UniFIDES: Universal Fractional Integro-Differential Equation Solvers

TL;DR

UniFIDES is a machine learning platform capable of efficiently solving a broad range of fractional and integer-order integro-differential equations, facilitating applications across science and engineering without specialized adjustments.

Contribution

This paper introduces UniFIDES, a universal ML-based solver for fractional and integer integro-differential equations, addressing nonlinear challenges without ad hoc modifications.

Findings

Accurately solves diverse FIDEs in science and engineering

Works in both forward and inverse problem settings

Demonstrates broad applicability across disciplines

Abstract

The development of data-driven approaches for solving differential equations has been followed by a plethora of applications in science and engineering across a multitude of disciplines and remains a central focus of active scientific inquiry. However, a large body of natural phenomena incorporates memory effects that are best described via fractional integro-differential equations (FIDEs), in which the integral or differential operators accept non-integer orders. Addressing the challenges posed by nonlinear FIDEs is a recognized difficulty, necessitating the application of generic methods with immediate practical relevance. This work introduces the Universal Fractional Integro-Differential Equation Solvers (UniFIDES), a comprehensive machine learning platform designed to expeditiously solve a variety of FIDEs in both forward and inverse directions, without the need for ad hoc manipulation of the equations. The effectiveness of UniFIDES is demonstrated through a collection of integer-order and fractional problems in science and engineering. Our results highlight UniFIDES' ability to accurately solve a wide spectrum of integro-differential equations and offer the prospect of using machine learning platforms universally for discovering and describing dynamical and complex systems.