论文标题
通过人工神经网络的可变投影方法对部分微分方程的数值近似
Numerical Approximation of Partial Differential Equations by a Variable Projection Method with Artificial Neural Networks
论文作者
论文摘要
我们提出了一种基于可变投影（VARPRO）框架和人工神经网络（ANN）的线性和非线性PDE的方法。对于线性PDE，在搭配点上执行边界/初始值问题会导致有关网络系数的可分离非线性最小二乘问题。我们通过VARPRO方法重新制定了这一问题，以消除线性输出层系数，从而导致有关隐藏层系数的问题减少了。首先，通过非线性最小二乘法解决了减少的问题，以确定隐藏层系数，然后通过线性最小二乘方法计算输出层系数。对于非线性PDE，在搭配点上执行边界/初始值问题会导致不可分开的非线性最小二乘问题，这阻止了此类问题的VARPRO策略。为了启用非线性PDE的VARPRO方法，我们首先使用特定形式的线性化将牛顿迭代的问题线性化。线性化系统由VARPRO框架与ANN一起解决。牛顿迭代的收敛后，网络系数为原始非线性问题提供了解决方案字段的表示。我们提供了具有线性和非线性PDE的足够数值示例，以证明此处方法的性能。对于光滑的场解决方案，随着套在一起点的数量或输出层系数的数量增加，当前方法的误差会呈指数减小。我们将当前方法与先前工作中的ELM方法进行了比较。在相同的条件和网络配置下，当前方法的精度明显优于ELM方法。
We present a method for solving linear and nonlinear PDEs based on the variable projection (VarPro) framework and artificial neural networks (ANN). For linear PDEs, enforcing the boundary/initial value problem on the collocation points leads to a separable nonlinear least squares problem about the network coefficients. We reformulate this problem by the VarPro approach to eliminate the linear output-layer coefficients, leading to a reduced problem about the hidden-layer coefficients only. The reduced problem is solved first by the nonlinear least squares method to determine the hidden-layer coefficients, and then the output-layer coefficients are computed by the linear least squares method. For nonlinear PDEs, enforcing the boundary/initial value problem on the collocation points leads to a nonlinear least squares problem that is not separable, which precludes the VarPro strategy for such problems. To enable the VarPro approach for nonlinear PDEs, we first linearize the problem with a Newton iteration, using a particular form of linearization. The linearized system is solved by the VarPro framework together with ANNs. Upon convergence of the Newton iteration, the network coefficients provide the representation of the solution field to the original nonlinear problem. We present ample numerical examples with linear and nonlinear PDEs to demonstrate the performance of the method herein. For smooth field solutions, the errors of the current method decrease exponentially as the number of collocation points or the number of output-layer coefficients increases. We compare the current method with the ELM method from a previous work. Under identical conditions and network configurations, the current method exhibits an accuracy significantly superior to the ELM method.
