This work establishes rigorous first-of-its-kind upper bounds on the generalization error for the method of approximating solutions to the (d+1)-dimensional incompressible Navier-Stokes equations by training depth-2 neural networks trained via the unsupervised Physics-Informed Neural Network (PINN) framework. This is achieved by bounding the Rademacher complexity of the PINN risk. For appropriately weight bounded net classes our derived generalization bounds do not explicitly depend on the network width and our framework characterizes the generalization gap in terms of the fluid's kinematic viscosity and loss regularization parameters. In particular, the resulting sample complexity bounds are dimension-independent. Our generalization bounds suggest using novel activation functions for solving fluid dynamics. We provide empirical validation of the suggested activation functions and the corresponding bounds on a PINN setup solving the Taylor-Green vortex benchmark.
Figures & tables
Figure 1: The left column is a snapshot of the true solution ( t=0 for the left plot and t=1 for the right plot) and the right column is the neural net prediction. The rows represent the three components of the solution, u,v , and p . The domain is (x,y)∈[0,2]2 , with the viscosity set to 10−2 and the density set to 1 .
Figure 2: A scatter plot showing the relation between the generalization error upperbound given in Theorem 1 and the generalization error calculated after training of the tanh3 activated PINNs. Each point is labeled by the Nr/N0 ratio at which the net was trained. On the left, the viscosity is set to ν=0.001 and on the right ν=0.01 .