论文标题
分数Fokker-Planck方程的自适应深度密度近似
Adaptive deep density approximation for fractional Fokker-Planck equations
论文作者
论文摘要
在这项工作中，我们提出了基于求解分数fokker-Planck方程（FPE）的归一化流量的自适应深度学习方法。 FPE的解是概率密度函数（PDF）。传统的基于网格的方法是无效的，因为无界的计算域，大量尺寸和非局部分数操作员。为此，我们用基于流动的深层生成模型Sightifiend Krnet诱导的明确PDF模型表示解决方案，该模型从简单分布到目标分布构建了传输图。我们考虑了两种近似分数拉普拉斯的方法。一种方法是蒙特卡洛近似。另一种方法是构建具有高斯径向基函数（GRBF）的辅助模型，以近似解决方案，以便我们可以利用高斯的分数laplacian在分析上已知的事实。基于这两种不同的方法来近似分数拉普拉斯式，我们提出了两个模型MCNF和GRBFNF，以近似固定的FPE和MCTNF近似于时间依赖性的FPE。为了进一步提高准确性，我们会改进训练集和近似解决方案。提出了各种数值示例，以证明我们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 the unbounded computation domain, a large number of dimensions and the nonlocal fractional operator. To this end, we represent the solution with an explicit PDF model induced by a flow-based deep generative model, simplified KRnet, 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, MCNF and GRBFNF, to approximate stationary FPEs and MCTNF 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.
