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5. field

torchfsm.field.diffused_noise ¤

diffused_noise(
    mesh: Union[
        Sequence[tuple[float, float, int]],
        MeshGrid,
        FourierMesh,
    ],
    diffusion_coef: float = 1.0,
    zero_centered: bool = True,
    unit_variance: bool = True,
    unit_magnitude: bool = True,
    device: Optional[torch.device] = None,
    dtype: Optional[torch.dtype] = None,
    n_batch: int = 1,
    n_channel: int = 1,
) -> SpatialTensor["B C H ..."]

Generate a diffused noise field. The noise is generated by integrating a Laplacian operator with a random initial condition. The diffusion coefficient controls the amount of diffusion applied to the noise.

Parameters:

Name Type Description Default
mesh Union[Sequence[tuple[float, float, int]], MeshGrid, FourierMesh]

The mesh to generate the noise on. If a sequence is provided, it should be in the form of [(x_min, x_max, n_points), ...].

 required
diffusion_coef float

The diffusion coefficient. Default is 1.0.

 1.0
zero_centered bool

If True, the noise will be zero-centered. Default is True.

 True
unit_variance bool

If True, the noise will have unit variance. Default is True.

 True
unit_magnitude bool

If True, the noise will have unit magnitude. Default is True.

 True
device Optional[device]

The device to generate the noise on. Default is None.

 None
dtype Optional[dtype]

The data type of the generated noise. Default is None.

 None
n_batch int

The number of batches. Default is 1.

 1
n_channel int

The number of channels. Default is 1.

 1

Returns:

Type Description
SpatialTensor['B C H ...']

SpatialTensor["B C H ...]: The generated diffused noise field.
Source code in torchfsm/field.py 
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def diffused_noise(
    mesh: Union[Sequence[tuple[float, float, int]], MeshGrid, FourierMesh],
    diffusion_coef: float = 1.0,
    zero_centered: bool = True,
    unit_variance: bool = True,
    unit_magnitude: bool = True,
    device: Optional[torch.device] = None,
    dtype: Optional[torch.dtype] = None,
    n_batch: int = 1,
    n_channel: int = 1,
) -> SpatialTensor["B C H ..."]:
    r"""
    Generate a diffused noise field.
        The noise is generated by integrating a Laplacian operator with a random initial condition.
        The diffusion coefficient controls the amount of diffusion applied to the noise.

    Args:
        mesh (Union[Sequence[tuple[float, float, int]], MeshGrid, FourierMesh]): The mesh to generate the noise on.
            If a sequence is provided, it should be in the form of [(x_min, x_max, n_points), ...].
        diffusion_coef (float): The diffusion coefficient. Default is 1.0.
        zero_centered (bool): If True, the noise will be zero-centered. Default is True.
        unit_variance (bool): If True, the noise will have unit variance. Default is True.
        unit_magnitude (bool): If True, the noise will have unit magnitude. Default is True.
        device (Optional[torch.device]): The device to generate the noise on. Default is None.
        dtype (Optional[torch.dtype]): The data type of the generated noise. Default is None.
        n_batch (int): The number of batches. Default is 1.
        n_channel (int): The number of channels. Default is 1.

    Returns:
        SpatialTensor["B C H ...]: The generated diffused noise field.
    """
    if device is None and (isinstance(mesh, FourierMesh) or isinstance(mesh, MeshGrid)):
        device = mesh.device
    if dtype is None and (isinstance(mesh, FourierMesh) or isinstance(mesh, MeshGrid)):
        dtype = mesh.dtype
    u_0 = torch.randn(
        *mesh_shape(mesh, n_batch=n_batch, n_channel=n_channel),
        device=device,
        dtype=dtype
    )
    diffusion = diffusion_coef * Laplacian()
    u_0 = diffusion.integrate(u_0, dt=1, step=1, mesh=mesh)
    if zero_centered:
        u_0 = u_0 - u_0.mean()
    if unit_variance:
        u_0 = u_0 / u_0.std()
    if unit_magnitude:
        u_0 = u_0 / u_0.abs().max()
    return u_0

torchfsm.field.kolm_force ¤

kolm_force(
    x: torch.Tensor,
    drag_coef: float = -0.1,
    k: float = 4.0,
    length_scale: float = 1.0,
) -> Operator

Generate cosine force field for 2d kolmogorov flow in vorticity form. It is defined as \(a \omega - k cos (k l x)\)

Parameters:

Name Type Description Default
x Tensor

The input tensor.

 required
drag_coef float

The drag coefficient \(a\). Default is -0.1.

