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In the field of equation learning, exhaustively considering all possibleequations derived from a basis function dictionary is infeasible. Sparseregression and greedy algorithms have emerged as popular approaches to tacklethis challenge. However, the presence of multicollinearity poses difficultiesfor sparse regression techniques, and greedy steps may inadvertently excludeterms of the true equation, leading to reduced identification accuracy. In thisarticle, we present an approach that strikes a balance betweencomprehensiveness and efficiency in equation learning. Inspired by stepwiseregression, our approach combines the coefficient of determination, $R^2$, andthe Bayesian model evidence, $p(\boldsymbol y|\mathcal M)$, in a novel way. Ourprocedure is characterized by a comprehensive search with just a minorreduction of the model space at each iteration step. With two flavors of ourapproach and the adoption of $p(\boldsymbol y|\mathcal M)$ for bi-directionalstepwise regression, we present a total of three new avenues for equationlearning. Through three extensive numerical experiments involving randompolynomials and dynamical systems, we compare our approach against fourstate-of-the-art methods and two standard approaches. The results demonstratethat our comprehensive search approach surpasses all other methods in terms ofidentification accuracy. In particular, the second flavor of our approachestablishes an efficient overfitting penalty solely based on $R^2$, whichachieves highest rates of exact equation recovery.

Improved identification accuracy in equation learning via comprehensive $\boldsymbol{R^2}$-elimination and Bayesian model selection