Inverse Problems¶

DeltaFlow can turn a pretrained unconditional velocity field into a posterior sampler for inverse problems (reconstructing a signal \(x\) from a noisy measurement \(y = A(x) + \varepsilon\)) without retraining the backbone.

Design, wrap don't rewrite¶

PosteriorSolver wraps an existing ODE solver (Euler, Heun, ...) and injects a measurement-likelihood gradient at every step, so the base integrator is reused. This follows FlowDPS and Flower, which both modify the sampling-time ODE and never touch the pretrained field.

Per step, given state \(x_t\) and velocity \(v_t = v_\theta(x_t, t)\):

[ \hat{x}0 = \text{Tweedie}(x_t, v_t, t), \qquad g = \nabla}\big[-\log p(y \mid \hat{x0)\big], ] [ xt) - \eta\, g, ]}t} = \text{BaseSolver.step}(x_t, t, \mathrm{d

with \(\eta\) the guidance_scale. The gradient is computed by autograd.

Usage¶

from deltaflow.solvers import EulerSolver, PosteriorSolver
from deltaflow.inverse import GaussianLikelihood, BlurOperator

operator = BlurOperator(kernel_size=9, sigma=2.0)
likelihood = GaussianLikelihood(operator=operator, y=measurement, sigma=0.05)

solver = PosteriorSolver(
    base_solver=EulerSolver(model),
    likelihood=likelihood,
    guidance_scale=1.0,
)
recon = solver.sample(torch.randn_like(x_init), n_steps=100)

Measurement operators¶

deltaflow.inverse ships common linear operators:

Operator	Task
`IdentityOperator`	Denoising
`MaskOperator`	Inpainting
`BlurOperator`	Deblurring
`DownsampleOperator`	Super-resolution

Latent-space problems¶

If the velocity field operates on VAE latents while the measurement operator is defined on pixels, pass a decoder to the likelihood object, and the gradient is pulled back through the decoder automatically, no extra bookkeeping.

Tweedie decomposition¶

The map \((x_t, v_t, t) \mapsto \hat{x}_0\) is provided by BaseTweedie. Use LinearTweedie for a linear-interpolant (rectified-flow) backbone and VPTweedie for a variance-preserving one. It must match the training path.