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.

翻译：本文提出了一种利用物理信息机器学习框架求解分布式阶分数阶微分方程的新方法。该方法的核心理念是在训练阶段扩展支持向量回归算法，以逼近控制方程的未知解。通过将分布式阶泛函方程嵌入SVR框架，我们将物理定律直接融入学习过程。为进一步提升计算效率，我们采用Gegenbauer正交多项式作为核函数，利用其分数阶微分特性简化问题表述。最终，将SVR的优化问题转化为二次规划问题或其对偶形式下的正定系统进行求解。通过对基于Caputo定义的分布式阶分数阶微分方程（涵盖常微分与偏微分情形）进行一系列数值实验，验证了所提方法的有效性。