通过路径积分学习的随机有效动力学的重整化群层级

中文摘要 AI 辅助

本研究将空间中的随机重整化群（RG）与时间演化的路径积分描述相结合，给出了粗粒化动力学的空间层级以及时空路径上的分布。RG被定义为一个扩散过程，应用尺度相关的拉普拉斯阻尼并附加高斯噪声。每个空间尺度上的时间演化被表述为Onsager-Machlup作用量，其中包含漂移项（即预测器）和由RG确定振幅的白噪声。这些动力学的预测器通过最小化RG与路径积分描述之间路径分布的Kullback-Leibler散度来优化。最优预测器包含连接空间尺度的得分函数，因此预测器和得分是多尺度路径公式的两个方面。该公式统一了使用预测器的模拟、使用得分的无条件生成以及使用两者的超分辨率。预测器没有封闭形式，因此通过神经网络在两种实现中计算，这两种实现仅在得分计算方式上有所不同。第一种实现通过路径分布的自动微分获得得分，忠实于该公式。第二种实现通过去噪训练将得分作为额外的网络输出，计算成本较低。这两种实现在两个代表性多尺度系统（Kolmogorov流和双时间尺度Lorenz-96模型）的数值实验中得到验证。

英文摘要

This study combines a stochastic renormalization group (RG) in space with a path-integral description of the time evolution, giving a spatial hierarchy of coarse-grained dynamics together with the distribution over spatiotemporal paths. The RG is defined as a diffusion process that applies scale-dependent Laplacian damping together with additive Gaussian noise. The time evolution at each spatial scale is formulated as an Onsager--Machlup action, with a drift (i.e., the predictor) and white noise whose amplitude is fixed by the RG. The predictor of these dynamics is optimized by minimizing the Kullback--Leibler divergence between the path distributions from the RG and from the path-integral description. The optimal predictor contains the score function that connects the spatial scales, so the predictor and the score are two aspects of the same multiscale path formulation. The formulation unifies simulation using the predictor, unconditional generation using the score, and super-resolution using both. The predictor has no closed form, so it is computed by a neural network in two realizations that differ only in how the score is computed. The first realization obtains the score by automatic differentiation of the path distribution, remaining faithful to that formulation. The second obtains the score as an additional network output trained by denoising, at a lower computational cost. These two realizations are validated through numerical experiments on two representative multiscale systems, the Kolmogorov flow and the two-timescale Lorenz-96 model.

发表机构

• Research Institute for Earth and Information Sciences, Japan Agency for Marine-Earth Science and Technology（地球与信息科学研究所，日本海洋地球科学技术局）
• Aeolus Labs（埃俄罗斯实验室）

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