# Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields

arXiv: https://arxiv.org/abs/2608.18004 · HTML: https://mlsanity.com/paper/2608.18004/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Flow matching, time series and science, one of 14 directions in Diffusion models.
- Builds on 30 of its 47 references that are in the map.
- First posted on arXiv 18 Aug 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 30)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
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- [Generative Modeling by Estimating Gradients of the Data Distribution](https://mlsanity.com/paper/1907.05600/) (2019)
- [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)
- [A Baseline for Detecting Misclassified and Out-of-Distribution Examples in Neural Networks](https://mlsanity.com/paper/1610.02136/) (2016)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [Energy-based Out-of-distribution Detection](https://mlsanity.com/paper/2010.03759/) (2020)
- [Watch your Up-Convolution: CNN Based Generative Deep Neural Networks are Failing to Reproduce Spectral Distributions](https://mlsanity.com/paper/2003.01826/) (2020)
- [Practical and Asymptotically Exact Conditional Sampling in Diffusion Models](https://mlsanity.com/paper/2306.17775/) (2023)
- [Reduce, Reuse, Recycle: Compositional Generation with Energy-Based Diffusion Models and MCMC](https://mlsanity.com/paper/2302.11552/) (2023)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Learning Energy-Based Models by Diffusion Recovery Likelihood](https://mlsanity.com/paper/2012.08125/) (2020)
- [On the Anatomy of MCMC-Based Maximum Likelihood Learning of Energy-Based Models](https://mlsanity.com/paper/1903.12370/) (2019)
- [FlowDPS: Flow-Driven Posterior Sampling for Inverse Problems](https://mlsanity.com/paper/2503.08136/) (2025)
- [Functional Flow Matching](https://mlsanity.com/paper/2305.17209/) (2023)
- [Statistical Efficiency of Score Matching: The View from Isoperimetry](https://mlsanity.com/paper/2210.00726/) (2022)
- [Physics-Constrained Flow Matching: Sampling Generative Models with Hard Constraints](https://mlsanity.com/paper/2506.04171/) (2025)
- [From Zero to Turbulence: Generative Modeling for 3D Flow Simulation](https://mlsanity.com/paper/2306.01776/) (2023)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [Learning Iterative Reasoning through Energy Diffusion](https://mlsanity.com/paper/2406.11179/) (2024)
- [Deep Directed Generative Models with Energy-Based Probability Estimation](https://mlsanity.com/paper/1606.03439/) (2016)
- [Generative Latent Neural PDE Solver using Flow Matching](https://mlsanity.com/paper/2503.22600/) (2025)
- [Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling](https://mlsanity.com/paper/2504.10612/) (2025)
- [Learning Energy-Based Models in High-Dimensional Spaces with Multi-scale Denoising Score Matching](https://mlsanity.com/paper/1910.07762/) (2019)
- [Chance-constrained Flow Matching for High-Fidelity Constraint-aware Generation](https://mlsanity.com/paper/2509.25157/) (2025)
- [Energy-based models for inverse imaging problems](https://mlsanity.com/paper/2507.12432/) (2025)
- [Energy-Based Flow Matching for Generating 3D Molecular Structure](https://mlsanity.com/paper/2508.18949/) (2025)
- [Flow marching for a generative PDE foundation model](https://mlsanity.com/paper/2509.18611/) (2025)
- [An energy-based model approach to rare event probability estimation](https://mlsanity.com/paper/2310.04082/) (2023)

## Abstract (arXiv)

Probabilistic modeling of physical fields benefits from both a data-driven prior and known physical structure such as the governing equations. Energy-based models (EBMs) are a natural fit since energies compose additively, which enables augmenting physics information during inference. However, EBMs have been difficult to train and sample from due to the intractable partition function. We show in this work that flow matching models with a potential-induced velocity yield an explicit scalar energy at all transport times, whose gradient is exactly the converted learned score and which recovers the marginal negative log-density at the population optimum. The time-dependent energy functions are obtained purely from the matching regression objective on an independent linear Gaussian interpolation, without a variational form or additional MCMC steps, and the sampling retains the flow ODE. Access to the energy function from a trained model serves three roles: energy-corrected data generation, energy as a scoring function for out-of-distribution (OOD) detection, and energy compositional posterior sampling for inverse problems. In particular, we show the explicit energy permits general MCMC samplers in the predictor-corrector sampling framework, reducing PDE residual and spectral distance compared to the flow ODE baseline. Furthermore, we demonstrate utilizing the data energy and physics-based energy (e.g., PDE residuals) as complementary mechanisms to improve detection accuracy for OOD tasks. In addition, we explore the connection to MCMC-based inference for inverse problems by composing the energy with a quadratic observational likelihood that yields a posterior energy, used as an explicitly chosen family of inference-time targets.