Physics-informed neural networks with adaptive localized artificial viscosity

Physics-informed Neural Network (PINN) is a promising tool that has been applied in variety of physical phenomena described by partial differential equations (PDE). However, it observed PINNs are difficult to train certain “stiff” problems, which include various nonlinear hyperbolic PDEs display shocks their solutions. Recent studies added diffusion term the PDE, and an artificial viscosity (AV) value was manually tuned allow solve these problems. In this paper, we propose three approaches address problem, none rely on priori definition value. The first method learns global AV value, whereas other two learn localized values around shocks, means parametrized map or residual-based map. We proposed methods inviscid Burgers equation Buckley-Leverett equation, latter being classical problem Petroleum Engineering. results show able both small accurate shock location improve approximation error over nonadaptive alternative method.