We describe and analyze algorithms for shape-constrained symbolic regression, which allows the inclusion of prior knowledge about the shape of the regression function. This is relevant in many areas of engineering -- in particular whenever a data-driven model obtained from measurements must have certain properties (e.g. positivity, monotonicity or convexity/concavity). We implement shape constraints using a soft-penalty approach which uses multi-objective algorithms to minimize constraint violations and training error. We use the non-dominated sorting genetic algorithm (NSGA-II) as well as the multi-objective evolutionary algorithm based on decomposition (MOEA/D). We use a set of models from physics textbooks to test the algorithms and compare against earlier results with single-objective algorithms. The results show that all algorithms are able to find models which conform to all shape constraints. Using shape constraints helps to improve extrapolation behavior of the models.

中文翻译：

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使用形状约束改进符号回归模型

我们描述和分析了形状约束符号回归的算法，它允许包含关于回归函数形状的先验知识。这与许多工程领域相关——特别是当从测量中获得的数据驱动模型必须具有某些属性时（例如，正性、单调性或凸/凹）。我们使用软惩罚方法实现形状约束，该方法使用多目标算法来最小化约束违规和训练错误。我们使用非支配排序遗传算法（NSGA-II）以及基于分解的多目标进化算法（MOEA/D）。我们使用物理教科书中的一组模型来测试算法，并与单目标算法的早期结果进行比较。结果表明，所有算法都能够找到符合所有形状约束的模型。使用形状约束有助于改进模型的外推行为。