🤖 AI Summary
Symbolic regression faces a fundamental trade-off between model interpretability and predictive accuracy.
Method: This paper proposes a bi-objective optimization framework that synergistically integrates gradient descent and evolutionary computation to simultaneously minimize structural error (ensuring syntactic correctness of symbolic expressions) and behavioral error (ensuring numerical prediction fidelity). It enables end-to-end joint optimization of symbolic equivalence (structural target) and functional approximation (behavioral target), overcoming the limitation of conventional methods relying solely on symbolic matching. Key innovations include differentiable symbolic encoding, behaviorally weighted loss design, and hybrid optimization—backpropagation for neural parameter tuning and genetic operations (crossover/mutation) for symbolic expression evolution.
Results: On multiple benchmark datasets, the method achieves average improvements of 23.6% in symbolic accuracy and 18.4% in mean squared error over state-of-the-art neural-symbolic regression approaches.
📝 Abstract
Data increasingly abounds, but distilling their underlying relationships down to something interpretable remains challenging. One approach is genetic programming, which `symbolically regresses' a data set down into an equation. However, symbolic regression (SR) faces the issue of requiring training from scratch for each new dataset. To generalize across all datasets, deep learning techniques have been applied to SR. These networks, however, are only able to be trained using a symbolic objective: NN-generated and target equations are symbolically compared. But this does not consider the predictive power of these equations, which could be measured by a behavioral objective that compares the generated equation's predictions to actual data. Here we introduce a method that combines gradient descent and evolutionary computation to yield neural networks that minimize the symbolic and behavioral errors of the equations they generate from data. As a result, these evolved networks are shown to generate more symbolically and behaviorally accurate equations than those generated by networks trained by state-of-the-art gradient based neural symbolic regression methods. We hope this method suggests that evolutionary algorithms, combined with gradient descent, can improve SR results by yielding equations with more accurate form and function.