 -0.1
k float

The wave number. Default is 4.0.

 4.0
length_scale float

The length scale \(l\). Default is 1.0.

 1.0
Source code in torchfsm/field.py 
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def kolm_force(
    x: torch.Tensor,
    drag_coef: float = -0.1,
    k: float = 4.0,
    length_scale: float = 1.0,
) -> Operator:
    r"""
    Generate cosine force field for 2d kolmogorov flow in vorticity form.
        It is defined as $a \omega - k cos (k l x)$

    Args:
        x (torch.Tensor): The input tensor.
        drag_coef (float): The drag coefficient $a$. Default is -0.1.
        k (float): The wave number. Default is 4.0.
        length_scale (float): The length scale $l$. Default is 1.0.
    """


    return drag_coef * ImplicitSource() - ExplicitSource(
        k * torch.cos(k * length_scale * x)
    )

torchfsm.field.wave_1d ¤

wave_1d(
    x: SpatialTensor["B C H ..."],
    min_k: int = 1,
    max_k: int = 5,
    min_amplitude: float = 0.5,
    max_amplitude: float = 1.0,
    n_polynomial: int = 5,
    zero_mean: bool = False,
    mean_shift_coef=0.3,
    batched: bool = False,
) -> SpatialTensor["B C H ..."]

Generate a 1D wave field with multiple harmonics.

Parameters:

Name Type Description Default
x SpatialTensor['B C H ...']

The input tensor.

 required
min_k int

The minimum wave number. Default is 1.

 1
max_k int

The maximum wave number. Default is 5.

 5
min_amplitude float

The minimum amplitude. Default is 0.5.

 0.5
max_amplitude float

The maximum amplitude. Default is 1.0.

 1.0
n_polynomial int

The number of polynomial terms. Default is 5.

 5
zero_mean bool

If True, the mean of the wave will be zero. Default is False.

 False
mean_shift_coef float

The coefficient for mean shift. Default is 0.3.

 0.3
batched bool

If True, the input tensor is batched. Default is False.

 False

Returns:

Type Description
SpatialTensor['B C H ...']

SpatialTensor["B C H ..."]: The generated wave field.
Source code in torchfsm/field.py 
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def wave_1d(
    x: SpatialTensor["B C H ..."],
    min_k: int = 1,
    max_k: int = 5,
    min_amplitude: float = 0.5,
    max_amplitude: float = 1.0,
    n_polynomial: int = 5,
    zero_mean: bool = False,
    mean_shift_coef=0.3,
    batched: bool = False,
) -> SpatialTensor["B C H ..."]:
    r"""
    Generate a 1D wave field with multiple harmonics.

    Args:
        x (SpatialTensor["B C H ..."]): The input tensor.
        min_k (int): The minimum wave number. Default is 1.
        max_k (int): The maximum wave number. Default is 5.
        min_amplitude (float): The minimum amplitude. Default is 0.5.
        max_amplitude (float): The maximum amplitude. Default is 1.0.
        n_polynomial (int): The number of polynomial terms. Default is 5.
        zero_mean (bool): If True, the mean of the wave will be zero. Default is False.
        mean_shift_coef (float): The coefficient for mean shift. Default is 0.3.
        batched (bool): If True, the input tensor is batched. Default is False.

    Returns:
        SpatialTensor["B C H ..."]: The generated wave field.
    """
    x_new = x / x.max() * torch.pi * 2
    y = torch.zeros_like(x)
    if not batched:
        x_new = x_new.unsqueeze(0)
        y = y.unsqueeze(0)
    batch = x_new.shape[0]
    shape = [batch, n_polynomial] + [1] * (x_new.dim() - 2)
    k = torch.randint(min_k, max_k + 1, shape, device=x.device, dtype=x.dtype)
    amplitude = (
        torch.rand(*shape, device=x.device, dtype=x.dtype)
        * (max_amplitude - min_amplitude)
        + min_amplitude
    )
    shift = torch.rand(*shape, device=x.device, dtype=x.dtype) * torch.pi * 2
    for i in range(n_polynomial):
        y += amplitude[:, i : i + 1, ...] * torch.sin(
            k[:, i : i + 1, ...] * (x_new + shift[:, i : i + 1, ...])
        )
    if not zero_mean:
        value_shift = torch.rand(
            [batch] + [1] * (x_new.dim() - 1), device=x.device, dtype=x.dtype
        )
        value_shift = (value_shift - 0.5) * 2 * (
            max_amplitude - min_amplitude
        ) * mean_shift_coef + min_amplitude
        y += value_shift
    if not batched:
        y = y.squeeze(0)
    return y