The Principia Machina/Tools/PySR: Difference between revisions

Revision as of 00:59, 6 October 2026

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 $ℒ (ϕ, \partial_{μ} ϕ)$ 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 ( $δ S = 0$ ), PySR directly extracts analytical scalar Lagrangians from observational state trajectories.
Pareto Complexity Optimization: Automatically balances accuracy against expression complexity using score metrics:

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

Custom Operator Ingestion: Supports specialized physical operators (e.g., spatial gradients $\nabla ϕ$ , d'Alembertian operators $◻ ϕ$ , or relativistic dot products $u^{μ} u_{μ}$ ).

SVD Integration Pipeline

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

@@ Line 3: / Line 3: @@
 == 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 [[The Principia Machina|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?
+'''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 [[The Principia Machina|Principia Machina]] architecture, PySR addresses the inverse theoretical problem: given a field dataset or trajectory simulation, what analytical scalar expression or Lagrangian density <math>\mathcal{L}(q, \dot{q}, t)</math> or <math>\mathcal{L}(\phi, \partial_\mu \phi)</math> 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.
+* '''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 (<math>\delta S = 0</math>), PySR directly extracts analytical scalar Lagrangians from observational state trajectories.
 * '''Pareto Complexity Optimization:''' Automatically balances accuracy against expression complexity using score metrics:
-$$\text{Score} = -\frac{\Delta \log(\text{MSE})}{\Delta \text{Complexity}}$$
+<math display="block">\text{Score} = -\frac{\Delta \log(\text{MSE})}{\Delta \text{Complexity}}</math>
 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 $u^\mu u_\mu$).
+* '''Custom Operator Ingestion:''' Supports specialized physical operators (e.g., spatial gradients <math>\nabla \phi</math>, d'Alembertian operators <math>\Box \phi</math>, or relativistic dot products <math>u^\mu u_\mu</math>).
 == SVD Integration Pipeline ==
-. '''Data Generation:''' Extract spatial perturbation profiles from vacuum disturbance simulations.
+# '''Data Generation:''' Extract spatial perturbation profiles from vacuum disturbance simulations.
-. '''Operator Set Specification:''' Define admissible operators $\{+, -, \times, \div, \nabla, \partial_t, \text{exp}\}$.
+# '''Operator Set Specification:''' Define admissible operators <math>\{+, -, \times, \div, \nabla, \partial_t, \text{exp}\}</math>.
-. '''Variational Loss Enforcement:''' Evaluate candidates against action stationarity constraints.
+# '''Variational Loss Enforcement:''' Evaluate candidates against action stationarity constraints.
-. '''Analytical Verification:''' Pass discovered symbolic candidate expressions to [[The Principia Machina/Tools/Cadabra2|Cadabra2]] for formal Noether symmetry evaluation.
+# '''Analytical Verification:''' Pass discovered symbolic candidate expressions to [[The Principia Machina/Tools/Cadabra2|Cadabra2]] for formal Noether symmetry evaluation.

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Revision as of 00:59, 6 October 2026

Overview & Architectural Fit

Mathematical Evaluation for TPM & SVD

SVD Integration Pipeline