Operator learning is a data-driven approximation of mappings between infinite-dimensional function spaces, such as the solution operators of partial differential equations. Kernel-based operator learning can offer accurate, theoretically justified approximations that require less training than standard methods. However, they can become computationally prohibitive for large training sets and can be sensitive to noise. We propose a regularized random Fourier feature (RRFF) approach, coupled with a finite element reconstruction map (RRFF-FEM), for learning operators from noisy data. The method uses random features drawn from multivariate Student's $t$ distributions, together with frequency-weighted Tikhonov regularization that suppresses high-frequency noise. We establish high-probability bounds on the extreme singular values of the associated random feature matrix and show that when the number of features $N$ scales like $m \log m$ with the number of training samples $m$, the system is well-conditioned, which yields estimation and generalization guarantees. Detailed numerical experiments on benchmark PDE problems, including advection, Burgers', Darcy flow, Helmholtz, Navier-Stokes, and structural mechanics, demonstrate that RRFF and RRFF-FEM are robust to noise and achieve improved performance with reduced training time compared to the unregularized random feature model, while maintaining competitive accuracy relative to kernel and neural operator tests.

翻译：算子学习是无限维函数空间之间映射（例如偏微分方程解算子）的数据驱动逼近方法。基于核的算子学习能够提供精确的、理论可证的逼近，且比标准方法需要更少的训练。然而，对于大规模训练集，其计算成本可能过高，且对噪声敏感。本文提出一种正则化随机傅里叶特征方法，结合有限元重构映射，用于从含噪数据中学习算子。该方法采用从多元学生$t$分布中抽取的随机特征，并辅以频率加权的Tikhonov正则化来抑制高频噪声。我们建立了相关随机特征矩阵极端奇异值的高概率界，并证明当特征数量$N$与训练样本数$m$满足$N \sim m \log m$的比例关系时，系统是良态的，从而获得估计与泛化保证。在包括平流、Burgers、达西流、Helmholtz、Navier-Stokes及结构力学在内的基准PDE问题上的详细数值实验表明，RRFF与RRFF-FEM对噪声具有鲁棒性，相较于未正则化的随机特征模型，能以更少的训练时间实现性能提升，同时相对于核方法与神经算子测试保持具有竞争力的精度。