广义朗之万方程中记忆核的学习
Learning Memory Kernels in Generalized Langevin Equations
AI总结：
本研究提出一种学习广义朗之万方程记忆核的新方法，结合正则化Prony方法与带RKHS正则化的Sobolev范数损失回归，在指数加权L²空间性能更优，经数值验证在多权重设置下均优于现有估计器。
摘要：
我们提出一种学习广义朗之万方程（Generalized Langevin Equations）中记忆核的新方法。该方法首先利用正则化Prony方法从轨迹数据估计相关函数，随后在基于Sobolev范数的损失函数上结合RKHS（再生核希尔伯特空间）正则化进行回归。我们的方法在指数加权L²空间中保证了更优的性能，其核估计误差由相关函数的估计误差控制。通过数值算例，我们证明了该估计器相较于其他依赖L²损失函数的回归估计器，以及一种由逆拉普拉斯变换得到的估计器的优越性，数值算例凸显了其在不同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项和漂移项应用的示例。
英文摘要：
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.
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