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Doob's Lagrangian: A Sample-Efficient Variational Approach to Transition Path Sampling
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Proposes a simulation-free variational Doob's h-transform objective with a boundary-respecting Gaussian (mixture) path parameterization for transition path sampling in Brownian/Langevin dynamics.
Problem
Transition path sampling under rare-event conditioning requires simulating many trajectories to estimate Doob's h-transform, which is extremely sample-inefficient when the target endpoint is rarely reached.
Approach
Formulate Doob's h-transform as a variational least-action (Lagrangian) problem over endpoint-conditioned marginals, parameterized by a neural-network Gaussian (or mixture) path that satisfies boundary conditions by construction, enabling simulation-free training with analytic drift from the Fokker-Planck constraint.
Key results
- On Müller-Brown, the method uses 1.28M potential evaluations versus 1.03B for fixed-length MCMC while reaching a similar mean log-likelihood (858.50 vs 866.56).
- On alanine dipeptide, the method (internal coordinates, 5 mixtures) reaches a MinMax energy of -15.95 with 38.4M-51.2M evaluations, where exact-state MCMC is intractable (>10B evaluations).
- On Chignolin, the Cartesian model samples a feasible folding transition within 25.6M potential energy evaluations.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use a simulation-free variational path method instead of two-way MCMC shooting | Müller-Brown potential transition path sampling · Müller-Brown potential | potential evaluations: 1.03B -> 1.28M | Helped |
| use a simulation-free variational path method instead of two-way MCMC shooting | Müller-Brown potential transition path sampling · Müller-Brown potential | Log-Likelihood: 866.56 $\pm$ 17.00 -> 858.50 $\pm$ 17.61 | No clear effect |
| use internal coordinates instead of Cartesian coordinates | Doob's Lagrangian · Alanine dipeptide (exact) | Max Energy: 726.40 $\pm$ 0.07 -> -14.62 $\pm$ 0.02 | Helped |
| add Gaussian mixture paths (ablation) | Doob's Lagrangian in internal coordinates · Alanine dipeptide (exact) | MinMax Energy: -14.67 -> -15.54; -15.95 | Helped |
| add Gaussian mixture paths (ablation) | Doob's Lagrangian in Cartesian coordinates · Alanine dipeptide (exact) | MinMax Energy: 726.18 -> 513.72; 247.96 | Helped |
| replace the neural-network parameterization of Gaussian statistics with piecewise linear splines (ablation) | Doob's Lagrangian · synthetic transition path learning | W1 distance of learned marginals | Hurt |
| use a single Gaussian path instead of Gaussian mixture paths (ablation) | Doob's Lagrangian · symmetric high-energy-barrier potential | ability to model multiple transition path modes | Hurt |
Surprising findings
- Spline parameterizations of the marginals were inferior to the neural network parameterization even after an order of magnitude more potential evaluations.
Limitations
- Endpoints A and B are restricted to point masses with Gaussian noise rather than arbitrary sets.
- Fixed-length paths; variable-length paths not supported.
- Relies on AMBER14/OpenMM energy evaluations for molecular systems, with cost dominated by energy calls.
Experimental setup
Datasets: Müller-Brown potential alanine dipeptide (AMBER14 force field) Chignolin
Benchmarks: Müller-Brown potential alanine dipeptide Chignolin
Metrics: number of potential evaluations maximum energy minmax energy log-likelihood of paths
Baselines: two-way shooting MCMC variable-length MCMC fixed-length MCMC
Builds on: Doob's h-transform action matching conditional flow matching bridge matching
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