSEr-LLM Edit and achieves among the strongest controllability across all tasks. Furthermore, we find that LaSEr-Edit, especially LaSEr-EBM Edit, performs well even when multiple constraints are controlled simultaneously.

Hye Ryung Son, Saehee Eom, Mooho Song +1

Graduate School of Data Science Seoul National University · Georgia Institute of Technology

Large Language Model Safety Localization Spatiotemporal Fields