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.

翻译：数据驱动的微分方程求解方法的发展，已在众多科学与工程领域催生了大量应用，并依然是当前科学研究的一个核心焦点。然而，大量自然现象包含记忆效应，这些效应最适合通过分数阶积分-微分方程（FIDEs）来描述，其中的积分或微分算子接受非整数阶。解决非线性FIDEs带来的挑战是一个公认的难题，需要应用具有直接实际意义的通用方法。本文介绍了通用分数阶积分-微分方程求解器（UniFIDES），这是一个全面的机器学习平台，旨在快速求解各种正向和反向的FIDEs，而无需对方程进行特定的专门处理。通过一系列科学与工程中的整数阶和分数阶问题，验证了UniFIDES的有效性。我们的结果突显了UniFIDES能够精确求解广泛的积分-微分方程，并为普遍使用机器学习平台来发现和描述动态复杂系统提供了前景。