Girsanov Reweighting for Uncertainty Propagation in Rare-Event Kinetics

Technical Summary: Girsanov Reweighting for Uncertainty Propagation in Rare-Event Kinetics

Problem Statement
Machine-learning interatomic potentials (MLIPs) have emerged as a powerful tool for sampling rare events in molecular dynamics (MD), offering near ab initio accuracy at a fraction of the computational cost. However, the uncertainty associated with these models remains a significant challenge. Existing uncertainty quantification (UQ) approaches predominantly focus on point-wise quantities (energies and forces) or equilibrium thermodynamic observables. They fail to address the propagation of parameter uncertainties to kinetic observables, such as the averaged committor probability $\langle p_{A\to B} \rangle_{\lambda_{\partial A}}$ and the resulting reaction rate $k_{AB}$. Directly propagating these uncertainties via standard Monte Carlo methods—sampling new parameter sets and re-running expensive rare-event simulations for each realization—is computationally prohibitive.

Methodology
The authors propose a framework that couples rare-event sampling methods, specifically Adaptive Multilevel Splitting (AMS), with Girsanov reweighting to propagate MLIP parameter uncertainty to kinetic observables without resampling trajectories.

1. Theoretical Framework: The dynamics are modeled using underdamped Langevin equations. The rare-event probability $p$ is defined as the committor probability averaged over the exit distribution $\lambda_{\partial A}$ on the boundary of the reactant state $A$.
2. Uncertainty Quantification (UQ): The study utilizes the Point-wise Optimal Parameter Set (POPS) framework to generate a posterior distribution $\pi(d\theta)$ over the MLIP parameters $\theta$. This approach is chosen to account for model misspecification errors, which standard Bayesian regression often fails to capture due to the near-deterministic nature of ab initio training data.
3. Girsanov Reweighting: The core of the methodology relies on the Girsanov theorem to compute the likelihood ratio between path-space probability measures under a reference potential $V(x, \theta_L)$ and a perturbed potential $V(x, \theta)$.
   • Full Estimator ($\hat{p}_G$): An exact estimator derived by reweighting the ensemble of reactive trajectories sampled at $\theta_L$ using the likelihood ratio $M(X, \theta_L, \theta)$. This ratio depends on the difference in forces and the initial state distribution.
   • Cumulant Estimator ($\hat{p}_C$): To address the computational cost of calculating the Fisher Information Matrix (FIM) required for the full estimator (especially for deep learning models like MACE), the authors derive a first-order cumulant expansion. This approximation relies on the empirical mean of the "score" vector $s(X)$, significantly reducing computational overhead while maintaining numerical stability.
4. Reaction Rate Extension: The framework extends to the reaction rate $k_{AB}$ via Hill's relation ($k_{AB} = \phi_A \langle p_{A\to B} \rangle_{\lambda_{\partial A}}$). The authors argue that under the assumption that MLIPs are highly accurate within metastable basins (where the exit frequency $\phi_A$ is sampled), the uncertainty in $k_{AB}$ is dominated by the uncertainty in the committor probability $p$.

Key Contributions

• Path-Space Reweighting for Kinetics: The paper introduces the first application of Girsanov reweighting to propagate parametric uncertainty specifically to kinetic observables (committor probabilities), bridging a gap previously limited to thermodynamic quantities.
• Efficient Estimators: The derivation of the first-order cumulant estimator ($\hat{p}_C$) allows for the analytical propagation of uncertainty in high-dimensional parameter spaces without the prohibitive cost of computing full path-space Jacobians or re-running simulations.
• Validation on Complex Systems: The methodology is validated on three distinct systems: a dimer in a Weeks-Chandler-Andersen (WCA) solvent, a rugged Müller-Brown potential, and the conformational transition of butane using MACE foundation models.

Results

• Sensitivity Analysis: On the dimer and Müller-Brown systems, both the full Girsanov estimator and the cumulant estimator accurately captured the sensitivity of the rare-event probability to parameter variations, with the cumulant estimator showing excellent coverage even for substantial parameter deviations.
• Butane Conformational Dynamics: In the application to butane using MACE models, the framework successfully distinguished between algorithmic (epistemic) noise and model misspecification uncertainty.
  • For the more accurate MACE-OMAT-0 model, the uncertainty-aware distribution remained centered near the Maximum Likelihood Estimate (MLE), with the reference value well-covered.
  • For the less accurate MACE-MP-0a model, the framework correctly identified significant model bias, resulting in a wider uncertainty distribution that still encompassed the reference "ground truth" (MACE-MPA-0).
• Computational Efficiency: The reweighting approach demonstrated a speedup of approximately 4000$\times$ compared to the brute-force approach of re-running full AMS simulations for each parameter realization. The reweighting step itself added only a marginal computational cost (approx. 30 minutes) to the initial AMS run.
• Stability: The log-weights used in the reweighting process remained concentrated around zero (typically within $(-2, 2)$), confirming that the variance inflation typically associated with Girsanov reweighting was mitigated by the short duration of reactive trajectories and bounded potential differences.

Significance and Claims
The paper claims that path-space reweighting provides an efficient and rigorous route for propagating MLIP uncertainty to rare-event kinetics. By leveraging the unbiased nature of splitting estimators and the mathematical rigor of Girsanov's theorem, the framework enables the construction of uncertainty-aware probability distributions for reaction rates without the need for costly resampling.

The authors modestly note that the extension to reaction rates relies on the assumption that MLIPs are highly accurate within metastable basins, an assumption supported by active learning literature but requiring further investigation for complex reactions where the dividing surface might lie outside these regions. They conclude that this approach effectively isolates algorithmic variance from model misspecification, providing tight and reliable confidence intervals for kinetic observables derived from uncertain surrogate models.