SIAM Review, Volume 66, Issue 3, Page 535-571, May 2024.
Supervised operator learning centers on the use of training data, in the form of input-output pairs, to estimate maps between infinite-dimensional spaces. It is emerging as a powerful tool to complement traditional scientific computing, which may often be framed in terms of operators mapping between spaces of functions. Building on the classical random features methodology for scalar regression, this paper introduces the function-valued random features method. This leads to a supervised operator learning architecture that is practical for nonlinear problems yet is structured enough to facilitate efficient training through the optimization of a convex, quadratic cost. Due to the quadratic structure, the trained model is equipped with convergence guarantees and error and complexity bounds, properties that are not readily available for most other operator learning architectures. At its core, the proposed approach builds a linear combination of random operators. This turns out to be a low-rank approximation of an operator-valued kernel ridge regression algorithm, and hence the method also has strong connections to Gaussian process regression. The paper designs function-valued random features that are tailored to the structure of two nonlinear operator learning benchmark problems arising from parametric partial differential equations. Numerical results demonstrate the scalability, discretization invariance, and transferability of the function-valued random features method.

中文翻译：

---

使用随机特征的算子学习：科学计算的工具


《SIAM 评论》，第 66 卷，第 3 期，第 535-571 页，2024 年 5 月。

监督算子学习的重点是使用输入输出对形式的训练数据来估计无限维空间之间的映射。它正在成为补充传统科学计算的强大工具，传统科学计算通常可以根据函数空间之间映射的运算符来构建。基于标量回归的经典随机特征方法，本文引入了函数值随机特征方法。这导致了一种监督算子学习架构，该架构对于非线性问题来说是实用的，但其结构足以通过凸二次成本的优化来促进有效的训练。由于二次结构，训练后的模型具有收敛保证以及误差和复杂性界限，这些属性对于大多数其他算子学习架构来说是不容易获得的。所提出的方法的核心是构建随机算子的线性组合。事实证明，这是算子值核岭回归算法的低秩近似，因此该方法也与高斯过程回归有很强的联系。本文设计了函数值随机特征，这些特征针对参数偏微分方程产生的两个非线性算子学习基准问题的结构进行了定制。数值结果证明了函数值随机特征方法的可扩展性、离散化不变性和可传递性。