Physics-informed Deep Learning for simultaneous Surrogate Modelling and PDE-constrained Optimization

We model the flow around an airfoil with a physics-informed neural network (PINN) while simultaneously optimizing the airfoil geometry to maximize its lift-to-drag ratio. The parameters of the airfoil shape are provided as inputs to the PINN and the multidimensional search space of shape parameters is populated with collocation points to ensure that the Navier-Stokes equations are approximately satisfied throughout. We use the fact that the PINN is automatically differentiable to calculate gradients of the lift-to-drag ratio with respect to the parameter values. This allows us to use the L-BFGS gradient-based optimization algorithm, which is more efficient than non-gradient-based algorithms. We train the PINN with adaptive sampling of collocation points, such that the accuracy of the solution is enhanced along the optimization trajectory. We demonstrate this method on two examples: one that optimizes a single parameter, and another that optimizes eleven parameters. The method is successful and, by comparison with conventional CFD, we find that the velocity and pressure fields have small pointwise errors and that the method converges to optimal parameters. We find that different PINNs converge to slightly different parameters, reflecting the fact that there are many closely-spaced local minima when using stochastic gradient descent. This method can be applied relatively easily to other optimization problems and avoids the difficult process of writing adjoint codes. As knowledge about how to train PINNs improves and hardware dedicated to neural networks becomes faster, this method of simultaneous training and optimization with PINNs could become easier and faster than using adjoint codes.