Time-invariant hp-variational physics informed neural network to solvethe pipe conveying fluid equation

Physics-informed neural networks (PINN) are machine-learning methods that leverage the physics expressed in the partial differential equations (PDE) to extract information from high-dimensional data generated through experiments. Petrov-Galerkin hp-variational physics-informed neural networks (hp-VPINN) formulate the solution based on domain decomposition and projection onto space of high-order polynomials allowing the network to construct both global and local approximations. The integrand in the variational form lowers the order of the differential operators of the PDE which are incorporated into the loss function of the network, hence reducing the training cost of the network. However, such a method has only been tested on simple transient and academic problems, and in the current literature, all solutions to time-dependent problems still depend on specific initial conditions. Hence, the network must be re-trained for different initial conditions every time, which is expensive and not practical for the same system. In the present work, we propose a time-invariant hp-VPINN to solve the linear equation for small lateral displacements of the pipe conveying fluid (PCF). In the linear regime, the fluid flow inside the pipe induces added damping and stabilizes the system. The added damping changes as the flow speed increases until it becomes negative, and the system becomes unstable by flutter for a cantilevered PCF. In the proposed work, the temporal terms will be discretized using Euler backward method where the network will be trained using random previous time steps to predict the next time step. Once training is complete, the network will be used as a time integrator where it will be given a random initial time step, then it will use step output as an input for the next time step eliminating the dependency on the time history of the solution. We will test the predictability of our trained model by comparing the network outputs with experimental data for different flow speeds and initial conditions. We anticipate that our model will perform as a general solver for the linear PCF equation for any flow speed and initial conditions. For future work, the model will be further developed to solve the nonlinear PCF equations in two and three dimensions where the system undergoes through two-and three-dimensional limit cycles and chaotic motion. Finally, we plan to create a real-time model that gives real-time predictions without depending on the long-term history of the system.

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