In recent years, machine learning has become an interdisciplinary research hotspot in nonlinear science and artificial intelligence. Nonlinear integro-differential equations (IDEs), as an essential mathematical model in science and engineering, often face challenges in forward problem analysis and inverse problem solving due to the complexity of their kernel functions. This paper proposes a machine learning framework that combines degenerate kernel schemes to solve the IDEs of mathematical models in nonlinear science, including forward problems and inverse problems. Herein, the general smooth continuous IDEs are first approximated to the IDEs with degenerate kernels, and the equivalent nonlinear differential equations are obtained by introducing the auxiliary differential operators with new boundary conditions to replace the integral ones. For the functions to be solved in the original IDEs and the new functions in the auxiliary differential operators, the independent full connection deep neural networks (FCDNN) are established. By constructing the loss function based on the equivalent nonlinear differential equations and all boundary conditions and initial conditions, this machine learning is trained to realize the solution of the nonlinear IDEs forward problem by using the Adam optimizer. By constructing new loss components based on the measured values of the function, the IDEs inverse problems can be further solved by machine learning, such as unknown parameters or source items in IDEs. Detailed numerical analysis of the forward problem, inverse problem, and some high-dimensional problems of the nonlinear IDEs shows that the proposed machine learning has high accuracy and universality for such nonlinear problems. In addition, the effects of the characteristics of the solution, the network framework, the forms of activation function and loss function, and physical information distribution points on the convergence of the machine learning method are discussed in detail. The universal machine learning solution for nonlinear IDEs can serve the applications of IDEs in science and engineering. The data and code accompanying this paper are publicly available at .

中文翻译：

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具有简并核格式的非线性积分微分方程的机器学习


近年来，机器学习已成为非线性科学和人工智能领域的跨学科研究热点。非线性积分微分方程（IDE）作为科学和工程中重要的数学模型，由于其核函数的复杂性，在正问题分析和逆问题求解中经常面临挑战。本文提出了一种结合简并核方案的机器学习框架来解决非线性科学中数学模型的IDE，包括正向问题和逆向问题。这里，首先将一般光滑连续IDE近似为具有退化核的IDE，并通过引入具有新边界条件的辅助微分算子来代替积分算子，得到等效的非线性微分方程。针对原有IDE中需要求解的函数和辅助微分算子中的新函数，建立了独立的全连接深度神经网络（FCDNN）。通过构造基于等效非线性微分方程和所有边界条件和初始条件的损失函数，训练该机器学习以使用 Adam 优化器实现非线性 IDE 前向问题的求解。通过根据函数的测量值构造新的损失分量，可以通过机器学习进一步解决 IDE 逆问题，例如 IDE 中的未知参数或源项。对非线性集成开发环境的正向问题、逆向问题和一些高维问题的详细数值分析表明，所提出的机器学习对于此类非线性问题具有较高的准确性和普适性。 此外，还详细讨论了解的特性、网络框架、激活函数和损失函数的形式以及物理信息分布点对机器学习方法收敛性的影响。非线性IDE通用机器学习解决方案可以服务于IDE在科学和工程领域的应用。本文随附的数据和代码可在 上公开获取。