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.

翻译：监督式算子学习的核心在于利用输入-输出对形式的训练数据来估计无限维空间之间的映射关系。它正逐渐成为补充传统科学计算的强大工具——传统科学计算常可表述为函数空间之间的算子映射问题。本文在经典标量回归随机特征方法的基础上，提出了函数值随机特征方法。由此构建的监督式算子学习架构既适用于非线性问题，又具有足够的结构化特性，可通过优化凸二次代价函数实现高效训练。得益于二次结构，训练后的模型具备收敛保证及误差与复杂度界限，这些性质在大多数其他算子学习架构中难以直接获得。该方法的核心在于构建随机算子的线性组合，这实际上是对算子值核岭回归算法的低秩逼近，因此该方法也与高斯过程回归存在紧密联系。本文针对参数化偏微分方程衍生的两个非线性算子学习基准问题，设计了符合其结构特性的函数值随机特征。数值实验验证了函数值随机特征方法的可扩展性、离散不变性及可迁移性。