Machine Learning for Nonlinear Integro-Differential Equations with Degenerate Kernel Scheme

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 highdimensional 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 https://github.com/Lihui0626/PINN-for-IDEs.