The Principia Machina/Tools/PySINDy

PySINDy: Sparse Identification of Equations from Data, Without an Action

Added to the survey on 2026-10-06 by the Research Toolkit's research pass. The scores below are that pass's assessment from the evidence cited under References, on the survey's four axes and scale; they are not the original survey's (2026-09-26), and no rule of the Principia Machina reads them. See the additions on the front page.

Overview & Architectural Fit

PySINDy identifies differential equations from data by sparse regression over a library of candidate functions: it seeks the fewest terms that reproduce the measured time derivatives, $\dot{X}\approx \Theta (X)\Xi$ with $\Xi$ sparse. Variants handle partial differential equations (PDE-FIND) and noisy data (weak forms). It is written in Python in the style of scikit-learn.

Mathematical Evaluation for TPM

• Lagrangian / Action (3/10): Finds equations of motion, not actions; variational structure has to be imposed through custom libraries.
• Field / PDE (6.5/10): PDE-FIND and weak-form libraries identify partial differential equations from data in space and time.
• Relativity (1/10): Nothing specific to relativity.
• Quantum / Data (8/10): Learning from data is the core.

Integration Pipeline

1. Identify equations from the data with no action assumed.
2. Ask whether they admit a Lagrangian: test them against the Helmholtz conditions of the inverse problem of the calculus of variations, or compare them with the Euler-Lagrange equations of a Lagrangian found by a Lagrangian network or PySR.
3. Read agreement as support for the action hypothesis, and disagreement as a pointer to dissipation, a missing variable or a biased library.

Status, Licence and Limits

• Status: Active. Version 2.1.0 on 8 January 2026 (PyPI), 2.0.0 on 22 August 2025.
• Licence: MIT.
• Home: https://github.com/dynamicslab/pysindy
• Limits: Results depend heavily on the candidate library and on how derivatives are estimated from noisy data. A sparse model is not necessarily the true one.

On the Principia Machina

A tool runs only where the sponsor installed and bound it, and what it returns is untrusted input, vetted before use; its numbers are never values of the site's calculator, TPC. A law it finds is not a register record: it can enter only as the kernel of a new container, and the data it was fitted to can never be the outside value of that law's verification. See Agent/Tools and the operator's guide.

References

• PYSINDY2022 – Kaptanoglu A., de Silva B., Fasel U., Kaheman K., Goldschmidt A., Callaham J., Delahunt C., Nicolaou Z., Champion K., Loiseau J.-C., Kutz J., Brunton S. (2022). PySINDy: A comprehensive Python package for robust sparse system identification. JOSS 7(69), 3994. doi:10.21105/joss.03994.
• SINDY16 – Brunton, S. L., Proctor, J. L., Kutz, J. N. (2016). Discovering governing equations from data by sparse identification of nonlinear dynamical systems. PNAS 113, 3932–3937. doi:10.1073/pnas.1517384113.
• PDEFIND2017 – Rudy S.H., Brunton S.L., Proctor J.L., Kutz J.N. (2017). Data-driven discovery of partial differential equations. Science Advances 3(4), e1602614. doi:10.1126/sciadv.1602614.