We propose a numerical scheme based on Random Projection Neural Networks (RPNN) for the solution of Ordinary Differential Equations (ODEs) with a focus on stiff problems. In particular, we use an Extreme Learning Machine, a single-hidden layer Feedforward Neural Network with Radial Basis Functions which widths are uniformly distributed random variables, while the values of the weights between the input and the hidden layer are set equal to one. The numerical solution is obtained by constructing a system of nonlinear algebraic equations, which is solved with respect to the output weights using the Gauss-Newton method. For our illustrations, we apply the proposed machine learning approach to solve two benchmark stiff problems, namely the Rober and the van der Pol ones (the latter with large values of the stiffness parameter), and we perform a comparison with well-established methods such as the adaptive Runge-Kutta method based on the Dormand-Prince pair, and a variable-step variable-order multistep solver based on numerical differentiation formulas, as implemented in the \texttt{ode45} and \texttt{ode15s} MATLAB functions, respectively. We show that our proposed scheme yields good numerical approximation accuracy without being affected by the stiffness, thus outperforming in same cases the \texttt{ode45} and \texttt{ode15s} functions. Importantly, upon training using a fixed number of collocation points, the proposed scheme approximates the solution in the whole domain in contrast to the classical time integration methods.
翻译:我们提出了一个基于随机投影神经网络的数值方案( RPNN), 以解决普通差异方程式( ODEs) 的普通差异方程式( ODEs ) 。 特别是, 我们使用一个极端学习机器, 是一个单层隐藏层向神经网络, 带有半边基函数, 宽度是均匀分布的随机变量, 而输入和隐藏层之间的权重则被设定为等于一个。 数字解决方案是通过构建一个非线性代数方程式( RPNN) 系统获得的。 这个系统通过使用 Gaus- Newton 方法解决了输出权重的对比。 对于我们的插图, 我们采用拟议的机器学习方法来解决两个基准硬性硬性问题, 即 Rober 和 van der Pol 系统( 后者含有坚硬度参数的较大值), 我们与基于 Dormand- Prind对子的适应 Runge- Kutta方法, 以及基于数字差异公式的可变性可变式数级数级数解数解解解解数解解解解。 rode45} 和rodealnicalendAB 方法, 显示我们的拟议正正数的正正值=x=xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx