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.

我们介绍了一种新的学习内存核函数的方法，通过一种正则化的 Prony 方法从轨迹数据中估计相关函数，然后利用一种基于 Sobolev 范数损失函数的回归和 RKHS 正规化来提高性能，我们证明了我们的估计器相对于依赖于 L2 损失函数的其他回归估计器和由逆拉普拉斯变换导出的估计器的优越性，并通过多个权重参数选择的数值例子来突出显示其一致优势，此外，我们提供了方程中力和漂移项的应用示例。

泛化朗之万方程中的学习记忆核

Learning Memory Kernels in Generalized Langevin Equations

Regularization plays a pivotal role in ill-posed machine learning and inverse
problems. However, the fundamental comparative analysis of various
regularization norms remains open. We establish a small noise analysis
framework to assess the effects of norms in Tikhonov and RKHS regularizations,
in the context of ill-posed linear inverse problems with Gaussian noise. This
framework studies the convergence rates of regularized estimators in the small
noise limit and reveals the potential instability of the conventional
L2-regularizer. We solve such instability by proposing an innovative class of
adaptive fractional RKHS regularizers, which covers the L2 Tikhonov and RKHS
regularizations by adjusting the fractional smoothness parameter. A surprising
insight is that over-smoothing via these fractional RKHSs consistently yields
optimal convergence rates, but the optimal hyper-parameter may decay too fast
to be selected in practice.

本文通过建立一个小噪声分析框架，比较了 Tikhonov 和 RKHS 正则化问题中各种正则化规范的效果，发现 L2 正则化器存在潜在的不稳定性，并通过提出一种新颖的自适应分数 RKHS 正则化器来解决其不稳定性，实验结果表明，过度平滑可以提高收敛速度，但最优超参数难以实际选择。