Probability Flow ODE driven diffusion posterior sampling for large scale inverse problem with two-stage measurement alignment

Diffusion-based posterior sampling combines learned diffusion priors with measurement likelihoods to solve inverse problems. However, existing methods typically evaluate measurement guidance using a predicted clean sample or posterior mean rather than the current diffusion state, which can introduce biased likelihood gradients under noisy observations and nonlinear forward operators. They also often require hundreds or thousands of reverse-time steps, resulting in expensive likelihood evaluations. We propose probability flow ODE posterior sampling (PF-ODE), a deterministic framework that applies measurement likelihood guidance at the current diffusion state directly into the sampler, eliminating the reliance on posterior mean estimate. This high-order probability flow ODE solver through Strang splitting works in guided sampling while reduces discretization error and likelihood evaluations. For challenging nonlinear inverse problems, we further introduce a two-stage guidance strategy: phase-based guidance first mitigates cycle skipping and establishes a robust initialization, followed by proximal optimization using full measurements to enforce data consistency and recover fine-scale structures. Experiments on noisy image inpainting, sparse-view CT, and seismic full-waveform inversion demonstrate improved reconstruction quality with fewer likelihood evaluations than existing diffusion posterior sampling methods.