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.

翻译：我们提出了一种用于学习广义朗之万方程中记忆核的新方法。该方法首先利用正则化的Prony方法从轨迹数据中估计相关函数，随后通过基于Sobolev范数的损失函数结合再生核希尔伯特空间（RKHS）正则化进行回归。我们的方法保证了指数加权L²空间内的性能提升，其中核估计误差受限于估计相关函数的误差。通过数值算例，我们展示了本估计量相较于其他依赖L²损失函数的回归估计量以及基于逆拉普拉斯变换的估计量具有一致性优势，且该优势在不同权重参数选择下均得以保持。此外，我们还提供了包含力项和漂移项在方程中应用的示例。