You can use KAN to solve fluid dynamics problems like the Navier-Stokes equations by treating the neural network as a Physics-Informed Neural Network (PINN). This involves defining a loss function that combines the residuals of the partial differential equations (PDEs) with boundary condition losses.
Key Steps:
- Define the Model: Initialize a
KAN model where the output dimensions correspond to the physical variables (e.g., u, v, and p for velocity components and pressure). - Compute Derivatives: Use
torch.autograd.functional.jacobian and custom Hessian implementations to compute the spatial derivatives required by the Navier-Stokes equations from the model's predictions. - Formulate Loss:
- PDE Residuals: Calculate the continuity equation and momentum equations (x and y) using the computed derivatives.
- Boundary Conditions (BC): Add losses for no-slip conditions, inlet velocity, and outlet pressure.
- Total Loss:
total_loss = torch.mean(pde_residuals) + bc_loss.
- Optimization: Use the
LBFGS optimizer, which is often effective for PINN training tasks.
Note: This implementation is a community contribution and has not been officially verified by the KAN authors.
import torch
from kan import KAN, LBFGS
# 1. Initialize KAN model
# width=[2,3,3,3] means 2 inputs (x, y) and 3 outputs (u, v, p)
model = KAN(width=[2,3,3, 3], grid=5, k=10, grid_eps=1.0, noise_scale_base=0.25)
# 2. Define the loss function (simplified logic)
def navier_stokes_residuals(coords):
y_pred = model(coords)
# ... compute gradients/hessians using autograd ...
# ... compute continuity, x_momentum, y_momentum residuals ...
# ... compute boundary condition losses ...
return total_loss
# 3. Train using LBFGS
optimizer = LBFGS(model.parameters(), lr=1, history_size=10, line_search_fn="strong_wolfe")
def closure():
optimizer.zero_grad()
loss = navier_stokes_residuals(coordinates)
loss.backward()
return loss
optimizer.step(closure)
