Study of 1^st Order Numerical Scheme Physics Informed Neural Network (N-PINN) for 1D Riemann Problem

Recent research works for solving partial differential equations (PDEs) with deep neural networks (DNNs) have demonstrated that spatiotemporal function approximators defined by auto-differentiation are effective for approximating nonlinear problems, e.g. the Burger’s equation, heat conduction equations, Allen-Cahn and other reaction-diffusion equations, and Navier-Stokes equation. Meanwhile, researchers apply automatic differentiation in physics-informed neural network (PINN) to solve nonlinear hyperbolic systems based on conservation laws with highly discontinuous transition, such as Riemann problem, by inverse problem formulation in data-driven approach. However, it remains a challenge for forward methods using DNNs without knowing part of the solution to resolve discontinuities in nonlinear conservation laws. In this study, we incorporate 1^st order numerical schemes into DNNs to set up functional approximator instead of auto-differentiation from traditional deep learning framework e.g. TensorFlow package, which improves the effectiveness of capturing discontinuities in Riemann problems with constraint conditions e.g. boundary conditions and initial conditions being applied. If partial data in shockwave region of the solution is adopted for numerical physics-informed neural network (N-PINN), the results of predictions are more effective than traditional PINN set up by automatic differentiation.