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.
翻译:在本文中,我们提出并研究以神经网络为基础的解决高度四孔多孔中程方程式(QPME)的方法。介绍了这种非线性PDE的三种变式配方:一种强有力的配方和两种薄弱的配方。对于强度配方,解决办法直接与神经网络进行参数化,并通过尽量减少PDE残留物进行优化。可以证明,优化问题的趋同保证了近似溶方在1美元意义上的趋同。弱质配方是在Brenier,Y.,2020年之后产生的,这是QPME非常薄弱的解决方法的特点。具体地说,这些配有中间功能,这些功能与神经网络相配,并受过优化弱质配方的训练。还进一步进行了广泛的数字测试,以调查低度和高度的每种配方的利和弊端。这是在解决高度非线性PDE与以神经网络为基础的方法的界线上所作的初步探索,我们希望这些方法可以为今后的调查提供一些有用的经验。