Improved Identification Accuracy in Equation Learning via Comprehensive $\boldsymbol{R^2}$-Elimination and Bayesian Model Selection

Abstract

In the field of equation learning, exhaustively considering all possible combinations derived from a basis function dictionary is infeasible. Sparse regression and greedy algorithms have emerged as popular approaches to tackle this challenge. However, the presence of strong collinearities poses difficulties for sparse regression techniques, and greedy steps may inadvertently exclude important components of the true equation, leading to reduced identification accuracy. In this article, we present a novel algorithm that strikes a balance between comprehensiveness and efficiency in equation learning. Inspired by stepwise regression, our approach combines the coefficient of determination, $R^2$, and the Bayesian model evidence, $p(y|\mathcal{M})$, in a novel way. Through three extensive numerical experiments involving random polynomials and dynamical systems, we compare our method against two standard approaches, four state-of-the-art methods, and bidirectional stepwise regression incorporating $p(y|\mathcal{M})$. The results demonstrate that our less greedy algorithm surpasses all other methods in terms of identification accuracy. Furthermore, we discover a heuristic approach to mitigate the overfitting penalty associated with $R^2$ and propose an equation learning procedure solely based on $R^2$, which achieves high rates of exact equation recovery.