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

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 this http URL 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 this http URL 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.