用物理约束的机器学习解分数阶微分方程，提升精度与效率。
A Physics-Informed Machine Learning Approach for Solving Distributed Order Fractional Differential Equations
- 将物理定律嵌入支持向量回归框架，直接约束解的结构。
- 采用盖根鲍尔多项式作核函数，简化分数阶导数计算。
- 适用于含分布式阶分数阶导数的偏微分方程求解场景。
本文提出一种基于物理信息机器学习的新方法，用于求解分布式阶分数阶微分方程。核心思路是将支持向量回归（SVR）算法扩展，用于在训练阶段逼近控制方程的未知解。通过将分布式阶函数方程嵌入SVR框架，实现物理规律对学习过程的直接约束。为提高计算效率，采用具有分数阶微分性质的盖根鲍尔正交多项式作为核函数，简化问题建模。最终的SVR优化问题可转化为二次规划或其对偶形式下的正定系统求解。通过一系列数值实验验证了该方法在基于Caputo定义的分布式阶分数阶微分方程（含常微分与偏微分）上的有效性。
原文摘要 · Abstract (English)
This paper introduces a novel methodology for solving distributed-order fractional differential equations using a physics-informed machine learning framework. The core of this approach involves extending the support vector regression (SVR) algorithm to approximate the unknown solutions of the governing equations during the training phase. By embedding the distributed-order functional equation into the SVR framework, we incorporate physical laws directly into the learning process. To further enhance computational efficiency, Gegenbauer orthogonal polynomials are employed as the kernel function, capitalizing on their fractional differentiation properties to streamline the problem formulation. Finally, the resulting optimization problem of SVR is addressed either as a quadratic programming problem or as a positive definite system in its dual form. The effectiveness of the proposed approach is validated through a series of numerical experiments on Caputo-based distributed-order fractional differential equations, encompassing both ordinary and partial derivatives.
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