标题： 从连续视角的机器学习 显示英文标题

标题： Machine Learning from a Continuous Viewpoint

摘要： 我们提出了一种机器学习的连续表述，将其视为变分问题和微分-积分方程的问题，在经典数值分析的精神下进行处理。 我们证明了传统的机器学习模型和算法，如随机特征模型、两层神经网络模型和残差神经网络模型，都可以以某种缩放形式作为不同连续表述的不同离散化而恢复。 我们还展示了从这种连续表述自然产生的新模型的例子，如基于流的随机特征模型，以及新算法的例子，如平滑粒子法和谱方法。 我们讨论了在这个框架下如何研究泛化误差和隐式正则化等问题。 显示英文摘要

摘要： We present a continuous formulation of machine learning, as a problem in the calculus of variations and differential-integral equations, in the spirit of classical numerical analysis. We demonstrate that conventional machine learning models and algorithms, such as the random feature model, the two-layer neural network model and the residual neural network model, can all be recovered (in a scaled form) as particular discretizations of different continuous formulations. We also present examples of new models, such as the flow-based random feature model, and new algorithms, such as the smoothed particle method and spectral method, that arise naturally from this continuous formulation. We discuss how the issues of generalization error and implicit regularization can be studied under this framework.