# Data-to-Energy Stochastic Dynamics arXiv: https://arxiv.org/abs/2509.26364 ยท HTML: https://mlsanity.com/paper/2509.26364/ ## Key facts - Cited by 6 papers in the map (data through 14 Oct 2026). - Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models. - Builds on 43 of its 65 references that are in the map. - First posted on arXiv 30 Sep 2025. ## Placed in - Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/) - Deep reinforcement learning methods / [RL with diffusion and flow models](https://mlsanity.com/deeprl/rl-with-diffusion-and-flow-models/) ## Builds on (most cited first, 30 of 43) - [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020) - [High-Resolution Image Synthesis with Latent Diffusion Models](https://mlsanity.com/paper/2112.10752/) (2021) - [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014) - [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020) - [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021) - [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) - [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016) - [Flow Straight and Fast: Learning to Generate and Transfer Data with Rectified Flow](https://mlsanity.com/paper/2209.03003/) (2022) - [Analyzing and Improving the Image Quality of StyleGAN](https://mlsanity.com/paper/1912.04958/) (2019) - [Spectral Normalization for Generative Adversarial Networks](https://mlsanity.com/paper/1802.05957/) (2018) - [DiffWave: A Versatile Diffusion Model for Audio Synthesis](https://mlsanity.com/paper/2009.09761/) (2020) - [Alias-Free Generative Adversarial Networks](https://mlsanity.com/paper/2106.12423/) (2021) - [Stochastic Interpolants: A Unifying Framework for Flows and Diffusions](https://mlsanity.com/paper/2303.08797/) (2023) - [Large Language Diffusion Models](https://mlsanity.com/paper/2502.09992/) (2025) - [Improving and generalizing flow-based generative models with minibatch optimal transport](https://mlsanity.com/paper/2302.00482/) (2023) - [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021) - [Movie Gen: A Cast of Media Foundation Models](https://mlsanity.com/paper/2410.13720/) (2024) - [WaveGrad: Estimating Gradients for Waveform Generation](https://mlsanity.com/paper/2009.00713/) (2020) - [Diffusion Schr"odinger Bridge with Applications to Score-Based Generative Modeling](https://mlsanity.com/paper/2106.01357/) (2021) - [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013) - [Trajectory balance: Improved credit assignment in GFlowNets](https://mlsanity.com/paper/2201.13259/) (2022) - [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021) - [Path Integral Sampler: a stochastic control approach for sampling](https://mlsanity.com/paper/2111.15141/) (2021) - [Neural Stochastic Differential Equations: Deep Latent Gaussian Models in the Diffusion Limit](https://mlsanity.com/paper/1905.09883/) (2019) - [Denoising Diffusion Samplers](https://mlsanity.com/paper/2302.13834/) (2023) - [An optimal control perspective on diffusion-based generative modeling](https://mlsanity.com/paper/2211.01364/) (2022) - [Solving Schr"odinger Bridges via Maximum Likelihood](https://mlsanity.com/paper/2106.02081/) (2021) - [The Diffusion Duality](https://mlsanity.com/paper/2506.10892/) (2025) - [Neural Optimal Transport](https://mlsanity.com/paper/2201.12220/) (2022) ## Built on by (most cited first, 6 of 6) - [FLAC: Maximum Entropy RL via Kinetic Energy Regularized Bridge Matching](https://mlsanity.com/paper/2602.12829/) (2026) - [Discrete diffusion samplers and bridges: Off-policy algorithms and applications in latent spaces](https://mlsanity.com/paper/2602.05961/) (2026) - [LatentFlow: A General Framework for Conditioning Stochastic Processes](https://mlsanity.com/paper/2607.12922/) (2026) - [Entropy Across the Bridge: Conditional-Marginal Discretization for Flow and Schr"odinger Samplers](https://mlsanity.com/paper/2605.16126/) (2026) - [Tilted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2609.34642/) (2026) - [Amortised Inference through One-Step Implicit Sampling](https://mlsanity.com/paper/2610.12679/) (2026) ## Abstract (arXiv) The Schr"odinger bridge problem is concerned with finding a stochastic dynamical system bridging two marginal distributions that minimises a certain transportation cost. This problem, which represents a generalisation of optimal transport to the stochastic case, has received attention due to its connections to diffusion models and flow matching, as well as its applications in the natural sciences. However, all existing algorithms allow to infer such dynamics only for cases where samples from both distributions are available. In this paper, we propose the first general method for modelling Schr"odinger bridges when one (or both) distributions are given by their unnormalised densities, with no access to data samples. Our algorithm relies on a generalisation of the iterative proportional fitting (IPF) procedure to the data-free case, inspired by recent developments in off-policy reinforcement learning for training of diffusion samplers. We demonstrate the efficacy of the proposed data-to-energy IPF on synthetic problems, finding that it can successfully learn transports between multimodal distributions. As a secondary consequence of our reinforcement learning formulation, which assumes a fixed time discretisation scheme for the dynamics, we find that existing data-to-data Schr"odinger bridge algorithms can be substantially improved by learning the diffusion coefficient of the dynamics. Finally, we apply the newly developed algorithm to the problem of sampling posterior distributions in latent spaces of generative models, thus creating a data-free image-to-image translation method. Code: https://github.com/mmacosha/d2e-stochastic-dynamics