Uncertainty-Calibrated Spatiotemporal Field Diffusion with Sparse Supervision

TL;DR

This paper introduces SOLID, a diffusion-based framework that accurately reconstructs and forecasts physical fields from sparse, time-varying sensor data, providing well-calibrated uncertainty estimates without dense field training.

Contribution

SOLID is a novel end-to-end diffusion model trained solely on sparse observations, enabling reliable spatiotemporal field reconstruction and uncertainty quantification without dense data or pre-imputation.

Findings

Achieves up to tenfold reduction in probabilistic error.

Provides calibrated uncertainty maps with  > 0.7.

Operates effectively under severe sparsity conditions.

Abstract

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 (\r{ho} > 0.7) under severe sparsity.