Not Every Term Adds New Structure: Sobolev Novelty for Symbolic Regression

Symbolic Regression (SR) plays a central role in scientific knowledge discovery by distilling mathematical equations from observational data. Most existing SR methods function within a bi-level optimization framework: an outer loop that searches for the discrete equation structure, and an inner loop that optimizes the continuous parameters of that structure. Crucially, parameter-fitting quality directly determines a structure's score and thus the outer-loop search. However, nonlinear operators make the inner loop highly non-convex, and budget-driven reliance on fast local solvers (e.g., BFGS) often yields poor local minima and underestimated scores for correct structures. This ``Good Structure, Bad Score'' phenomenon becomes a key bottleneck, degrading efficiency and misguiding the search away from the true equation. To resolve this, we propose SAGE-Fit (Structure-Aware and Semantics-Guided Evaluator for Symbolic Regression), an SR-native fitting framework that exploits the dual native priors of symbolic expressions. By capitalizing on the structural and semantic priors unique to SR, we design tailored modules for each property, thereby effectively mitigating this optimization bottleneck. Extensive experiments demonstrate that our approach, as a plug-and-play module, significantly enhances evaluation fidelity and universally improves the performance of various SR systems.

Symbolic regression aims to recover closed-form equations from observations, providing interpretable models for scientific discovery. Existing approaches struggle to combine flexible structural search with efficient inference. Search-based methods can refine expression structure but often rely on costly combinatorial optimization with random initialization. Pretrained neural models generate formulas almost instantly, but their predictions often contain symbolic errors. We introduce MOSAIC-SR, which uses a pretrained Transformer to propose multiple initial sketches. These sketches initialize searches in several promising regions, avoiding random starts in the vast expression space. Each search jointly recovers structure and constants through scale-aware constant optimization and local symbolic repair. We evaluate MOSAIC-SR on the SRSD-Feynman dataset with and without dummy variables and on six additional benchmarks. MOSAIC-SR obtains the highest symbolic solution rate on every dataset while ranking among the top two methods in predictive accuracy. This advantage persists in the presence of irrelevant dummy inputs. The results show that learned priors can focus search on promising equation structures, and that numerical optimization and symbolic repair are important for recovery.

A fundamental challenge in symbolic regression (SR) is efficiently recovering complex mathematical expressions from observational data. Although this problem is NP-hard, many expressions of practical interest decompose naturally into combinations of nonlinear feature modules, concentrating structural complexity into a small number of reusable components. Here, we introduce FePySR, a two-stage framework that reduces the SR search space by extracting valid features prior to equation search. FePySR first employs a heterogeneous neural network to constrain observational data to a set of candidate expressions, then performs structural optimization within this refined expression space using PySR. Across five standard benchmarks, FePySR outperforms state-of-the-art methods by achieving higher equation recovery rates. On a set of 75 highly complex synthesized equations, FePySR recovers 36 equations, while producing substantially smaller mean squared errors on the remaining unrecovered cases, with reduced computation time compared to PySR. FePySR's first stage also maintains consistent performance under varying numbers of selected top features and increasing levels of noise in the observational data. Applied to ordinary differential equations governing biological systems, FePySR successfully identifies governing equations in 24 out of 100 tests where PySR recovers none. Taken together, FePySR is a generalizable framework that can enhance the SR solvers, enabling the efficient and reliable recovery of symbolic expressions across scientific domains.