Adaptive Protection for Evolutionary Feature Construction in Symbolic Regression with Application to Credit Classification

Symbolic regression via genetic programming routinely fails on small, wide datasets - a regime common in clinical-trial monitoring, biostatistics, and engineering pilot studies - by converging on bloated, overfit expressions that exploit correlation rather than prediction. We present Evolutional Math, an open-source genetic programming system that combines four design choices to yield compact, interpretable formulas in this regime. First, fitness is measured by R-squared on held-out cross-validation folds rather than Pearson correlation on the training set, eliminating single-variable shortcuts that correlate but mis-scale. Second, a multi-island architecture runs independent populations seeded with distinct operator subsets (algebraic, logarithmic, trigonometric, and full) with ring-topology migration every M generations, preventing the search from collapsing into one region of formula space. Third, a structural deduplication scheme treats formulas differing only in constants as equivalent, so the elite archive contains structurally distinct candidates rather than near-duplicate variants. Fourth, top-k individuals undergo numerical constant refinement via scipy L-BFGS-B after each migration phase, decoupling structure search from parameter fitting. We evaluate the system on synthetic benchmarks of the form log(x_i) * x_j / (x_k * c), trigonometric mixtures, and an anonymized clinical site-monitoring dataset with 24 rows and approximately 290 candidate numeric features. The system consistently recovers compact ground-truth structures with R-squared at or above 0.99 within tens of thousands of unique formula evaluations. A reference implementation is released under a noncommercial source-available license.

Mathematical formulas serve as a language through which humans communicate with nature. Discovering mathematical laws from scientific data to describe natural phenomena has been a long-standing pursuit of humanity for centuries. In the field of artificial intelligence, this challenge is known as the symbolic regression problem. Among existing symbolic regression approaches, Genetic Programming (GP) based on evolutionary algorithms remains one of the most classical and widely adopted methods. GP simulates the evolutionary process across generations through genetic mutation and crossover. However, mutations and crossovers in GP are entirely random. While this randomness effectively mimics natural evolution, it inevitably produces both beneficial and detrimental variations. If there existed a metaphorical God capable of foreseeing which genetic mutations or crossovers would yield superior outcomes and performing targeted gene editing accordingly, the efficiency of evolution could be substantially improved. Motivated by this idea, we propose in this paper a symbolic regression approach based on gene editing, termed GESR. In GESR, we trained two "hands of God" (two BERT models). Among them, the first leverages the BERT's masked language modeling capability to guide the mutation of genes (expression symbols). The other BERT model guides the crossover of individual genes by predicting the crossover point. Experimental results demonstrate that GESR significantly improves computational efficiency compared with traditional GP algorithms and achieves strong overall performance across multiple symbolic regression tasks.

In memetic symbolic regression, candidate structures are compared after coefficient calibration, so the calibration protocol itself contributes to evolutionary selection. Standard centralized calibration evaluates each structure at its pooled-sample optimum, ignoring how stable this calibration is under covariate shifts, and can thus favor structures whose fit relies on sample-specific coefficients. We propose Dirichlet-Sinkhorn Constant Averaging (DSCA), a calibration strategy that partitions the optimization data into equally sized subsets with different covariate distributions, calibrates each candidate independently on every partition, and evaluates it at the mean of the resulting parameters. We show that the excess loss of DSCA relative to centralized calibration vanishes at the population level for correctly specified, identifiable expressions, whereas under misspecification it persists when partition-specific calibrations do not aggregate to the pooled optimum. On synthetic benchmarks and ten real-world datasets, DSCA improves functional recovery and the accuracy-complexity trade-off over centralized Broyden-Fletcher-Goldfarb-Shanno and Levenberg-Marquardt calibration, under selection by negative log-likelihood and by the Akaike and Bayesian information criteria. Mechanism analyses associate the DSCA excess loss with the generalization gap and show that the effect is not reproduced by repeated centralized fitting. These results indicate that controlled heterogeneous calibration provides a complementary source of selection pressure in symbolic-regression search.