Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations
Adaptive Deep Density Approximation for Fractional Fokker–Planck Equations
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DOI：
10.1007/s10915-023-02379-z
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发表时间：
2022-10
影响因子：
2.5
通讯作者：
Li Zeng;X. Wan;Tao Zhou
中科院分区：
文献类型：
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作者：
Li Zeng;X. Wan;Tao Zhou
在这项工作中，我们提出了基于归一化流的自适应深度学习方法来求解分数福克 - 普朗克方程（FPE）。 FPE 的解是概率密度函数 (PDF)。传统的基于网格的方法由于无界计算域、大量维度和非局部分数算子而无效。为此，我们用基于流的深度生成模型诱导的显式 PDF 模型来表示解决方案，该模型构建了从简单分布到目标分布的传输图。我们考虑两种方法来近似分数拉普拉斯。一种方法是蒙特卡罗近似。另一种方法是使用高斯径向基函数 (GRBF) 构建辅助模型来近似解，这样我们就可以利用高斯的分数拉普拉斯算子通过分析已知的事实。基于这两种不同的分数拉普拉斯近似方法，我们提出了两种近似稳态 FPE 的模型和一种近似时间相关 FPE 的模型。为了进一步提高准确性，我们交替细化训练集和近似解。提出了各种数值示例来证明我们的自适应深度密度方法的有效性。
In this work, we propose adaptive deep learning approaches based on normalizing flows for solving fractional Fokker–Planck equations (FPEs). The solution of a FPE is a probability density function (PDF). Traditional mesh-based methods are ineffective because of a unbounded computation domain, a large number of dimensions and a nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, which constructs a transport map from a simple distribution to the target distribution. We consider two methods to approximate the fractional Laplacian. One method is the Monte Carlo approximation. The other method is to construct an auxiliary model with Gaussian radial basis functions (GRBFs) to approximate the solution such that we may take advantage of the fact that the fractional Laplacian of a Gaussian is known analytically. Based on these two different ways for the approximation of the fractional Laplacian, we propose two models to approximate stationary FPEs and one model to approximate time-dependent FPEs. To further improve the accuracy, we refine the training set and the approximate solution alternately. A variety of numerical examples is presented to demonstrate the effectiveness of our adaptive deep density approaches.