We introduce a novel approach for decomposing and learning every scale of a given multiscale objective function in $\mathbb{R}^d$, where $d\ge 1$. This approach leverages a recently demonstrated implicit bias of the optimization method of gradient descent by Kong and Tao, which enables the automatic generation of data that nearly follow Gibbs distribution with an effective potential at any desired scale. One application of this automated effective potential modeling is to construct reduced-order models. For instance, a deterministic surrogate Hamiltonian model can be developed to substantially soften the stiffness that bottlenecks the simulation, while maintaining the accuracy of phase portraits at the scale of interest. Similarly, a stochastic surrogate model can be constructed at a desired scale, such that both its equilibrium and out-of-equilibrium behaviors (characterized by auto-correlation function and mean path) align with those of a damped mechanical system with the original multiscale function being its potential. The robustness and efficiency of our proposed approach in multi-dimensional scenarios have been demonstrated through a series of numerical experiments. A by-product of our development is a method for anisotropic noise estimation and calibration. More precisely, Langevin model of stochastic mechanical systems may not have isotropic noise in practice, and we provide a systematic algorithm to quantify its covariance matrix without directly measuring the noise. In this case, the system may not admit closed form expression of its invariant distribution either, but with this tool, we can design friction matrix appropriately to calibrate the system so that its invariant distribution has a closed form expression of Gibbs.

翻译：本文提出了一种创新方法，用于分解和学习定义在$\mathbb{R}^d$（其中$d\ge 1$）上的给定多尺度目标函数的每个尺度。该方法利用了Kong和Tao近期揭示的梯度下降优化方法的隐式偏置特性，能够自动生成近似服从吉布斯分布（并附带任意所需尺度下有效势）的数据。该自动有效势建模的一个应用是构建降阶模型。例如，可开发确定性哈密顿替代模型，在显著缓解瓶颈模拟的刚性的同时，保持感兴趣尺度下相图的准确性。类似地，可在目标尺度上构建随机替代模型，使其平衡态与非平衡态行为（通过自相关函数和平均路径表征）均与以原始多尺度函数为势的阻尼机械系统保持一致。通过一系列数值实验，验证了本方法在多维场景下的鲁棒性与高效性。本研究的附带成果是一种各向异性噪声估计与校准方法。具体而言，随机机械系统的朗之万模型在实际中可能不具备各向同性噪声，我们提供了一套系统算法，无需直接测量噪声即可量化其协方差矩阵。在此情形下，系统亦可能不存在不变分布的闭式表达式，但借助该工具，可设计适当的摩擦矩阵对系统进行校准，使其不变分布具有吉布斯形式的闭式表达式。