论文标题
通过深度学习解决非本地fokker-Planck方程
Solving Non-local Fokker-Planck Equations by Deep Learning
论文作者
论文摘要
物理知识的神经网络（PINN）最近成为在各种初始和边界条件下的大型部分偏微分方程的强大求解器。在本文中，我们提出了陷入困境的陷阱细菌，物理信息的神经网络与最近开发的改良梯形规则合并，用于准确评估分数laplacian并解决2d和3d中的空间分流fokker-planck方程。我们详细描述了修改的梯形规则并验证二阶精度。我们证明，通过使用低$ \ natercal {l}^2 $相对误差在各种数值示例上预测解决方案，陷阱细菌具有高表达能力。我们还使用局部指标（例如绝对误差和相对误差）来分析可以进一步改进的地方。我们提出了一种有效的方法，可以提高陷阱 - 细菌在局部指标上的性能，前提是对真实解决方案的高保真模拟进行了物理观察。除了适应性和网状独立性等深度学习求解器的通常优势外，Trapz-pinn还能够用（0,2）$的任意$α\ in nutiantal laplacian求解PDE，并专门研究矩形域。它还具有推广到更高维度的潜力。
Physics-informed neural networks (PiNNs) recently emerged as a powerful solver for a large class of partial differential equations under various initial and boundary conditions. In this paper, we propose trapz-PiNNs, physics-informed neural networks incorporated with a modified trapezoidal rule recently developed for accurately evaluating fractional laplacian and solve the space-fractional Fokker-Planck equations in 2D and 3D. We describe the modified trapezoidal rule in detail and verify the second-order accuracy. We demonstrate trapz-PiNNs have high expressive power through predicting solution with low $\mathcal{L}^2$ relative error on a variety of numerical examples. We also use local metrics such as pointwise 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 $α\in (0,2)$ and specializes on rectangular domain. It also has potential to be generalized into higher dimensions.
