# Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling

arXiv: https://arxiv.org/abs/2410.07974 · HTML: https://mlsanity.com/paper/2410.07974/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 84 references that are in the map.
- First posted on arXiv 10 Oct 2024.

## Builds on (most cited first, 27 of 27)

- [Flow Matching for Generative Modeling](https://mlsanity.com/paper/2210.02747/) (2022)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [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)
- [Likelihood Training of Schr"odinger Bridge using Forward-Backward SDEs Theory](https://mlsanity.com/paper/2110.11291/) (2021)
- [Diffusion Schr"odinger Bridge Matching](https://mlsanity.com/paper/2303.16852/) (2023)
- [Diffusion-based Molecule Generation with Informative Prior Bridges](https://mlsanity.com/paper/2209.00865/) (2022)
- [Let us Build Bridges: Understanding and Extending Diffusion Generative Models](https://mlsanity.com/paper/2208.14699/) (2022)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Action Matching: Learning Stochastic Dynamics from Samples](https://mlsanity.com/paper/2210.06662/) (2022)
- [Diffusion Bridge Mixture Transports, Schr"odinger Bridge Problems and Generative Modeling](https://mlsanity.com/paper/2304.00917/) (2023)
- [DeePMD-kit: A deep learning package for many-body potential energy representation and molecular dynamics](https://mlsanity.com/paper/1712.03641/) (2017)
- [Non-Denoising Forward-Time Diffusions](https://mlsanity.com/paper/2312.14589/) (2023)
- [SE(3) Equivariant Graph Neural Networks with Complete Local Frames](https://mlsanity.com/paper/2110.14811/) (2021)
- [Simulating Diffusion Bridges with Score Matching](https://mlsanity.com/paper/2111.07243/) (2021)
- [A Computational Framework for Solving Wasserstein Lagrangian Flows](https://mlsanity.com/paper/2310.10649/) (2023)
- [Generalized Schr"odinger Bridge Matching](https://mlsanity.com/paper/2310.02233/) (2023)
- [From Molecules to Materials: Pre-training Large Generalizable Models for Atomic Property Prediction](https://mlsanity.com/paper/2310.16802/) (2023)
- [Stochastic Optimal Control for Collective Variable Free Sampling of Molecular Transition Paths](https://mlsanity.com/paper/2207.02149/) (2022)
- [Aligned Diffusion Schr"odinger Bridges](https://mlsanity.com/paper/2302.11419/) (2023)
- [A reinforcement learning approach to rare trajectory sampling](https://mlsanity.com/paper/2005.12890/) (2020)
- [Augmented Bridge Matching](https://mlsanity.com/paper/2311.06978/) (2023)
- [Reinforcement learning of rare diffusive dynamics](https://mlsanity.com/paper/2105.04321/) (2021)
- [Molecular Latent Space Simulators](https://mlsanity.com/paper/2007.00728/) (2020)
- [Transition Path Sampling with Boltzmann Generator-based MCMC Moves](https://mlsanity.com/paper/2312.05340/) (2023)
- [Generative methods for sampling transition paths in molecular dynamics](https://mlsanity.com/paper/2205.02818/) (2022)
- [Conditioning Normalizing Flows for Rare Event Sampling](https://mlsanity.com/paper/2207.14530/) (2022)

