Statistical Operator Learning for Dynamical Systems
Learning transfer operators, Koopman operators and infinitesimal generators from trajectory data.
tklearn is a Python library for statistical learning of linear evolution operators associated with dynamical systems.
The package provides scalable algorithms for estimating transfer operators, Koopman operators and infinitesimal generators directly from trajectory data, without requiring access to the underlying equations of motion. It combines modern statistical learning with structured numerical linear algebra to recover spectral decompositions, eigenfunctions and dynamical modes that characterize long-term behavior of stochastic and deterministic systems.
The current release implements the Toeplitz-based spectral estimation framework introduced in
Toeplitz Based Spectral Methods for Data-Driven Dynamical Systems
Vladimir R. Kostic, Karim Lounici, Massimiliano Pontil (2026).
While originally motivated by this work, the long-term goal of tklearn is to become a general-purpose toolbox for statistical operator learning.
Current functionality includes
- Transfer and Koopman operator estimation
- Toeplitz spectral filtering
- Reduced Rank Regression (RRR)
- Kernel and finite-dimensional estimators
- Spectral decomposition
- Koopman mode decomposition
- Multi-step prediction
- Numerical linear algebra utilities
- Example notebooks and benchmark problems
Future releases will include generator learning, resolvent methods and pseudospectral estimation.
Clone the repository
git clone https://github.com/vladi-iit/tklearn.git
cd tklearnInstall
pip install -e .or
pip install .tklearn/
├── docs/ Documentation
├── notebooks/ Example notebooks
├── src/
│ └── tklearn/
│ ├── estimators/ Statistical operator estimators
│ ├── nn/ Neural network modules
│ ├── spectral_filters.py
│ ├── linalg.py
│ ├── numerical_solvers.py
│ └── ...
├── tests/
└── pyproject.toml
The repository contains several self-contained demonstrations:
- 1D Langevin dynamics
- Duffing oscillator
- Low-rank pseudospectral estimation
These notebooks illustrate the complete workflow from trajectory generation to spectral estimation and visualization.
The methods implemented in tklearn are designed for applications including
- Molecular dynamics
- Statistical physics
- Computational chemistry
- Climate science
- Fluid dynamics
- Time-series analysis
- Reinforcement learning
- Robotics and control
- Computational biology
The project is under active development.
Current emphasis is on robust implementations of operator-learning algorithms together with reproducible benchmark problems. The API may evolve as additional estimators and numerical methods are incorporated.
If you use this software in your research, please cite
@article{kostic2026toeplitz,
title={Toeplitz Based Spectral Methods for Data-Driven Dynamical Systems},
author={Kostic, Vladimir R. and Lounici, Karim and Pontil, Massimiliano},
year={2026}
}A complete BibTeX entry will be added upon publication.
MIT License