Internal variable reformulation of Volterra integro-differential equations with exponential kernels for physics-informed neural networks

We developed a physics-informed neural network (PINN) framework for solving integro-differential equations (IDEs), with particular emphasis on Volterra-type problems with exponentially decaying kernels. While PINNs provide a flexible approach for incorporating physical laws without mesh-based discretization, the treatment of convolution integral terms remains computationally demanding. To address this issue, we introduced internal variables for exponential kernels, transforming the original IDE into an equivalent system of differential equations, eliminating the need for explicit quadrature, and significantly reducing computational cost and memory requirements. The proposed method incorporates both differential and integral operators within the PINN framework. Numerical results demonstrate that the method maintains accuracy while significantly improving computational efficiency. It also extends naturally to inverse problems, where viscoelastic parameters are accurately identified from sparse observations.