# From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach

arXiv: https://arxiv.org/abs/2406.09028 · HTML: https://mlsanity.com/paper/2406.09028/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 50 references that are in the map.
- First posted on arXiv 13 Jun 2024.

## Builds on (most cited first, 12 of 12)

- [VAMPnets: Deep learning of molecular kinetics](https://mlsanity.com/paper/1710.06012/) (2017)
- [Data-driven approximation of the Koopman generator: Model reduction, system identification, and control](https://mlsanity.com/paper/1909.10638/) (2019)
- [Learning Dynamical Systems via Koopman Operator Regression in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2205.14027/) (2022)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Chasing Collective Variables using Autoencoders and biased trajectories](https://mlsanity.com/paper/2104.11061/) (2021)
- [Solving eigenvalue PDEs of metastable diffusion processes using artificial neural networks](https://mlsanity.com/paper/2110.14523/) (2021)
- [Learning the Infinitesimal Generator of Stochastic Diffusion Processes](https://mlsanity.com/paper/2405.12940/) (2024)
- [Transition Path Sampling with Boltzmann Generator-based MCMC Moves](https://mlsanity.com/paper/2312.05340/) (2023)
- [The Galerkin method beats Graph-Based Approaches for Spectral Algorithms](https://mlsanity.com/paper/2306.00742/) (2023)
- [Kernelized Diffusion maps](https://mlsanity.com/paper/2302.06757/) (2023)
- [Reweighted Manifold Learning of Collective Variables from Enhanced Sampling Simulations](https://mlsanity.com/paper/2207.14554/) (2022)
- [Data-driven path collective variables](https://mlsanity.com/paper/2312.13868/) (2023)

## Built on by (most cited first, 6 of 6)

- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [kooplearn: A Scikit-Learn Compatible Library of Algorithms for Evolution Operator Learning](https://mlsanity.com/paper/2512.21409/) (2025)
- [Adaptive Kernel Selection for Kernelized Diffusion Maps](https://mlsanity.com/paper/2604.18402/) (2026)
- [Demystifying Linear Operator Learning for Control Systems](https://mlsanity.com/paper/2609.19428/) (2026)
- [Multi-Task Learning of Conditional Mean Operators: applications to dynamical systems and uncertainty quantification](https://mlsanity.com/paper/2609.35429/) (2026)
- [Langevin-Informed Transfer Learning: Replacing Target Samples by Black-Box Feedback](https://mlsanity.com/paper/2610.01522/) (2026)

## Abstract (arXiv)

We investigate learning the eigenfunctions of evolution operators for time-reversal invariant stochastic processes, a prime example being the Langevin equation used in molecular dynamics. Many physical or chemical processes described by this equation involve transitions between metastable states separated by high potential barriers that can hardly be crossed during a simulation. To overcome this bottleneck, data are collected via biased simulations that explore the state space more rapidly. We propose a framework for learning from biased simulations rooted in the infinitesimal generator of the process and the associated resolvent operator. We contrast our approach to more common ones based on the transfer operator, showing that it can provably learn the spectral properties of the unbiased system from biased data. In experiments, we highlight the advantages of our method over transfer operator approaches and recent developments based on generator learning, demonstrating its effectiveness in estimating eigenfunctions and eigenvalues. Importantly, we show that even with datasets containing only a few relevant transitions due to sub-optimal biasing, our approach recovers relevant information about the transition mechanism.