# Variational approach for learning Markov processes from time series data

arXiv: https://arxiv.org/abs/1707.04659 · HTML: https://mlsanity.com/paper/1707.04659/

## Key facts

- Cited by 52 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jul 2017.

## Built on by (most cited first, 30 of 52)

- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deeptime: a Python library for machine learning dynamical models from time series data](https://mlsanity.com/paper/2110.15013/) (2021)
- [Sharp Spectral Rates for Koopman Operator Learning](https://mlsanity.com/paper/2302.02004/) (2023)
- [Simulate Time-integrated Coarse-grained Molecular Dynamics with Multi-Scale Graph Networks](https://mlsanity.com/paper/2204.10348/) (2022)
- [Learning invariant representations of time-homogeneous stochastic dynamical systems](https://mlsanity.com/paper/2307.09912/) (2023)
- [Implicit Transfer Operator Learning: Multiple Time-Resolution Surrogates for Molecular Dynamics](https://mlsanity.com/paper/2305.18046/) (2023)
- [Transferable Generative Models Bridge Femtosecond to Nanosecond Time-Step Molecular Dynamics](https://mlsanity.com/paper/2510.07589/) (2025)
- [Learning the Infinitesimal Generator of Stochastic Diffusion Processes](https://mlsanity.com/paper/2405.12940/) (2024)
- [A Control Theory of Predictability in Latent World Models](https://mlsanity.com/paper/2607.10362/) (2026)
- [GraphVAMPNet, using graph neural networks and variational approach to markov processes for dynamical modeling of biomolecules](https://mlsanity.com/paper/2201.04609/) (2022)
- [Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups](https://mlsanity.com/paper/2410.14477/) (2024)
- [Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systems](https://mlsanity.com/paper/2505.18671/) (2025)
- [Reaction coordinate flows for model reduction of molecular kinetics](https://mlsanity.com/paper/2309.05878/) (2023)
- [Latent Representation and Simulation of Markov Processes via Time-Lagged Information Bottleneck](https://mlsanity.com/paper/2309.07200/) (2023)
- [ResKoopNet: Learning Koopman Representations for Complex Dynamics with Spectral Residuals](https://mlsanity.com/paper/2501.00701/) (2025)
- [Koopman-Equivariant Gaussian Processes](https://mlsanity.com/paper/2502.06645/) (2025)
- [A Note on Learning Rare Events in Molecular Dynamics using LSTM and Transformer](https://mlsanity.com/paper/2107.06573/) (2021)
- [Learning stochastic dynamics and predicting emergent behavior using transformers](https://mlsanity.com/paper/2202.08708/) (2022)
- [Using pretrained graph neural networks with token mixers as geometric featurizers for conformational dynamics](https://mlsanity.com/paper/2409.19838/) (2024)
- [Efficient Parametric SVD of Koopman Operator for Stochastic Dynamical Systems](https://mlsanity.com/paper/2507.07222/) (2025)
- [Scalable Spatio-Temporal SE(3) Diffusion for Long-Horizon Protein Dynamics](https://mlsanity.com/paper/2602.02128/) (2026)
- [Protein Language Model Embeddings Improve Generalization of Implicit Transfer Operators](https://mlsanity.com/paper/2602.11216/) (2026)
- [Trustworthy Koopman Operator Learning: Invariance Diagnostics and Error Bounds](https://mlsanity.com/paper/2603.15091/) (2026)
- [Transfer operators on graphs: Spectral clustering and beyond](https://mlsanity.com/paper/2305.11766/) (2023)
- [Pushing the Limits of All-Atom Geometric Graph Neural Networks: Pre-Training, Scaling and Zero-Shot Transfer](https://mlsanity.com/paper/2410.21683/) (2024)
- [Learning dynamical systems from data: Gradient-based dictionary optimization](https://mlsanity.com/paper/2411.04775/) (2024)
- [Understanding recent deep-learning techniques for identifying collective variables of molecular dynamics](https://mlsanity.com/paper/2307.00365/) (2023)
- [Flow Matching for Optimal Reaction Coordinates of Biomolecular System](https://mlsanity.com/paper/2408.17139/) (2024)
- [Machine Learning of Slow Collective Variables and Enhanced Sampling via Spatial Techniques](https://mlsanity.com/paper/2412.20868/) (2024)
- [How deep is your network? Deep vs. shallow learning of transfer operators](https://mlsanity.com/paper/2509.19930/) (2025)

## Abstract (arXiv)

Inference, prediction and control of complex dynamical systems from time series is important in many areas, including financial markets, power grid management, climate and weather modeling, or molecular dynamics. The analysis of such highly nonlinear dynamical systems is facilitated by the fact that we can often find a (generally nonlinear) transformation of the system coordinates to features in which the dynamics can be excellently approximated by a linear Markovian model. Moreover, the large number of system variables often change collectively on large time- and length-scales, facilitating a low-dimensional analysis in feature space. In this paper, we introduce a variational approach for Markov processes (VAMP) that allows us to find optimal feature mappings and optimal Markovian models of the dynamics from given time series data. The key insight is that the best linear model can be obtained from the top singular components of the Koopman operator. This leads to the definition of a family of score functions called VAMP-r which can be calculated from data, and can be employed to optimize a Markovian model. In addition, based on the relationship between the variational scores and approximation errors of Koopman operators, we propose a new VAMP-E score, which can be applied to cross-validation for hyper-parameter optimization and model selection in VAMP. VAMP is valid for both reversible and nonreversible processes and for stationary and non-stationary processes or realizations.