  We introduce a novel approach for learning memory kernels in Generalized
Langevin Equations. This approach initially utilizes a regularized Prony method
to estimate correlation functions from trajectory data, followed by regression
over a Sobolev norm-based loss function with RKHS regularization. Our method
guarantees improved performance within an exponentially weighted L^2 space,
with the kernel estimation error controlled by the error in estimated
correlation functions. We demonstrate the superiority of our estimator compared
to other regression estimators that rely on L^2 loss functions and also an
estimator derived from the inverse Laplace transform, using numerical examples
that highlight its consistent advantage across various weight parameter
selections. Additionally, we provide examples that include the application of
force and drift terms in the equation.
