A Physics-Informed Machine Learning Approach for Solving Distributed Order Fractional Differential Equations
Alireza Afzal Aghaei
TL;DR
This paper develops a physics-informed machine learning framework to solve distributed-order fractional differential equations (DOFDEs) by extending least-squares support vector regression (LS-SVR) with Gegenbauer kernels. The method embeds the DOFDE into the regression objective, uses Gauss-Legendre quadrature to approximate the distributed-order integral, and leverages Caputo derivative properties for efficient evaluation of fractional derivatives. A dual formulation yields a positive-definite system that provides the solution in kernel form $\hat{u}(t)=\sum_{i=1}^N \beta_i \mathcal{L}K(t,t_i)$ with $K(t,t_i)=\sum_{j=0}^d G_j^{(\lambda)}(t) G_j^{(\lambda)}(t_i)$. Numerical experiments on both 1D and 2D DOFDEs demonstrate high accuracy when exact solutions are known and credible residual behavior otherwise, validating the approach and highlighting its computational advantages. The work advances PIML for fractional systems by coupling fractional basis kernels with a tractable optimization framework, enabling efficient and accurate solutions for problems with memory effects across multiple time scales.
Abstract
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.