A Physics-Informed Machine Learning Approach for Solving Distributed Order Fractional Differential Equations
This addresses computational challenges in fractional calculus for researchers in applied mathematics and engineering, though it appears incremental as it builds on existing SVR and physics-informed methods.
The paper tackles solving distributed-order fractional differential equations by developing a physics-informed machine learning approach that extends support vector regression with Gegenbauer polynomial kernels, achieving validation through numerical experiments on Caputo-based equations.
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.