Explaining the Explainer: Understanding the Inner Workings of Transformer-based Symbolic Regression Models

Following their success across many domains, transformers have also proven effective for symbolic regression (SR); however, the internal mechanisms underlying their generation of mathematical operators remain largely unexplored. Although mechanistic interpretability has successfully identified circuits in language and vision models, it has not yet been applied to SR. In this article, we introduce PATCHES, an evolutionary circuit discovery algorithm that identifies compact and correct circuits for SR. Using PATCHES, we isolate 28 circuits, providing the first circuit-level characterisation of an SR transformer. We validate these findings through a robust causal evaluation framework based on key notions such as faithfulness, completeness, and minimality. Our analysis shows that mean patching with performance-based evaluation most reliably isolates functionally correct circuits. In contrast, we demonstrate that direct logit attribution and probing classifiers primarily capture correlational features rather than causal ones, limiting their utility for circuit discovery. Overall, these results establish SR as a high-potential application domain for mechanistic interpretability and propose a principled methodology for circuit discovery.

Enhancing Generalization in Evolutionary Feature Construction for Symbolic Regression through Vicinal Jensen Gap Minimization

Genetic programming-based feature construction has achieved significant success in recent years as an automated machine learning technique to enhance learning performance. However, overfitting remains a challenge that limits its broader applicability. To improve generalization, we prove that vicinal risk, estimated through noise perturbation or mixup-based data augmentation, is bounded by the sum of empirical risk and a regularization term-either finite difference or the vicinal Jensen gap. Leveraging this decomposition, we propose an evolutionary feature construction framework that jointly optimizes empirical risk and the vicinal Jensen gap to control overfitting. Since datasets may vary in noise levels, we develop a noise estimation strategy to dynamically adjust regularization strength. Furthermore, to mitigate manifold intrusion-where data augmentation may generate unrealistic samples that fall outside the data manifold-we propose a manifold intrusion detection mechanism. Experimental results on 58 datasets demonstrate the effectiveness of Jensen gap minimization compared to other complexity measures. Comparisons with 15 machine learning algorithms further indicate that genetic programming with the proposed overfitting control strategy achieves superior performance.

Introns and Templates Matter: Rethinking Linkage in GP-GOMEA

GP-GOMEA is among the state-of-the-art for symbolic regression, especially when it comes to finding small and potentially interpretable solutions. A key mechanism employed in any GOMEA variant is the exploitation of linkage, the dependencies between variables, to ensure efficient evolution. In GP-GOMEA, mutual information between node positions in GP trees has so far been used to learn linkage. For this, a fixed expression template is used. This however leads to introns for expressions smaller than the full template. As introns have no impact on fitness, their occurrences are not directly linked to selection. Consequently, introns can adversely affect the extent to which mutual information captures dependencies between tree nodes. To overcome this, we propose two new measures for linkage learning, one that explicitly considers introns in mutual information estimates, and one that revisits linkage learning in GP-GOMEA from a grey-box perspective, yielding a measure that needs not to be learned from the population but is derived directly from the template. Across five standard symbolic regression problems, GP-GOMEA achieves substantial improvements using both measures. We also find that the newly learned linkage structure closely reflects the template linkage structure, and that explicitly using the template structure yields the best performance overall.

Mixture of Concept Bottleneck Experts

Concept Bottleneck Models (CBMs) promote interpretability by grounding predictions in human-understandable concepts. However, existing CBMs typically fix their task predictor to a single linear or Boolean expression, limiting both predictive accuracy and adaptability to diverse user needs. We propose Mixture of Concept Bottleneck Experts (M-CBEs), a framework that generalizes existing CBMs along two dimensions: the number of experts and the functional form of each expert, exposing an underexplored region of the design space. We investigate this region by instantiating two novel models: Linear M-CBE, which learns a finite set of linear expressions, and Symbolic M-CBE, which leverages symbolic regression to discover expert functions from data under user-specified operator vocabularies. Empirical evaluation demonstrates that varying the mixture size and functional form provides a robust framework for navigating the accuracy-interpretability trade-off, adapting to different user and task needs.

Robust Machine Learning Framework for Reliable Discovery of High-Performance Half-Heusler Thermoelectrics

Machine learning (ML) can facilitate efficient thermoelectric (TE) material discovery essential to address the environmental crisis. However, ML models often suffer from poor experimental generalizability despite high metrics. This study presents a robust workflow, applied to the half-Heusler (hH) structural prototype, for figure of merit (zT) prediction, to improve the generalizability of ML models. To resolve challenges in dataset handling and feature filtering, we first introduce a rigorous PCA-based splitting method that ensures training and test sets are unbiased and representative of the full chemical space. We then integrate Bayesian hyperparameter optimization with k-best feature filtering across three architectures-Random Forest, XGBoost, and Neural Networks - while employing SISSO symbolic regression for physical insight and comparison. Using SHAP and SISSO analysis, we identify A-site dopant concentration (xA'), and A-site Heat of Vaporization (HVA) as the primary drivers of zT besides Temperature (T). Finally, a high-throughput screening of approximately 6.6x10^8 potential compositions, filtered by stability constraints, yielded several novel high-zT candidates. Breaking from the traditional focus of improving test RMSE/R^2 values of the models, this work shifts the attention on establishing the test set a true proxy for model generalizability and strengthening the often neglected modules of the existing ML workflows for the data-driven design of next-generation thermoelectric materials.

