SCINN: Solution Character-Informed Neural Network for Scalar Conservation Laws

The physics-informed neural network (PINN) faces significant challenges when approximating solutions to conservation laws, particularly in ensuring conservation and accurately resolving discontinuities. To address these limitations, we propose solution character-informed neural network (SCINN), a novel framework that incorporates the boundedness constraint, implicit solution form, and Rankine–Hugoniot condition into the loss function, thereby enforcing conservation properties. Furthermore, we integrate a physics-based adaptive refinement (PAR) strategy to dynamically prioritize training near discontinuities, substantially improving the network’s ability to capture sharp gradients. Numerical experiments are conducted on benchmark problems, including the inviscid Burgers equation, the Lighthill-Whitham-Richards (LWR) traffic flow model and the Buckley-Leverett problem. The results show that compared to existing PINN-based methods, SCINN delivers outstanding performance in solving scalar conservation laws with wave interactions and non-convex non-concave flux functions. Compared to conventional PINN, SCINN yields a maximum reduction of 98.7% in mean squared error (MSE).