The Principia Machina/Tools/PySR

PySR / SymbolicRegression.jl: High-Performance Symbolic Discovery

Overview & Architectural Fit

PySR (interfaced via Python to the high-performance Julia backend SymbolicRegression.jl) is an open-source symbolic regression framework utilizing Multi-Population Genetic Algorithms and Pareto Frontier optimization. In the Principia Machina architecture, PySR addresses the inverse theoretical problem: given a field dataset or trajectory simulation, what analytical scalar expression or Lagrangian density $ℒ(q,\dot{q},t)$ or $ℒ(\varphi ,{\partial }_{\mu }\varphi )$ governs the system?

Mathematical Evaluation for TPM & SVD

• Lagrangian & Action Discovery (9.5/10): PySR can search candidate functional spaces for invariant quantities. By setting loss functions to penalize violations of Euler-Lagrange dynamics or Hamilton's principle ($\delta S=0$), PySR directly extracts analytical scalar Lagrangians from observational state trajectories.
• Pareto Complexity Optimization: Automatically balances accuracy against expression complexity using score metrics:

$$\text{Score}=-\frac{\Delta \log (\text{MSE})}{\Delta \text{Complexity}}$$ This directly prevents the discovery of unphysical "overfitted" mathematical artifacts.

• Custom Operator Ingestion: Supports specialized physical operators (e.g., spatial gradients $\mathrm{\nabla }\varphi$, d'Alembertian operators $◻\varphi$, or relativistic dot products $u^{\mu }{u}_{\mu }$).

SVD Integration Pipeline

1. Data Generation: Extract spatial perturbation profiles from vacuum disturbance simulations.
2. Operator Set Specification: Define admissible operators $\{+,-,\times ,÷,\mathrm{\nabla },{\partial }_t,\text{exp}\}$.
3. Variational Loss Enforcement: Evaluate candidates against action stationarity constraints.
4. Analytical Verification: Pass discovered symbolic candidate expressions to Cadabra2 for formal Noether symmetry evaluation.