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Title

Numerical Analysis and Deep Learning Solver of the Non-local Fokker-Planck Equations

Creator

Jiang, Senbao

Date

2022

Description

This thesis is divided into three mutually connected parts. ...
Show moreThis thesis is divided into three mutually connected parts. In the first part, we introduce and analyze arbitrarily high-order quadrature rules for evaluating the two-dimensional singular integrals of the forms \begin{align*} I_{i,j} = \int_{\mathbb{R}^2}\phi(x)\frac{x_ix_j}{|x|^{2+\alpha}} \d x, \quad 0< \alpha < 2 \end{align*} where $i,j\in\{1,2\}$ and $\phi\in C_c^N$ for $N\geq 2$. This type of singular integrals and its quadrature rule appear in the numerical discretization of fractional Laplacian in non-local Fokker-Planck Equations in 2D. The quadrature rules are trapezoidal rules equipped with correction weights for points around singularity. We prove the order of convergence is $2p+4-\alpha$, where $p\in\mathbb{N}_{0}$ is associated with total number of correction weights. We present numerical experiments to validate the order of convergence of the proposed modified quadrature rules. In the second part, we propose and analyze a general arbitrarily high-order modified trapezoidal rule for a class of weakly singular integrals of the forms $I = \int_{\R^n}\phi(x)s(x)\d x$ in $n$ dimensions, where $\phi$ and $s$ is the regular and singular part respectively. The admissible class requires $s$ satisfies three hypotheses and is large enough to contain singular kernel of the form $P(x)/|x|^r,\ r > 0$ where $P(x)$ is any monomial with degree strictly less than $r$. The modified trapezoidal rule is the singularity-punctured trapezoidal rule plus correction terms involving the correction weights for grid points around singularity. Correction weights are determined by enforcing the quadrature rule to exactly evaluate some monomials and solving corresponding linear systems. A long-standing difficulty of these types of methods is establishing the non-singularity of the linear system, despite strong numerical evidence. By using an algebraic-combinatorial argument, we show the non-singularity always holds and prove the general order of convergence of the modified quadrature rule. We present numerical experiments to validate the order of convergence. In the final part, we propose \emph{trapz-PiNN}, a physics-informed neural network incorporated with a modified trapezoidal rule and solve the space-fractional Fokker-Planck equations in 2D and 3D. We verify the modified trapezoidal rule has the second-order accuracy for evaluating the fractional laplacian. We demonstrate trapz-PiNNs have high expressive power through predicting solutions with low $\mathcal{L}^2$ relative error on a variety of numerical examples. We also use local metrics such as point-wise absolute and relative errors to analyze where could be further improved. We present an effective method for improving performance of trapz-PiNN on local metrics, provided that physical observations of high-fidelity simulation of the true solution are available. Besides the usual advantages of the deep learning solvers such as adaptivity and mesh-independence, the trapz-PiNN is able to solve PDEs with fractional laplacian with arbitrary $\alpha\in (0,2)$ and specializes on rectangular domains. It also has potential to be generalized into higher dimensions.
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