# A Physics-Informed Machine Learning Approach for Solving Distributed Order Fractional Differential Equations

arXiv: https://arxiv.org/abs/2409.03507 · HTML: https://mlsanity.com/paper/2409.03507/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 57 references that are in the map.
- First posted on arXiv 5 Sep 2024.

## Builds on (most cited first, 3 of 3)

- [An Orthogonal Polynomial Kernel-Based Machine Learning Model for Differential-Algebraic Equations](https://mlsanity.com/paper/2401.14382/) (2024)
- [Analytically Embedding Differential Equation Constraints into Least Squares Support Vector Machines using the Theory of Functional Connections](https://mlsanity.com/paper/1812.05571/) (2018)
- [A least squares support vector regression for anisotropic diffusion filtering](https://mlsanity.com/paper/2202.00595/) (2022)

## Built on by (most cited first, 1 of 1)

- [Tri-Accel: Curvature-Aware Precision-Adaptive and Memory-Elastic Optimization for Efficient GPU Usage](https://mlsanity.com/paper/2508.16905/) (2025)

## Abstract (arXiv)

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.