A Physics Informed Neural Nets (PINNs) Approach for Solving the Euler Equations

This was a final project for ENM 531 where I worked with Keshav Vedula to solve the Euler equations using physics-informed neural networks (PINNs) with multiple model architectures and training procedures. Specifically, we implement a Mixture of Experts (MoE) version of the PINNs approach to solve the steady two-dimensional Euler equations for the oblique shock wave problem. Compared to the traditional PINNs approach, it is found that the MoE architecture offers a tremendous improvement in relative accuracy (reduced relative L2 error from 17.6% to 0.6%) for the same number of model parameters and fewer training iterations. Further marginal improvement is achieved by implementing a causal training procedure in the direction of the body surface. In general, we observe that the MoE architecture reduces the model’s sensitivity to hyperparameter tuning including the relative weighting of the various loss terms. We interpret this result as an improvement in the robustness of the PINNs approach for solving hyperbolic PDEs when coupled with the MoE architecture.

Some brief results are shown below

To learn more, check out our final report or presentation linked below! If you are interested in the code, please reach out to me via email.