Physics-informed neural network simulation of thermal cavity flow

Physics-informed neural networks (PINNs) are an emerging technology that can be used both in place of and in conjunction with conventional simulation methods. In this paper, we used PINNs to perform a forward simulation without leveraging known data. Our simulation was of a 2D natural convection-driven cavity using the vorticity-stream function formulation of the Navier-Stokes equations. We used both 2D simulations across the x and z domains at constant Rayleigh (Ra) numbers and 3D simulations across the x, z and Ra domains. The 3D simulation was tested for a PINN’s ability to learn solutions in a higher-dimensional space than standard simulations. The results were validated against published solutions at Ra values of 10 \(^{3}\) $_{}^3$ , 10 \(^{4}\) $_{}^4$ , 10 \(^{5}\) $_{}^5$ , and 10 \(^{6}\) $_{}^6$ . Both the 2D simulations and 3D simulations successfully matched the expected results. For the 2D cases, more training iterations were needed for the model to converge at higher Ra values (10 \(^5\) $_{}^5$ and 10 \(^6\) $_{}^6$ ) than at lower Ra (10 \(^3\) $_{}^3$ and 10 \(^4\) $_{}^4$ ) indicating increased nonlinear fluid-thermal coupling. The 3D case was also able to converge but, but it required more training than any of the 2D cases due to the curse of dimensionality. These results showed the validity of standard simulations via PINNs and the feasibility of higher-order parameter space solutions that are not possible using conventional methods. They also showcased the additional computational demand associated with increasing the dimensionality of the learned parameter space.