Burgers' PINNs with implicit Euler Transfer Learning

Burgers’ equation is a well-established test case in mathematical analysis and numerical simulations of convective-diffusive Partial Differential Equations (PDEs). In our study, we focus on the viscous Burgers equation with initial and Dirichlet boundary conditions. In this work, we present an alternative Physics-informed Neural Network with an implicit Euler transfer learning approach to solve the Burgers’ equation. The solution is sought as a sequence of Multilayer Perceptrons (MLPs), one per time step. As an iterative procedure, a first MLP learns the initial condition. Having an MLP that estimates the solution of a given time, its knowledge is transferred to a new MLP that learns the solution at a next time step by training to minimize the residual of the implicit Euler scheme. The methodology is discussed and applied to three classical test cases. In the first, we discuss the determination of the network architecture. We observe the problem of a false stationary state and present a strategy to overcome this issue. The second is a shock wave and the third is a rarefaction problem. The proposed approach allows us to deal with smaller neural network architectures in comparison to the classical PINN approaches. This has the potential to enhance computational performance.