Numerical solutions of integro-differential equations with machine learning

Objective: In this study, machine learning has been employed to obtain numerical solutions for linear and nonlinear Volterra and Fredholm integro-differential equations. For this purpose, appropriate artificial neural network models were examined by defining the necessary cost function for the problems. Material and Methods: Python programming language was preferred for the study due to its open-source nature. The PyTorch library was utilized for deep learning methods. The artificial neural network models used in the study include Multilayer Perceptron, Radial Basis Function Network, Recurrent Neural Network, LSTM and GRU. The aim was to compare the performance of these models considering their different structures and characteristics. Results: The performance of the defined cost function and the used artificial neural network models were evaluated by examining 12 different examples in order to assess the system's success in numerical solution. The results were shared with the readers in a comparative manner. Conclusion: Based on the successful results obtained, it is observed that the proposed study provides a good alternative to analytical methods and classical numerical solution methods in numerical computation.