We consider the use of extreme learning machines (ELM) for computational partial differential equations (PDE). In ELM the hidden-layer coefficients in the neural network are assigned to random values generated on $[-R_m,R_m]$ and fixed, where $R_m$ is a user-provided constant, and the output-layer coefficients are trained by a linear or nonlinear least squares computation. We present a method for computing the optimal value of $R_m$ based on the differential evolution algorithm. The presented method enables us to illuminate the characteristics of the optimal $R_m$ for two types of ELM configurations: (i) Single-Rm-ELM, in which a single $R_m$ is used for generating the random coefficients in all the hidden layers, and (ii) Multi-Rm-ELM, in which multiple $R_m$ constants are involved with each used for generating the random coefficients of a different hidden layer. We adopt the optimal $R_m$ from this method and also incorporate other improvements into the ELM implementation. In particular, here we compute all the differential operators involving the output fields of the last hidden layer by a forward-mode auto-differentiation, as opposed to the reverse-mode auto-differentiation in a previous work. These improvements significantly reduce the network training time and enhance the ELM performance. We systematically compare the computational performance of the current improved ELM with that of the finite element method (FEM), both the classical second-order FEM and the high-order FEM with Lagrange elements of higher degrees, for solving a number of linear and nonlinear PDEs. It is shown that the current improved ELM far outperforms the classical FEM. Its computational performance is comparable to that of the high-order FEM for smaller problem sizes, and for larger problem sizes the ELM markedly outperforms the high-order FEM.
翻译:我们考虑使用极端学习机器(ELM)来计算部分偏差方程式(PDE)。在ELM中,神经网络中的隐藏层系数被指定为在 $-R_m,R_m) 和固定生成的随机值,其中$_m美元是一个用户提供的常数,输出层系数由线性或非线性最低平方计算来培训。我们提出了一个方法,根据差异进化算法来计算最优值$_m美元。推出的方法使我们能够为两种类型的 ELM 配置显示最佳值$_m:(i) 单一-Rm-ELM,其中单R_m美元是用于生成所有隐藏层的随机系数,而产出层的多R_m-mcal系数则用于生成不同隐藏层的随机系数。我们采用了最优的 $_mper 方法,并将其他改进的改进值纳入 ELM 配置的当前配置的当前正轨值, EM- 高级运算的高级性能是远向电磁度显示前电磁度的高级电磁度。我们对前电磁度显示前电磁度的电磁的电磁度的电磁的电磁度,这里显示前电磁度的电磁的电磁度的电磁的电磁度的电磁性能的高级电磁性能。