Physics Informed Neural Networks (PINNs) for Fluid Mechanics

How do deep learning methods work? Can a deep learning method be used to obtain solutions to a physical problem by using its mathematical description? In other words, can a deep learning method solve an initial- or boundary-value problem with a small amount of data or none? The answer is “yes.” This approach—known as Physics-Informed Neural Networks (PINNs)—was first introduced by Raissi et al. (2019). We answer “yes,” but we also emphasize the authors’ caveat from the same 2019 paper: “…the proposed methods should not be viewed as replacements for classical numerical methods….” Because the question “How much data is needed?” remains open, PINN models can still depend on experimental or numerical data.

In PINNs, modifications are made to the loss function compared to Artificial Neural Networks (ANNs). While ANNs typically require large amounts of input data, in PINNs the role of data is largely taken over by the differential equations that define the initial-/boundary-value problem. In this talk, we will focus on the main differences between basic ANNs and PINNs and on how PINNs can be used for a physical problem, e.g., the Navier–Stokes equations.

First, for rotating-disk flows, we use a PINN to obtain the unperturbed base-flow velocity profiles directly from the governing similarity equations, enforcing the differential operators and boundary conditions in the loss. The PINN-predicted profiles are quantitatively compared against the classical von Kármán solutions, highlighting the effects of sampling strategy and loss weighting on accuracy. Second, for the hydrodynamic stability of parallel Poiseuille flow, we treat the Orr–Sommerfeld (OS) eigenvalue problem as an inverse PINN: the complex eigenvalue is a trainable parameter, and the corresponding eigenfunction is represented by the network. I will present eigenvalues and eigenfunctions for the first 15 modes at Re = 2000 and compare them with numerical solutions, providing a partial answer to the question of how much data is needed to recover these results.

Raissi, M., Perdikaris, P. and Karniadakis, G.E., 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational physics, 378, pp. 686-707.