Physics-Informed Neural Networks for 2nd order ODEs with sharp gradients

In this work, four different methods based on Physics-Informed Neural Networks (PINNs) for solving Differential Equations (DE) are compared: Classic-PINN that makes use of Deep Neural Networks (DNNs) to approximate the DE solution;Deep-TFC improves the efficiency of classic-PINN by employing the constrained expression from the Theory of Functional Connections (TFC) so to analytically satisfy the DE constraints;PIELM that improves the accuracy of classic-PINN by employing a single-layer NN trained via Extreme Learning Machine (ELM) algorithm;X-TFC, which makes use of both constrained expression and ELM. The last has been recently introduced to solve challenging problems affected by discontinuity, learning solutions in cases where the other three methods fail. The four methods are compared by solving the boundary value problem arising from the 1D Steady-State Advection-Diffusion Equation for different values of the diffusion coefficient. The solutions of the DEs exhibit steep gradients as the value of the diffusion coefficient decreases, increasing the challenge of the problem.