利用连分数学习外推：预测超导材料的临界温度
Learning to extrapolate using continued fractions: Predicting the critical temperature of superconductor materials
AI总结：
本文提出一种基于连分数迭代拟合与加性样条模型的多变量回归方法，用于在扩展域中进行外推预测，并以超导体临界温度预测任务验证其优于多种现有回归方法。
摘要：
在人工智能（AI）和机器学习（ML）领域，利用有限实例 $S={(\mathbf{x^{(i)}},y^{(i)})}$ 来近似未知目标函数 $y=f(\mathbf{x})$ 是一个常见目标，其中 $\mathbf{x^{(i)}} \in D$，$D$ 表示感兴趣的域。我们将 $S$ 称为训练集，旨在找到一个低复杂度的数学模型，能够对新的实例 $\mathbf{x}$ 有效近似该目标函数。因此，模型的泛化能力在一个单独的集合 $T=\{\mathbf{x^{(j)}}\} \subset D$ 上进行评估，其中 $T \neq S$，且通常 $T \cap S = \emptyset$，以评估其在训练集之外的表现。然而，某些应用不仅要求在原始域 $D$ 内进行准确近似，还要求在包含 $D$ 的扩展域 $D'$ 内进行准确近似。这在新结构设计场景中尤为重要，因为在这些场景中，最小化近似误差至关重要。例如，当通过数据驱动方法开发新材料时，AI/ML 系统可以作为代理函数提供有价值的见解来指导设计过程。因此，学习到的模型可以用于促进新实验室实验的设计。在本文中，我们提出了一种基于连分数迭代拟合并结合加性样条模型的多变量回归方法。我们将该方法的性能与已有技术进行了比较，包括 AdaBoost、核岭回归、线性回归、Lasso Lars、线性支持向量回归、多层感知机、随机森林、随机梯度下降和 XGBoost。为了评估这些方法，我们聚焦于该领域的一个重要问题：基于物理化学特征预测超导体的临界温度。
英文摘要：
In the field of Artificial Intelligence (AI) and Machine Learning (ML), the approximation of unknown target functions $y=f(\mathbf{x})$ using limited instances $S={(\mathbf{x^{(i)}},y^{(i)})}$, where $\mathbf{x^{(i)}} \in D$ and $D$ represents the domain of interest, is a common objective. We refer to $S$ as the training set and aim to identify a low-complexity mathematical model that can effectively approximate this target function for new instances $\mathbf{x}$. Consequently, the model's generalization ability is evaluated on a separate set $T=\{\mathbf{x^{(j)}}\} \subset D$, where $T \neq S$, frequently with $T \cap S = \emptyset$, to assess its performance beyond the training set. However, certain applications require accurate approximation not only within the original domain $D$ but also in an extended domain $D'$ that encompasses $D$. This becomes particularly relevant in scenarios involving the design of new structures, where minimizing errors in approximations is crucial. For example, when developing new materials through data-driven approaches, the AI/ML system can provide valuable insights to guide the design process by serving as a surrogate function. Consequently, the learned model can be employed to facilitate the design of new laboratory experiments. In this paper, we propose a method for multivariate regression based on iterative fitting of a continued fraction, incorporating additive spline models. We compare the performance of our method with established techniques, including AdaBoost, Kernel Ridge, Linear Regression, Lasso Lars, Linear Support Vector Regression, Multi-Layer Perceptrons, Random Forests, Stochastic Gradient Descent, and XGBoost. To evaluate these methods, we focus on an important problem in the field: predicting the critical temperature of superconductors based on physical-chemical characteristics.
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