print("n" + "="*80)
print("SECTION 4: DATA VISUALIZATION")
print("="*80)
def visualize_darcy_samples(
permeability: np.ndarray,
stress: np.ndarray,
n_samples: int = 3
):
"""Visualize Darcy circulate samples."""
fig, axes = plt.subplots(n_samples, 2, figsize=(10, 4 * n_samples))
for i in vary(n_samples):
im1 = axes[i, 0].imshow(permeability[i], cmap='viridis', origin='decrease')
axes[i, 0].set_title(f'Permeability Subject (Pattern {i+1})')
axes[i, 0].set_xlabel('x')
axes[i, 0].set_ylabel('y')
plt.colorbar(im1, ax=axes[i, 0], label="okay(x,y)")
im2 = axes[i, 1].imshow(stress[i], cmap='scorching', origin='decrease')
axes[i, 1].set_title(f'Stress Subject (Pattern {i+1})')
axes[i, 1].set_xlabel('x')
axes[i, 1].set_ylabel('y')
plt.colorbar(im2, ax=axes[i, 1], label="u(x,y)")
plt.tight_layout()
plt.savefig('darcy_samples.png', dpi=150, bbox_inches="tight")
plt.present()
print("✓ Saved visualization to 'darcy_samples.png'")
visualize_darcy_samples(perm_train[:3], press_train[:3])
print("n" + "="*80)
print("SECTION 5: FOURIER NEURAL OPERATOR (FNO)")
print("="*80)
"""
The Fourier Neural Operator (FNO) learns mappings between perform areas
by parameterizing the integral kernel in Fourier area.
Key perception: Convolution in bodily area = multiplication in Fourier area
The FNO layer consists of:
1. FFT to remodel to frequency area
2. Multiplication with learnable weights (preserving solely low-frequency modes)
3. Inverse FFT to remodel again
4. Residual reference to an area linear transformation
"""
class SpectralConv2d(nn.Module):
"""
2D Spectral Convolution Layer for FNO.
Performs convolution in Fourier area by:
1. Computing FFT of enter
2. Multiplying with advanced learnable weights
3. Computing inverse FFT
"""
def __init__(
self,
in_channels: int,
out_channels: int,
modes1: int,
modes2: int
):
tremendous().__init__()
self.in_channels = in_channels
self.out_channels = out_channels
self.modes1 = modes1
self.modes2 = modes2
self.scale = 1 / (in_channels * out_channels)
self.weights1 = nn.Parameter(
self.scale * torch.rand(in_channels, out_channels, modes1, modes2, dtype=torch.cfloat)
)
self.weights2 = nn.Parameter(
self.scale * torch.rand(in_channels, out_channels, modes1, modes2, dtype=torch.cfloat)
)
def compl_mul2d(self, enter: torch.Tensor, weights: torch.Tensor) -> torch.Tensor:
"""Advanced multiplication for batch of 2D tensors."""
return torch.einsum("bixy,ioxy->boxy", enter, weights)
def ahead(self, x: torch.Tensor) -> torch.Tensor:
batch_size = x.form[0]
x_ft = torch.fft.rfft2(x)
out_ft = torch.zeros(
batch_size, self.out_channels, x.measurement(-2), x.measurement(-1) // 2 + 1,
dtype=torch.cfloat, gadget=x.gadget
)
out_ft[:, :, :self.modes1, :self.modes2] =
self.compl_mul2d(x_ft[:, :, :self.modes1, :self.modes2], self.weights1)
out_ft[:, :, -self.modes1:, :self.modes2] =
self.compl_mul2d(x_ft[:, :, -self.modes1:, :self.modes2], self.weights2)
x = torch.fft.irfft2(out_ft, s=(x.measurement(-2), x.measurement(-1)))
return x
class FNOBlock(nn.Module):
"""
FNO Block combining spectral convolution with native linear rework.
output = σ(SpectralConv(x) + LocalLinear(x))
"""
def __init__(
self,
channels: int,
modes1: int,
modes2: int,
activation: str="gelu"
):
tremendous().__init__()
self.spectral_conv = SpectralConv2d(channels, channels, modes1, modes2)
self.local_linear = nn.Conv2d(channels, channels, 1)
self.activation = nn.GELU() if activation == 'gelu' else nn.ReLU()
def ahead(self, x: torch.Tensor) -> torch.Tensor:
return self.activation(self.spectral_conv(x) + self.local_linear(x))
class FourierNeuralOperator2D(nn.Module):
"""
Full 2D Fourier Neural Operator for studying operators.
Structure:
1. Carry enter to larger dimensional channel area
2. Apply a number of FNO blocks (spectral convolutions + residuals)
3. Venture again to output area
This learns the mapping: okay(x,y) -> u(x,y) for Darcy circulate
"""
def __init__(
self,
in_channels: int = 1,
out_channels: int = 1,
modes1: int = 12,
modes2: int = 12,
width: int = 32,
n_layers: int = 4,
padding: int = 9
):
tremendous().__init__()
self.modes1 = modes1
self.modes2 = modes2
self.width = width
self.padding = padding
self.fc0 = nn.Linear(in_channels + 2, width)
self.fno_blocks = nn.ModuleList([
FNOBlock(width, modes1, modes2) for _ in range(n_layers)
])
self.fc1 = nn.Linear(width, 128)
self.fc2 = nn.Linear(128, out_channels)
def get_grid(self, form: Tuple, gadget: torch.gadget) -> torch.Tensor:
"""Create normalized grid coordinates."""
batch_size, size_x, size_y = form[0], form[2], form[3]
gridx = torch.linspace(0, 1, size_x, gadget=gadget)
gridy = torch.linspace(0, 1, size_y, gadget=gadget)
gridx, gridy = torch.meshgrid(gridx, gridy, indexing='ij')
grid = torch.stack([gridx, gridy], dim=-1)
grid = grid.unsqueeze(0).repeat(batch_size, 1, 1, 1)
return grid
def ahead(self, x: torch.Tensor) -> torch.Tensor:
batch_size = x.form[0]
grid = self.get_grid(x.form, x.gadget)
x = x.permute(0, 2, 3, 1)
x = torch.cat([x, grid], dim=-1)
x = self.fc0(x)
x = x.permute(0, 3, 1, 2)
if self.padding > 0:
x = F.pad(x, [0, self.padding, 0, self.padding])
for block in self.fno_blocks:
x = block(x)
if self.padding > 0:
x = x[..., :-self.padding, :-self.padding]
x = x.permute(0, 2, 3, 1)
x = F.gelu(self.fc1(x))
x = self.fc2(x)
x = x.permute(0, 3, 1, 2)
return x
print("nCreating Fourier Neural Operator mannequin...")
fno_model = FourierNeuralOperator2D(
in_channels=1,
out_channels=1,
modes1=8,
modes2=8,
width=32,
n_layers=4,
padding=5
).to(gadget)
n_params = sum(p.numel() for p in fno_model.parameters() if p.requires_grad)
print(f"✓ FNO Mannequin created with {n_params:,} trainable parameters")