PINNs #2 : Beltrami Flow (Navier-Stokes, Re=10)

Solving the unsteady incompressible Navier-Stokes equations (Beltrami flow, Re=10) using Physics-Informed Neural Networks (PINNs) with DeepXDE.

PDE:
∂u/∂t + u·∂u/∂x + v·∂u/∂y = -∂p/∂x + (1/Re)·∇²u
∂v/∂t + u·∂v/∂x + v·∂v/∂y = -∂p/∂y + (1/Re)·∇²v
∂u/∂x + ∂v/∂y = 0

Exact solution (Beltrami):
u(x,y,t) = -cos(πx)·sin(πy)·exp(-2π²t/Re)
v(x,y,t) = sin(πx)·cos(πy)·exp(-2π²t/Re)
p(x,y,t) = -1/4·[cos(2πx) + cos(2πy)]·exp(-4π²t/Re)

Network: FNN [3→64→64→64→3] with tanh activation
Training: Adam (15k iterations) + L-BFGS (on GPU)

#NavierStokes #PINNs #DeepXDE #CFD #ScientificML #Beltrami #UnsteadyFlow #Simulation #deeplearning #mathshorts #appliedmaths

Видео PINNs #2 : Beltrami Flow (Navier-Stokes, Re=10) канала PDESimulation