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 approach 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.

翻译：我们提出了一种从广义朗之万方程中学习记忆核的新方法。该方法首先利用正则化Prony方法从轨迹数据中估计相关函数，随后基于Sobolev范数损失函数并结合RKHS正则化进行回归。我们的方法保证了在指数加权$L^2$空间中的性能提升，且记忆核估计误差受相关函数估计误差的控制。通过数值示例，我们展示了该估计器相较于依赖$L^2$损失函数的其他回归估计器以及基于逆拉普拉斯变换的估计器的优越性，突显了其在多种权重参数选择下的一致优势。此外，我们还提供了包含方程中力和漂移项应用的示例。