In the field of equation learning, exhaustively considering all possible equations 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 multicollinearity poses difficulties for sparse regression techniques, and greedy steps may inadvertently exclude terms of the true equation, leading to reduced identification accuracy. In this article, we present an approach 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, $p(\boldsymbol y|\mathcal M)$, in a novel way. Our procedure is characterized by a comprehensive search with just a minor reduction of the model space at each iteration step. With two flavors of our approach and the adoption of $p(\boldsymbol y|\mathcal M)$ for bi-directional stepwise regression, we present a total of three new avenues for equation learning. Through three extensive numerical experiments involving random polynomials and dynamical systems, we compare our approach against four state-of-the-art methods and two standard approaches. The results demonstrate that our comprehensive search approach surpasses all other methods in terms of identification accuracy. In particular, the second flavor of our approach establishes an efficient overfitting penalty solely based on $R^2$, which achieves highest rates of exact equation recovery.

方程学习方法中，综合考虑基函数字典内的所有可能方程是不可行的。稀疏回归和贪婪算法已成为应对这一挑战的流行方法。然而，多重共线性的存在给稀疏回归技术带来困难，而贪婪步骤可能会无意中排除真实方程的项，导致降低辨识准确性。本文提出一种在方程学习中平衡综合性与效率的方法，通过将判定系数和贝叶斯模型证据以新颖的方式相结合，经过全面搜索并在每次迭代步骤中只对模型空间进行轻微缩减。通过三个包含随机多项式和动力系统的大规模数值实验，与四种最先进方法和两种标准方法进行比较，结果表明我们的综合搜索方法在辨识准确性方面超过所有其他方法。特别是我们的第二种方法仅基于判定系数建立了一种高效的过拟合惩罚机制，实现了最高的确切方程恢复率。

通过全面 $oldsymbol{R^2}$ 淘汰和贝叶斯模型选择提高方程式学习中的识别准确性

通过全面 $oldsymbol{R^2}$ 淘汰和贝叶斯模型选择提高方程式学习中的识别准确性