Purely Discrete Symbolic Regression: EML Signal Decomposition via MCTS and Basin-Hopping

Abstract

Extracting closed-form analytical representations of temporal dynamics from noisy data remains a central challenge. Traditional symbolic regression suffers from highly irregular search spaces and "expression bloat." While the isomorphic Exp-Minus-Log (EML) operator unifies the search space, its deep nesting triggers gradient explosion, causing continuous approximations to stall at local minima. To address this, we propose the Ultimate EML Symbolic Decomposition Transform (U-ESDT), a white-box 1D signal decomposition framework. U-ESDT employs a dual-track global optimization engine: an outer Monte Carlo Tree Search (MCTS) conducts purely discrete, deterministic topology generation, while an inner Basin-Hopping algorithm achieves global parameter annealing. Enhanced by affine-wrapped variables and a two-stage refinement mechanism, U-ESDT optimally balances exploratory breadth and convergence depth. Evaluated on benchmark physical signals and the Feynman dataset, U-ESDT demonstrates unparalleled topological parsimony and noise robustness. It completely eradicates expression bloat while achieving predictive fidelities that match or exceed state-of-the-art heuristic models (e.g., PySR), offering a rigorous paradigm for deciphering the mathematical laws of non-stationary time series.