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NLSA: Operator methods for dynamical systems

The NLSA repo started out as a MATLAB implementation of the nonlinear Laplacian spectral analysis (NLSA) technique – a kernel method for spectral decomposition of time series data generated by dynamical systems. Over time, the scope of the repo was expanded to include various methods for feature extraction and prediction based on Koopman and transfer operators of dynamical systems, as well as "quantum-inspired" operator methods that combine ideas from ergodic theory and quantum mechanics. Eventually, a Python library implementing these methods was added. The Python portion of the codebase is currently under active development. The MATLAB portion is mainly in maintenance mode and receives occasional updates and bug fixes.

Features and methods implemented

  • Delay-coordinate maps.
  • Variable-bandwidth kernels.
  • Kernel normalizations (diffusion maps, bistochastic).
  • Kernel eigendecomposition and Nystrom out-of-sample evaluation.
  • Spectral decomposition of Koopman/transfer operators.
  • Kernel analog forecasting.
  • Quantum mechanical data assimilation.

Acknowledgment

Development of the NLSA Python library is funded by the U.S. Environmental Security Technology Certification Program under grant NH24-8392. Research on associated mathematical methods is supported by the U.S. Office of Naval Research (ONR) and Department of Energy under grants N00014-19-1-2421, N00014-21-1-2946, and DE-SC0025101. Previous grants supporting this work include U.S. National Science Foundation grants DMS-1521775, 1842538, and DMS-1854383 and ONR grants N00014-14-1-0150 and N00014-16-1-2649.

Screenshots

Extraction and forecasting of the El Nino Southern Oscillation (ENSO)

Koopman ENSO mode

Above: Koopman ENSO mode. Generated using Python/examples/enso/koopman_diffusion_regularization_kaf.py.

2026/27 super El Nino forecast

Above: Kernel analog forecast (KAF) of the Nino 3.4 index using sea surface temperature (SST) fields in the Nino 3.4 index as predictors. The forecast was initialized in June 1, 2026. It predicts a strong El Nino event peaking in December 2026, witn ENSO-neutral conditions returning around May 2027 and persisting through the end of the forecast interval in June 2028. Generated using Python/examples/enso/kernel_analog_forecasting.py.