Authors: Koki Ikeda, Masahiro Nomura, Ryoki Hamano
Organizations: CyberAgent Shibuya, Tokyo, Japan · Institute of Science Tokyo Meguro, Tokyo, Japan
Abstract
Symbolic regression (SR) aims to discover explicit mathematical expressions that explain observed data and is widely used in domains where interpretability is essential. Because interpretability requires expressions to reflect meaningful regularities, SR is sensitive to observations that deviate from the dominant relationship. Such irregular observations, or outliers, are common in real-world data and can hinder SR from identifying underlying regularities. Robust regression mitigates this by downweighting observations with large residuals. However, deciding which observations should be treated as outliers is often ambiguous and depends on user interpretation and domain knowledge, a perspective largely overlooked in existing SR studies. This motivates approaches that present multiple candidate expressions, allowing users to examine different residual patterns and choose expressions consistent with their expertise. We propose diversified residual symbolic regression (DRSR), which achieves high predictive accuracy while promoting diversity with respect to residual patterns based on the Quality-Diversity paradigm. DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.
Symbolic regression (SR) aims to discover underlying mathematical expressions from data while preserving interpretability. Most existing learning-based SR methods primarily optimize expressions from observations without explicitly exploiting their structural mathematical properties. AI Feynman introduced a complementary paradigm that leverages such properties to recursively decompose complex expressions, but its decomposition criteria cover only restricted structural forms and its treatment of nested composition can require brute-force search over candidate sub-expressions. Building on this paradigm, we propose Deep Divide-and-Reduce in Symbolic Regression (DDRSR), a mathematically grounded framework that systematically generalizes expression decomposition and variable reduction. DDRSR extends translational symmetry to coefficient- and exponent-interfered forms, enables variable separation under overlapping variables and additive constant offsets, and generalizes the identification of nested compositional structures. We further characterize an intrinsic non-identifiability limitation of decomposition when no effective variable separation is induced. Experiments across multiple symbolic regression algorithms and benchmark datasets show that DDRSR identifies a broader range of decomposable structures than AI Feynman and overall improves downstream regression accuracy and exact-expression recovery.
Symbolic regression (SR) discovers analytical equations from data, yielding glass-box models with directly interpretable formulas, unlike black-box methods that rely on unstable post-hoc tools such as SHAP or LIME. This transparency is crucial in clinical medicine and social science, but SR faces three challenges: high-dimensional inputs, principled selection of Pareto-front formulae, and data irregularities such as multicollinearity and class imbalance. We introduce DeepPySR, which addresses these issues with a dynamic variable-pruning schedule to remove irrelevant features during search, an exponential Pareto selection criterion that eliminates trade-offs between accuracy and complexity, and a multi-layer architecture for hierarchical symbolic composition. On four Feynman physics benchmarks and seven biomedical and social-science datasets, DeepPySR outperforms PySR and baselines on body fat (R2: 0.794 vs.\ 0.702), heart disease (F1: 0.898 vs.\ 0.787), student performance (R2: 0.964 vs.\ 0.948), and Raine BMI (R2: 0.525 vs.\ 0.370), producing interpretable formulas aligned with domain risk factors.
Symbolic Regression (SR) seeks to find succinct mathematical expressions that represent the fundamental relationships within data, providing interpretability and scientific understanding that exceeds that of black-box models. Nevertheless, traditional methods like Genetic Programming face challenges with scalability and are highly sensitive to noise, while sparse regression techniques such as SINDy rely significantly on predetermined feature libraries. In this work, we present a Neural Symbolic Regression (NSR) framework that treats neural networks as functional preconditioners for symbolic discovery. Our approach uses a decoupled pipeline: a neural network first learns a smooth, noise-robust approximation of the target function in an interaction- aware nonlinear feature space. LASSO is then applied to extract sparse, interpretable closed-form expressions. To improve predictive accuracy and symbolic fidelity by integrating distributed hyperparameter optimization with Ray Tune and ASHA scheduling. Experiments on the Nguyen benchmark suite show that our approach consistently outperforms SINDy and non-tuned neural baselines in RMSE, noise robustness, and out-of-distribution generalization. Ablation studies confirm the significance of feature interactions, neural depth, and tuning strategies. In general, this study presents a scalable and understandable neural-symbolic framework, creating a solid link between neural approximation and the discovery of sparse equations for scientific machine learning.