PIS: A Generalized Physical Inversion Solver for Arbitrary Sparse Observations via Set-Conditioned Diffusion

Abstract

Estimation of PDE-constrained physical parameters from limited indirect measurements is inherently ill-posed, particularly when observations are sparse, irregular, and constrained by real-world sensor placement. This challenge is ubiquitous in fields such as fluid mechanics, seismic inversion, and structural health monitoring. Existing deep and operator-learning models collapse under these conditions: fixed-grid assumptions fail, reconstruction deteriorates sharply, and inversion becomes unreliable with limited robustness and no uncertainty quantification (UQ).We propose the Physical Inversion Solver (PIS), a set-conditioned diffusion framework enabling inversion from truly arbitrary observation sets. PIS employs a Set Transformer-based encoder to handle measurements of any number or geometry, and a cosine-annealed sparsity curriculum for exceptional robustness. An accompanying information-theoretic analysis provides insight into the limits of inversion under extreme sparsity by revealing how observation entropy varies across physical systems.PIS is evaluated on three challenging PDE inverse problems: Darcy flow, wavefield inversion (Helmholtz), and structural health monitoring (Hooke's Law). Across all tasks and sparsity regimes -- including extreme cases with an observation rate of only $0.29\%$ -- existing operator-learning baselines fail to reconstruct meaningful fields, often diverging or collapsing entirely.In stark contrast, PIS remains stable and accurate, reducing inversion error by $12.28\%$--$88.73\%$ and reliably producing calibrated posterior samples. These samples accurately reflect both data scarcity and intrinsic physical ambiguity. These results position PIS as a powerful, general-purpose, and uniquely sparsity-resilient solution for physical inversion under arbitrary and severely undersampled observations.