# PIS: A Generalized Physical Inversion Solver for Arbitrary Sparse Observations via Set Conditioned Flow Matching

arXiv: https://arxiv.org/abs/2512.13732 · HTML: https://mlsanity.com/paper/2512.13732/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 63 references that are in the map.
- First posted on arXiv 14 Dec 2025.

## Builds on (most cited first, 21 of 21)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023)
- [Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1505.05424/) (2015)
- [Deep Convolutional Neural Network for Inverse Problems in Imaging](https://mlsanity.com/paper/1611.03679/) (2016)
- [Attentive Neural Processes](https://mlsanity.com/paper/1901.05761/) (2019)
- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models](https://mlsanity.com/paper/2305.19836/) (2023)
- [Likelihood-informed dimension reduction for nonlinear inverse problems](https://mlsanity.com/paper/1403.4680/) (2014)

## Abstract (arXiv)

The estimation of high-dimensional physical parameters constrained by partial differential equations (PDEs) from limited and indirect measurements is a highly ill-posed problem. Traditional methods face significant accuracy and efficiency bottlenecks, particularly when observations are sparse, irregularly sampled, and constrained by real-world sensor placement. We propose the Physical Inversion Solver (PIS), a unified framework that couples Set-Conditioned Flow Matching with a Cosine-Annealed Sparsity Curriculum (CASC) to enable stable inversion from arbitrary, off-grid sensors even under minimal guidance. By leveraging straight-path transport, PIS achieves instantaneous inference (50 NFEs), offering orders-of-magnitude speedup over iterative baselines. Extensive experiments demonstrate that PIS reduces error by up to 88.7% under extreme sparsity (<1%) across subsurface characterization, wave-based characterization, and structural health monitoring, while providing robust uncertainty quantification for optimal sensor placement.