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 $\mathcal{L}(q, \dot{q}, t)$ or $\mathcal{L}(\phi, \partial_\mu \phi)$ 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 $\nabla \phi$, d'Alembertian operators $\Box \phi$, 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 $\{+, -, \times, \div, \nabla, \partial_t, \text{exp}\}$. 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.