## Built on by (most cited first, 27 of 27)

- [No Trick, No Treat: Pursuits and Challenges Towards Simulation-free Training of Neural Samplers](https://mlsanity.com/paper/2502.06685/) (2025)
- [SDE Matching: Scalable and Simulation-Free Training of Latent Stochastic Differential Equations](https://mlsanity.com/paper/2502.02472/) (2025)
- [Consistent Sampling and Simulation: Molecular Dynamics with Energy-Based Diffusion Models](https://mlsanity.com/paper/2506.17139/) (2025)
- [Trust Region Constrained Measure Transport in Path Space for Stochastic Optimal Control and Inference](https://mlsanity.com/paper/2508.12511/) (2025)
- [Action-Minimization Meets Generative Modeling: Efficient Transition Path Sampling with the Onsager-Machlup Functional](https://mlsanity.com/paper/2504.18506/) (2025)
- [Conditioning Diffusions Using Malliavin Calculus](https://mlsanity.com/paper/2504.03461/) (2025)
- [Conditional Diffusion Guidance under Hard Constraint: A Stochastic Analysis Approach](https://mlsanity.com/paper/2602.05533/) (2026)
- [ConfRover: Simultaneous Modeling of Protein Conformation and Dynamics via Autoregression](https://mlsanity.com/paper/2505.17478/) (2025)
- [UniSim: A Unified Simulator for Time-Coarsened Dynamics of Biomolecules](https://mlsanity.com/paper/2506.03157/) (2025)
- [The Spacetime of Diffusion Models: An Information Geometry Perspective](https://mlsanity.com/paper/2505.17517/) (2025)
- [Control Consistency Losses for Diffusion Bridges](https://mlsanity.com/paper/2512.05070/) (2025)
- [Generalized Flow Matching for Transition Dynamics Modeling](https://mlsanity.com/paper/2410.15128/) (2024)
- [Functional Adjoint Sampler: Scalable Sampling on Infinite Dimensional Spaces](https://mlsanity.com/paper/2511.06239/) (2025)
- [Implicit Neural Representations for Chemical Reaction Paths](https://mlsanity.com/paper/2502.15843/) (2025)
- [Transferable Learning of Reaction Pathways from Geometric Priors](https://mlsanity.com/paper/2504.15370/) (2025)
- [FragmentFlow: Scalable Transition State Generation for Large Molecules](https://mlsanity.com/paper/2602.02310/) (2026)
- [Flows and Diffusions on the Neural Manifold](https://mlsanity.com/paper/2507.10623/) (2025)
- [Entangled Schr"odinger Bridge Matching](https://mlsanity.com/paper/2511.07406/) (2025)
- [Analytic Bridge Diffusions for Controlled Path Generation](https://mlsanity.com/paper/2605.02961/) (2026)
- [Intrinsic-Noise Consolidation: A Doob-Barrier-Conditioned Diffusion Turns Analog Device Noise into a Continual-Learning Resource](https://mlsanity.com/paper/2607.06924/) (2026)
- [Twisted Schr"odinger Bridge Matching](https://mlsanity.com/paper/2607.16987/) (2026)
- [Follow the MEP: Scalable Neural Representations for Minimum-Energy Path Discovery in Molecular Systems](https://mlsanity.com/paper/2504.16381/) (2025)
- [Machine-Learned Sampling of Conditioned Path Measures](https://mlsanity.com/paper/2506.01904/) (2025)
- [Stochastic Control Policies for Robust Molecular Transition Path Sampling](https://mlsanity.com/paper/2608.13800/) (2026)
- [Machine learning kinetics from molecular dynamics data](https://mlsanity.com/paper/2609.17736/) (2026)
- [Gaussian Flow Dynamics: Simulation-Free Neural SDE Learning Beyond One-Time Marginals](https://mlsanity.com/paper/2610.04390/) (2026)
- [Langevin Flow Maps: Efficient Molecular Dynamics and Transition Path Sampling](https://mlsanity.com/paper/2610.05998/) (2026)

## Abstract (arXiv)

Rare event sampling in dynamical systems is a fundamental problem arising in the natural sciences, which poses significant computational challenges due to an exponentially large space of trajectories. For settings where the dynamical system of interest follows a Brownian motion with known drift, the question of conditioning the process to reach a given endpoint or desired rare event is definitively answered by Doob's h-transform. However, the naive estimation of this transform is infeasible, as it requires simulating sufficiently many forward trajectories to estimate rare event probabilities. In this work, we propose a variational formulation of Doob's h-transform as an optimization problem over trajectories between a given initial point and the desired ending point. To solve this optimization, we propose a simulation-free training objective with a model parameterization that imposes the desired boundary conditions by design. Our approach significantly reduces the search space over trajectories and avoids expensive trajectory simulation and inefficient importance sampling estimators which are required in existing methods. We demonstrate the ability of our method to find feasible transition paths on real-world molecular simulation and protein folding tasks.