bijx.FreeTheoryScaling

class bijx.FreeTheoryScaling[source]

Bases: SpectrumScaling

Scaling bijection mapping white noise to free field theory.

Implements the momentum-space scaling transformation that converts Gaussian white noise into samples from a free scalar field theory.

Type: \(\mathbb{R}^{H \times W \times C} \to \mathbb{R}^{H \times W \times C}\)

Transform:

\[ \mathcal{F}^{-1}\left[\frac{1}{\sqrt{m^2 + \mathbf{k}^2}} \mathcal{F}[\xi]\right] \]

For a free scalar field with mass \(m\), the two-point correlation function in momentum space is

\[ \langle\tilde{\phi}(\mathbf{k})\tilde{\phi}^*(\mathbf{k}')\rangle = \frac{\delta(\mathbf{k}-\mathbf{k}')}{m^2 + \mathbf{k}^2}\,. \]

Parameters:

  • m2 (float | Variable) – Mass squared parameter (can be learnable nnx.Variable).

  • space_shape (tuple[int, ...]) – Spatial lattice dimensions.

  • channel_dim (int) – Number of channel dimensions.

  • finite_size (bool) – Whether to use lattice momenta (True) or continuum (False).

  • precompute_spectrum (bool) – Whether to precompute scaling factors.

  • half (bool) – Whether to use factor of 1/2 conventional in the action.

Note

Assumes periodic boundary conditions.

Example

>>> # Free scalar field with mass m=0.1 on 32x32 lattice
>>> m2 = 0.01
>>> scaling = FreeTheoryScaling(
...     m2, space_shape=(32, 32), finite_size=True
... )
>>> phi, log_det = scaling.forward(eps, log_density)  # eps ~ N(0,1)
__init__(m2, space_shape, channel_dim=0, finite_size=True, precompute_spectrum=True, half=True)[source]
Parameters:

  • m2 (float | Variable)

  • space_shape (tuple[int, ...])

  • channel_dim (int)

  • finite_size (bool)

  • precompute_spectrum (bool)

  • half (bool)

Methods

apply(x, log_density[, reverse])

Unified transformation method.

forward(x, log_density, **kwargs)

Apply forward transformation.

invert()

Create an inverted version of this bijection.

reverse(x, log_density, **kwargs)

Apply reverse (inverse) transformation.

scale(r[, reverse])

Apply Fourier-space scaling transformation.

spectrum_function(ks, m2)

Compute free field scaling factors from momenta and mass.

Attributes

scaling

Multiplicative factor in Fourier space.
spectrum_function(ks, m2)[source]

Compute free field scaling factors from momenta and mass.

Parameters:

  • ks – Momentum grid with shape (…, spatial_dim).

  • m2 – Mass squared parameter.

Returns:

Scaling factors for free field theory propagator.

property scaling

Multiplicative factor in Fourier space.