# Conditional flow matching for physics-constrained inverse problems with finite training data arXiv: https://arxiv.org/abs/2603.14135 ยท HTML: https://mlsanity.com/paper/2603.14135/ ## Key facts - Cited by 2 papers in the map (data through 14 Oct 2026). - Placed in Flow matching, time series and science, one of 14 directions in Diffusion models. - Builds on 33 of its 56 references that are in the map. - First posted on arXiv 14 Mar 2026. ## Placed in - Diffusion models / [Flow matching, time series and science](https://mlsanity.com/diffusion/flow-matching-time-series-and-science/) ## Builds on (most cited first, 30 of 33) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015) - [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022) - [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013) - [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016) - [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019) - [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014) - [Building Normalizing Flows with Stochastic Interpolants](https://mlsanity.com/paper/2209.15571/) (2022) - [Improving Diffusion Models for Inverse Problems using Manifold Constraints](https://mlsanity.com/paper/2206.00941/) (2022) - [Conditional Image Generation with Score-Based Diffusion Models](https://mlsanity.com/paper/2111.13606/) (2021) - [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024) - [Adversarial Regularizers in Inverse Problems](https://mlsanity.com/paper/1805.11572/) (2018) - [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023) - [DMPlug: A Plug-in Method for Solving Inverse Problems with Diffusion Models](https://mlsanity.com/paper/2405.16749/) (2024) - [CoCoGen: Physically-Consistent and Conditioned Score-based Generative Models for Forward and Inverse Problems](https://mlsanity.com/paper/2312.10527/) (2023) - [Flow Priors for Linear Inverse Problems via Iterative Corrupted Trajectory Matching](https://mlsanity.com/paper/2405.18816/) (2024) - [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025) - [Composing Normalizing Flows for Inverse Problems](https://mlsanity.com/paper/2002.11743/) (2020) - [Memorization and Regularization in Generative Diffusion Models](https://mlsanity.com/paper/2501.15785/) (2025) - [Stochastic Normalizing Flows for Inverse Problems: a Markov Chains Viewpoint](https://mlsanity.com/paper/2109.11375/) (2021) - [Solution of Physics-based Bayesian Inverse Problems with Deep Generative Priors](https://mlsanity.com/paper/2107.02926/) (2021) - [Flower: A Flow-Matching Solver for Inverse Problems](https://mlsanity.com/paper/2509.26287/) (2025) - [Regularising Inverse Problems with Generative Machine Learning Models](https://mlsanity.com/paper/2107.11191/) (2021) - [The efficacy and generalizability of conditional GANs for posterior inference in physics-based inverse problems](https://mlsanity.com/paper/2202.07773/) (2022) - [A dimension-reduced variational approach for solving physics-based inverse problems using generative adversarial network priors and normalizing flows](https://mlsanity.com/paper/2310.04690/) (2023) - [Physics-Constrained Fine-Tuning of Flow-Matching Models for Generation and Inverse Problems](https://mlsanity.com/paper/2508.09156/) (2025) ## Built on by (most cited first, 2 of 2) - [Probabilistic Forecasting of Localized Wildfire Spread Based on Conditional Flow Matching](https://mlsanity.com/paper/2603.26975/) (2026) - [Livin' on a Prior: Likelihood Score Approximation for Inverse Problems](https://mlsanity.com/paper/2609.34446/) (2026) ## Abstract (arXiv) This study presents a conditional flow matching framework for solving physics-constrained Bayesian inverse problems. In this setting, samples from the joint distribution of inferred variables and measurements are assumed available, while explicit evaluation of the prior and likelihood densities is not required. We derive a simple and self-contained formulation of both the unconditional and conditional flow matching algorithms, tailored specifically to inverse problems. In the conditional setting, a neural network is trained to learn the velocity field of a probability flow ordinary differential equation that transports samples from a chosen source distribution directly to the posterior distribution conditioned on observed measurements. This black-box formulation accommodates nonlinear, high-dimensional, and potentially non-differentiable forward models without restrictive assumptions on the noise model. We further analyze the behavior of the learned velocity field in the regime of finite training data. Under mild architectural assumptions, we show that overtraining can induce degenerate behavior in the generated conditional distributions, including variance collapse and a phenomenon termed selective memorization, wherein generated samples concentrate around training data points associated with similar observations. A simplified theoretical analysis explains this behavior, and numerical experiments confirm it in practice. We demonstrate that standard early-stopping criteria based on monitoring test loss effectively mitigate such degeneracy. The proposed method is evaluated on several physics-based inverse problems. We investigate the impact of different choices of source distributions, including Gaussian and data-informed priors. Across these examples, conditional flow matching accurately captures complex, multimodal posterior distributions while maintaining computational efficiency.