论文标题
基于神经网络的变异方法，用于在高维度求解二次多孔介质方程
Neural Network Based Variational Methods for Solving Quadratic Porous Medium Equations in High Dimensions
论文作者
论文摘要
在本文中，我们提出并研究基于神经网络的方法，用于用于高维二次多孔培养基（qpme）的解决方案。提出了该非线性PDE的三种变分配方：强大的配方和两个弱制剂。对于强制配方，该溶液通过神经网络直接进行参数化，并通过最大程度地减少PDE残差来进行优化。可以证明，优化问题的收敛性确保了$ l^1 $ sense中近似解决方案的收敛性。弱制剂是在Brenier，Y。，2020年得出的，其特征是qpme的非常弱的解决方案。具体而言，这些解决方案用中间功能表示，这些功能通过神经网络进行了参数化并经过训练以优化弱制剂。进一步进行了广泛的数值测试，以调查低维和高维度的每个配方的利弊。这是沿着使用基于神经网络的方法来解决高维非线性PDE的最初探索，我们希望这可以为将来的研究提供一些有用的经验。
In this paper, we propose and study neural network based methods for solutions of high-dimensional quadratic porous medium equation (QPME). Three variational formulations of this nonlinear PDE are presented: a strong formulation and two weak formulations. For the strong formulation, the solution is directly parameterized with a neural network and optimized by minimizing the PDE residual. It can be proved that the convergence of the optimization problem guarantees the convergence of the approximate solution in the $L^1$ sense. The weak formulations are derived following Brenier, Y., 2020, which characterizes the very weak solutions of QPME. Specifically speaking, the solutions are represented with intermediate functions who are parameterized with neural networks and are trained to optimize the weak formulations. Extensive numerical tests are further carried out to investigate the pros and cons of each formulation in low and high dimensions. This is an initial exploration made along the line of solving high-dimensional nonlinear PDEs with neural network based methods, which we hope can provide some useful experience for future investigations.
