Learning Continuous Fields with Hierarchical Latent Modeling and Sparse Context Encoding

Reconstructing continuous physical fields from sparse sensor observations is a fundamental challenge in scientific machine learning, with applications in climate, atmosphere, and ocean sciences. Implicit neural representations (INRs) provide a powerful coordinate-based framework, but existing sparse-to-dense INR approaches struggle to represent the multiscale spatiotemporal variability of physical fields. We present $\text{HiLS}(\text{Hierarchical Latent modeling and Sparse context encoding})$, an INR architecture built on two key innovations: $\text{(i)}\text{Sparse Context Encoding (SCE)}$ that uses sparse observations together with sparse coordinates with a latent cross-attention mechanism to compress high-dimensional fields into a latent embedding, enabling accurate reconstruction; $\text{(ii)}\text{Hierarchical Latent Modeling (HLM)}$ learns diverse spatial structures through a hierarchy of blocks built on rational basis functions. We evaluated $\text{HiLS}$ in $64$ configurations that span four geophysical datasets, four reconstruction tasks with fixed or variable sensors and locations, three sparsity levels, and temporal extrapolation. $\text{HiLS}$ achieves the best average rank across all experiments. Ablation studies confirm the contribution of each architectural component.