Using Physics-Informed Neural Networks for Solving 2<sup>nd</sup>-Order Volterra Integro-Differential Equation by DeepXDE Library

Many disciplines widely use deep learning (DL) as a key tool for investigating the behavior of various systems. Deep learning (DL) has recently been utilized to solve differential equations using physics-based input. Physics-informed neural networks (PINNs) are a novel DL model that excels at solving both inverse and forward non-linear PDE problems. PINNs may be trained as surrogate models for approximation solutions to the VIDE without label data by embedding the physical information outlined by PDEs in feedforward NNs. The objective of this study is to solve the second-order Volterra integral-differential equations (2nd-order VIDEs) for the first time using PINN and the DeepXDE library, and find and reduce the square of the relative error. After being converted 2nd-order VIDEs to a PDE, the series of approximated closed-form iterated solutions is obtained using the generic Gauss legendary quadrate. Furthermore, three instances are provided and resolved to demonstrate the dependability, effectiveness, and suitability of the suggested approach. Identified tanh and sin as the activation functions, with L-BFGS and L-BFGS-B as the more suitable optimization approaches for addressing second-order VIDEs via PINN. The training time for all examples was 160–280 seconds, and discovered that the square of the relative error varies between 0.1e-5 and 0.1e-7, which is an outstanding result. The results demonstrate the advantages of faster convergence and improved accuracy.