ECSEL: Explainable Classification via Signomial Equation Learning

We introduce ECSEL, an explainable classification method that learns formal expressions in the form of signomial equations, motivated by the observation that many symbolic regression benchmarks admit compact signomial structure. ECSEL directly constructs a structural, closed-form expression that serves as both a classifier and an explanation. On standard symbolic regression benchmarks, our method recovers a larger fraction of target equations than competing state-of-the-art approaches while requiring substantially less computation. Leveraging this efficiency, ECSEL achieves classification accuracy competitive with established machine learning models without sacrificing interpretability. Further, we show that ECSEL satisfies some desirable properties regarding global feature behavior, decision-boundary analysis, and local feature attributions. Experiments on benchmark datasets and two real-world case studies i.e., e-commerce and fraud detection, demonstrate that the learned equations expose dataset biases, support counterfactual reasoning, and yield actionable insights.

Knowledge-Informed Kernel State Reconstruction for Interpretable Dynamical System Discovery

Recovering governing equations from data is central to scientific discovery, yet existing methods often break down under noisy, partial observations, or rely on black-box latent dynamics that obscure mechanism. We introduce MAAT (Model Aware Approximation of Trajectories), a framework for symbolic discovery built on knowledge-informed Kernel State Reconstruction. MAAT formulates state reconstruction in a reproducing kernel Hilbert space and directly incorporates structural and semantic priors such as non-negativity, conservation laws, and domain-specific observation models into the reconstruction objective, while accommodating heterogeneous sampling and measurement granularity. This yields smooth, physically consistent state estimates with analytic time derivatives, providing a principled interface between fragmented sensor data and symbolic regression. Across twelve diverse scientific benchmarks and multiple noise regimes, MAAT substantially reduces state-estimation MSE for trajectories and derivatives used by downstream symbolic regression relative to strong baselines.

An Empirical Investigation of Neural ODEs and Symbolic Regression for Dynamical Systems

Accurately modelling the dynamics of complex systems and discovering their governing differential equations are critical tasks for accelerating scientific discovery. Using noisy, synthetic data from two damped oscillatory systems, we explore the extrapolation capabilities of Neural Ordinary Differential Equations (NODEs) and the ability of Symbolic Regression (SR) to recover the underlying equations. Our study yields three key insights. First, we demonstrate that NODEs can extrapolate effectively to new boundary conditions, provided the resulting trajectories share dynamic similarity with the training data. Second, SR successfully recovers the equations from noisy ground-truth data, though its performance is contingent on the correct selection of input variables. Finally, we find that SR recovers two out of the three governing equations, along with a good approximation for the third, when using data generated by a NODE trained on just 10% of the full simulation. While this last finding highlights an area for future work, our results suggest that using NODEs to enrich limited data and enable symbolic regression to infer physical laws represents a promising new approach for scientific discovery.

ROIDS: Robust Outlier-Aware Informed Down-Sampling

Informed down-sampling (IDS) is known to improve performance in symbolic regression when combined with various selection strategies, especially tournament selection. However, recent work found that IDS's gains are not consistent across all problems. Our analysis reveals that IDS performance is worse for problems containing outliers. IDS systematically favors including outliers in subsets which pushes GP towards finding solutions that overfit to outliers. To address this, we introduce ROIDS (Robust Outlier-Aware Informed Down-Sampling), which excludes potential outliers from the sampling process of IDS. With ROIDS it is possible to keep the advantages of IDS without overfitting to outliers and to compete on a wide range of benchmark problems. This is also reflected in our experiments in which ROIDS shows the desired behavior on all studied benchmark problems. ROIDS consistently outperforms IDS on synthetic problems with added outliers as well as on a wide range of complex real-world problems, surpassing IDS on over 80% of the real-world benchmark problems. Moreover, compared to all studied baseline approaches, ROIDS achieves the best average rank across all tested benchmark problems. This robust behavior makes ROIDS a reliable down-sampling method for selection in symbolic regression, especially when outliers may be included in the data set.

Beyond Error-Based Optimization: Experience-Driven Symbolic Regression with Goal-Conditioned Reinforcement Learning

Symbolic Regression aims to automatically identify compact and interpretable mathematical expressions that model the functional relationship between input and output variables. Most existing search-based symbolic regression methods typically rely on the fitting error to inform the search process. However, in the vast expression space, numerous candidate expressions may exhibit similar error values while differing substantially in structure, leading to ambiguous search directions and hindering convergence to the underlying true function. To address this challenge, we propose a novel framework named EGRL-SR (Experience-driven Goal-conditioned Reinforcement Learning for Symbolic Regression). In contrast to traditional error-driven approaches, EGRL-SR introduces a new perspective: leveraging precise historical trajectories and optimizing the action-value network to proactively guide the search process, thereby achieving a more robust expression search. Specifically, we formulate symbolic regression as a goal-conditioned reinforcement learning problem and incorporate hindsight experience replay, allowing the action-value network to generalize common mapping patterns from diverse input-output pairs. Moreover, we design an all-point satisfaction binary reward function that encourages the action-value network to focus on structural patterns rather than low-error expressions, and concurrently propose a structure-guided heuristic exploration strategy to enhance search diversity and space coverage. Experiments on public benchmarks show that EGRL-SR consistently outperforms state-of-the-art methods in recovery rate and robustness, and can recover more complex expressions under the same search budget. Ablation results validate that the action-value network effectively guides the search, with both the reward function and the exploration strategy playing critical roles.