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

• URL: http://arxiv.org/abs/2608.18004v1
• Date: Tue, 18 Aug 2026 16:53:08 GMT
• Title: Composing Flow-Matching Energies with Known Physics: Generation, OOD Detection, and Inversion on PDE Fields
• Abstract summary: Energy-based models (EBMs) are a natural fit since energies compose additively, which enables augmenting physics information during inference.<n>We show in this work that flow matching models with a potential-induced velocity yield an explicit energy at all transport times.<n>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.
• Score: 9.34545560117171
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Generating Physically Consistent Molecules with Energy-Based Models [6.4105959373235875]
  We introduce EBMol, an energy-based model that restores inductive bias by learning an atom-additive scalar potential without explicit simulation.<n>EBMol is the first EBM for 3D molecular generation to achieve state-of-the-art performance on QM9 and GEOM-Drugs.
  arXiv  Detail & Related papers  (2026-05-18T13:26:52Z)
• EFF-Grasp: Energy-Field Flow Matching for Physics-Aware Dexterous Grasp Generation [54.09734511705173]
  EFF-Grasp is a Flow-Matching synthesis-based framework for physics-aware dexterous grasp generation.<n>We reformulate grasp as a deterministic ordinary differential equation (ODE) process, which enables efficient and stable generation through smooth probability flows.<n> EFF-Grasp achieves superior performance in grasp quality and physical feasibility, while requiring substantially fewer sampling steps than diffusion-based baselines.
  arXiv  Detail & Related papers  (2026-03-17T06:11:56Z)
• Energy-Guided Flow Matching Enables Few-Step Conformer Generation and Ground-State Identification [45.52894539097255]
  We present EnFlow, a unified framework that couples flow matching with an explicitly learned energy model.<n>By incorporating energy-gradient guidance during sampling, our method steers trajectories toward lower-energy regions.<n>The learned energy function further enables efficient energy-based ranking of generated ensembles for accurate ground-state identification.
  arXiv  Detail & Related papers  (2025-12-27T14:00:22Z)
• Energy Matching: Unifying Flow Matching and Energy-Based Models for Generative Modeling [5.787270247104665]
  Current state-of-the-art generative models map noise to data distributions by matching flows or scores.<n>Here, we propose Energy Matching, a framework that endows flow-based approaches with the flexibility of EBMs.
  arXiv  Detail & Related papers  (2025-04-14T18:10:58Z)
• Energy-Weighted Flow Matching for Offline Reinforcement Learning [53.64306385597818]
  This paper investigates energy guidance in generative modeling, where the target distribution is defined as $q(mathbf x) propto p(mathbf x)exp(-beta mathcal E(mathcal x))$, with $p(mathbf x)$ being the data distribution and $mathcal E(mathcal x)$ as the energy function.<n>We introduce energy-weighted flow matching (EFM), a method that directly learns the energy-guided flow without the need for auxiliary models.<n>We extend this methodology to energy-weighted
  arXiv  Detail & Related papers  (2025-03-06T21:10:12Z)
• Iterated Energy-based Flow Matching for Sampling from Boltzmann Densities [11.850515912491657]
  We propose iterated energy-based flow matching (iEFM) to train continuous normalizing flow (CNF) models from unnormalized densities. Our results demonstrate that iEFM outperforms existing methods, showcasing its potential for efficient and scalable probabilistic modeling.
  arXiv  Detail & Related papers  (2024-08-29T04:06:34Z)
• Variational Potential Flow: A Novel Probabilistic Framework for Energy-Based Generative Modelling [10.926841288976684]
  We present a novel energy-based generative framework, Variational Potential Flow (VAPO) VAPO aims to learn a potential energy function whose gradient (flow) guides the prior samples, so that their density evolution closely follows an approximate data likelihood homotopy. Images can be generated after training the potential energy, by initializing the samples from Gaussian prior and solving the ODE governing the potential flow on a fixed time interval.
  arXiv  Detail & Related papers  (2024-07-21T18:08:12Z)
• On Feature Diversity in Energy-based Models [98.78384185493624]
  An energy-based model (EBM) is typically formed of inner-model(s) that learn a combination of the different features to generate an energy mapping for each input configuration. We extend the probably approximately correct (PAC) theory of EBMs and analyze the effect of redundancy reduction on the performance of EBMs.
  arXiv  Detail & Related papers  (2023-06-02T12:30:42Z)
• Sampling with Mollified Interaction Energy Descent [57.00583139477843]
  We present a new optimization-based method for sampling called mollified interaction energy descent (MIED) MIED minimizes a new class of energies on probability measures called mollified interaction energies (MIEs) We show experimentally that for unconstrained sampling problems our algorithm performs on par with existing particle-based algorithms like SVGD.
  arXiv  Detail & Related papers  (2022-10-24T16:54:18Z)
• An Energy-Based Prior for Generative Saliency [62.79775297611203]
  We propose a novel generative saliency prediction framework that adopts an informative energy-based model as a prior distribution. With the generative saliency model, we can obtain a pixel-wise uncertainty map from an image, indicating model confidence in the saliency prediction. Experimental results show that our generative saliency model with an energy-based prior can achieve not only accurate saliency predictions but also reliable uncertainty maps consistent with human perception.
  arXiv  Detail & Related papers  (2022-04-19T10:51:00Z)
• Particle Dynamics for Learning EBMs [83.59335980576637]
  Energy-based modeling is a promising approach to unsupervised learning, which yields many downstream applications from a single model. The main difficulty in learning energy-based models with the "contrastive approaches" is the generation of samples from the current energy function at each iteration. This paper proposes an alternative approach to getting these samples and avoiding crude MCMC sampling from the current model.
  arXiv  Detail & Related papers  (2021-11-26T23:41:07Z)
• Targeted free energy estimation via learned mappings [66.20146549150475]
  Free energy perturbation (FEP) was proposed by Zwanzig more than six decades ago as a method to estimate free energy differences. FEP suffers from a severe limitation: the requirement of sufficient overlap between distributions. One strategy to mitigate this problem, called Targeted Free Energy Perturbation, uses a high-dimensional mapping in configuration space to increase overlap.
  arXiv  Detail & Related papers  (2020-02-12T11:10:00Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.