We present two effective methods for solving high-dimensional partial differential equations (PDE) based on randomized neural networks. Motivated by the universal approximation property of this type of networks, both methods extend the extreme learning machine (ELM) approach from low to high dimensions. With the first method the unknown solution field in $d$ dimensions is represented by a randomized feed-forward neural network, in which the hidden-layer parameters are randomly assigned and fixed while the output-layer parameters are trained. The PDE and the boundary/initial conditions, as well as the continuity conditions (for the local variant of the method), are enforced on a set of random interior/boundary collocation points. The resultant linear or nonlinear algebraic system, through its least squares solution, provides the trained values for the network parameters. With the second method the high-dimensional PDE problem is reformulated through a constrained expression based on an Approximate variant of the Theory of Functional Connections (A-TFC), which avoids the exponential growth in the number of terms of TFC as the dimension increases. The free field function in the A-TFC constrained expression is represented by a randomized neural network and is trained by a procedure analogous to the first method. We present ample numerical simulations for a number of high-dimensional linear/nonlinear stationary/dynamic PDEs to demonstrate their performance. These methods can produce accurate solutions to high-dimensional PDEs, in particular with their errors reaching levels not far from the machine accuracy for relatively lower dimensions. Compared with the physics-informed neural network (PINN) method, the current method is both cost-effective and more accurate for high-dimensional PDEs.

翻译：我们提出了两种基于随机化神经网络的高维偏微分方程（PDE）求解方法。受此类网络通用逼近特性的启发，两种方法均将极限学习机（ELM）方法从低维推广至高维。第一种方法中，$d$维未知解场通过随机化前馈神经网络表示：隐层参数随机赋值并固定，输出层参数通过训练优化。在随机选取的内点/边界配置点上，强制满足PDE、边界/初始条件以及（针对方法局部变体的）连续性条件；通过最小二乘求解所得线性或非线性代数系统，为网络参数提供训练值。第二种方法基于函数连接理论近似变体（A-TFC）的约束表达式重新表述高维PDE问题，该变体避免了TFC项数随维度指数增长的问题。A-TFC约束表达式中的自由场函数由随机神经网络表示，其训练流程与第一种方法类似。我们通过大量数值仿真（涵盖多种高维线性/非线性稳态/动态PDE）验证方法性能。这些方法可精确求解高维PDE，特别是在相对低维情况下误差可接近机器精度。与物理信息神经网络（PINN）方法相比，本方法在高维PDE求解中兼具成本效益与更高精度。