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神经网络求解Boltzmann-BGK方程及其在微流中的应用
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Abstract:
本文提出一种基于神经网络的BGK方程求解方法，特别关注在微流问题中的应用。首先，通过引入灵活辅助分布函数构造BGK方程的降维模型，从而有效降低方程维度。其次，设计全连接神经网络架构高效逼近降维分布函数，以避免时空离散化。接着针对微流问题中复杂的Maxwell边界条件，提出特殊设计的损失函数进行处理。此外，利用多尺度输入策略和Maxwellian分裂技术以提升逼近效率。最后，通过对一维Couette流和二维矩形风管流两个经典问题进行数值实验，验证了该方法的有效性。
We consider the neural representation to solve the BGK equation, especially focusing on the application in microscopic flow problems. Firstly, a new dimension reduction model of the BGK equation with the flexible auxiliary distribution functions is deduced to reduce the problem dimension. Then, a fully connected neural network is utilized to approximate the dimension-reduced distribution with extremely high efficiency and to avoid discretization in space and time. A specially designed loss function is employed to deal with the Maxwell boundary conditions in microflow problems. Moreover, strategies such as multi-scale input and Maxwellian splitting are applied to further enhance the approximation efficiency. Finally, two classical numerical experiments, including one-dimensional Couette flow and two-dimensional duct flow, are studied to demonstrate the effectiveness of this neural representation method.
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