Spatio-temporal sensor data is often sparse, noisy, and irregular, and existing interpolation or learning methods struggle here because they either ignore governing PDEs or do not scale. We introduce FieldFormer, a transformer-based framework for mesh-free spatio-temporal field reconstruction that combines data-driven flexibility with physics-based structure. For each query, FieldFormer gathers a local neighborhood using a learnable velocity-scaled distance metric, enabling anisotropic adaptation to different propagation regimes. Neighborhoods are built efficiently via per-batch offset recomputation, and refined in an expectation-maximization style as the velocity scales evolve. Predictions are made by a local transformer encoder, and physics consistency is enforced through autograd-based PDE residuals and boundary-specific penalties. Across three benchmarks--a scalar anisotropic heat equation, a vector-valued shallow-water system, and a realistic advection-diffusion pollution simulation--FieldFormer consistently outperforms strong baselines by more than 40%. Our results demonstrate that FieldFormer enables accurate (RMSE<10^-2), efficient, and physically consistent field reconstruction from sparse (0. 4%-2%) and noisy (10%) data.
Bhardwaj et al. (Sat,) studied this question.