# Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervision

arXiv: https://arxiv.org/abs/2603.04431 · HTML: https://mlsanity.com/paper/2603.04431/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 42 of its 66 references that are in the map.
- First posted on arXiv 17 Feb 2026.

## Builds on (most cited first, 30 of 42)

- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Rethinking Attention with Performers](https://mlsanity.com/paper/2009.14794/) (2020)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Conditional Neural Processes](https://mlsanity.com/paper/1807.01613/) (2018)
- [Neural Processes](https://mlsanity.com/paper/1807.01622/) (2018)
- [Scale-MAE: A Scale-Aware Masked Autoencoder for Multiscale Geospatial Representation Learning](https://mlsanity.com/paper/2212.14532/) (2022)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Spatial-Temporal Fusion Graph Neural Networks for Traffic Flow Forecasting](https://mlsanity.com/paper/2012.09641/) (2020)
- [Representing Long-Range Context for Graph Neural Networks with Global Attention](https://mlsanity.com/paper/2201.08821/) (2022)
- [Adversarial Generation of Continuous Images](https://mlsanity.com/paper/2011.12026/) (2020)
- [DYffusion: A Dynamics-informed Diffusion Model for Spatiotemporal Forecasting](https://mlsanity.com/paper/2306.01984/) (2023)
- [Score-based Diffusion Models in Function Space](https://mlsanity.com/paper/2302.07400/) (2023)
- [On conditional diffusion models for PDE simulations](https://mlsanity.com/paper/2410.16415/) (2024)
- [Frequency-Aligned Knowledge Distillation for Lightweight Spatiotemporal Forecasting](https://mlsanity.com/paper/2507.02939/) (2025)
- [Neural Stochastic PDEs: Resolution-Invariant Learning of Continuous Spatiotemporal Dynamics](https://mlsanity.com/paper/2110.10249/) (2021)
- [Neural Point Process for Learning Spatiotemporal Event Dynamics](https://mlsanity.com/paper/2112.06351/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [Learning Spatial-Temporal Implicit Neural Representations for Event-Guided Video Super-Resolution](https://mlsanity.com/paper/2303.13767/) (2023)
- [Conditional Local Convolution for Spatio-temporal Meteorological Forecasting](https://mlsanity.com/paper/2101.01000/) (2021)
- [Deep Generative Data Assimilation in Multimodal Setting](https://mlsanity.com/paper/2404.06665/) (2024)

## Abstract (arXiv)

Physical fields are typically observed only at sparse, time-varying sensor locations, making forecasting and reconstruction ill-posed and uncertainty-critical. We present SOLID, a mask-conditioned diffusion framework that learns spatiotemporal dynamics from sparse observations alone: training and evaluation use only observed target locations, requiring no dense fields and no pre-imputation. Unlike prior work that trains on dense reanalysis or simulations and only tests under sparsity, SOLID is trained end-to-end with sparse supervision only. SOLID conditions each denoising step on the measured values and their locations, and introduces a dual-masking objective that (i) emphasizes learning in unobserved void regions while (ii) upweights overlap pixels where inputs and targets provide the most reliable anchors. This strict sparse-conditioning pathway enables posterior sampling of full fields consistent with the measurements, achieving up to an order-of-magnitude improvement in probabilistic error and yielding calibrated uncertainty maps (rho > 0.7) under severe sparsity.