A randomized neural network based Petrov–Galerkin method for approximating the solution of fractional order boundary value problems

IF 1.4 Q2 MATHEMATICS, APPLIED Results in Applied Mathematics Pub Date : 2024-08-01 DOI:10.1016/j.rinam.2024.100493
John P. Roop
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Abstract

This article presents the implementation of a randomized neural network (RNN) approach in approximating the solution of fractional order boundary value problems using a Petrov–Galerkin framework with Lagrange basis test functions. Traditional methods, like Physics Informed Neural Networks (PINNs), use standard deep learning techniques, which suffer from a computational bottleneck. In contrast, RNNs offer an alternative by employing a random structure with random coefficients, only solving for the output layer. We allow for the application of numerical analysis principles by using RNNs as trial functions and piecewise Lagrange polynomials as test functions. The article covers the construction and properties of the RNN basis, the definition and solution of fractional boundary value problems, and the implementation of the RNN Petrov–Galerkin method. We derive the stiffness matrix and solve it using least squares. Error analysis shows that the method meets the requirements of the Lax–Milgram lemma along with a Ceá inequality, ensuring optimal error estimates, depending on the regularity of the exact solution. Computational experiments demonstrate the method’s efficacy, including multiples cases with both regular and irregular solutions. The results highlight the utility of RNN-based Petrov–Galerkin methods in solving fractional differential equations with experimental convergence.

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基于随机神经网络的 Petrov-Galerkin 方法,用于近似求解分数阶边界值问题
本文介绍了一种随机神经网络(RNN)方法的实施情况,该方法使用带有拉格朗日基测试函数的彼得罗夫-加勒金框架来近似求解分数阶边界值问题。传统方法,如物理信息神经网络(PINNs),使用标准的深度学习技术,存在计算瓶颈。相比之下,RNNs 提供了另一种选择,即采用随机系数的随机结构,只对输出层进行求解。我们将 RNN 作为试验函数,将片断拉格朗日多项式作为测试函数,从而应用了数值分析原理。文章涉及 RNN 基础的构建和特性、分数边界值问题的定义和求解,以及 RNN Petrov-Galerkin 方法的实现。我们推导了刚度矩阵,并使用最小二乘法求解。误差分析表明,该方法符合 Lax-Milgram Lemma 以及 Ceá 不等式的要求,确保了最佳误差估计,这取决于精确解的正则性。计算实验证明了该方法的有效性,包括规则解和不规则解的多种情况。这些结果凸显了基于 RNN 的 Petrov-Galerkin 方法在求解分数微分方程时的实用性和实验收敛性。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
Results in Applied Mathematics
Results in Applied Mathematics Mathematics-Applied Mathematics
CiteScore
3.20
自引率
10.00%
发文量
50
审稿时长
23 days
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