Sparse Reconstruction by Direct Measurement-to-Field Generation in Function Space

Understanding physical systems often requires recovering their full spatial state from measurements available only at sparse locations. This reconstruction becomes challenging when physical boundaries are unknown, high-resolution training fields are unavailable, or important quantities cannot be measured directly. Sparse observations may also be consistent with multiple plausible states, leading to significant reconstruction uncertainty. Here we introduce Direct Measurement-to-Field Generation (\DMF{}), a functional rectified flow framework that learns how sparse measurements drive physical-field generation end to end. Instead of relying on grid-based rasterization or runtime guidance, \DMF{} jointly learns a sensor--global--sensor representation with the generative transport, dynamically fusing local grounding with global context to evaluate physical fields at arbitrary query coordinates. Across benchmarks involving unobserved boundaries, missing high-resolution training fields, and unmeasured physical quantities, \DMF{} outperforms alternative baselines by significantly reducing reconstruction errors and preserving the fidelity of physical properties. Furthermore, its stochastic ensemble generation reliably quantifies uncertainty while improving continuous ranked probability scores. These results extend sparse reconstruction from fixed-setting point estimation toward measurement-conditioned generation of physical fields under incomplete and changing information.