Reconstructing oscillatory wave fields from scattered sensors is a severely underdetermined inverse problem. Beyond the challenges of general physical-field reconstruction, wave responses are complex-valued, frequency-sensitive, and highly oscillatory, while costly simulation and sensing often leave only extreme-sparse observations. Existing low-rank, operator, and diffusion approaches are largely designed for real-valued, smoother fields; dense pixel-space diffusion is particularly inefficient for oscillatory complex fields and difficult to scale to 3D. We propose HarmoCore, which places a generative prior in a compact, continuous, and structured wave-field latent. HarmoCore represents joint real--imaginary channels with Functional Tucker cores over shared continuous spatial bases, learns a frequency-conditioned core diffusion prior, and performs Diffusion Posterior Sampling directly in core space. At fixed sensor coordinates, the multilinear decoder induces an explicit likelihood guidance operator, avoiding dense pixel-space correction. Optional target-equation residual guidance further promotes physical consistency. Experiments on 2D Helmholtz, 2D synthetic wave fields, and 3D Helmholtz show substantial gains under 1%--2% sensing while remaining practical in three dimensions.
Francesco Brandolin, Tariq Alkhalifahphysics.geo-ph cs.AI cs.LG
High-resolution velocity models are crucial for reservoir characterization and subsurface delineation. However, the band limited nature of our surface recorded data limits resolution. Utilizing well measurements to enhance the resolution of our subsurface models is an important objective. To this end, we present a diffusion-guided framework for structurally preconditioned velocity-model reconstruction from sparse well-log information. The proposed approach combines plane-wave PDE regularization, structurally preconditioned inversion, and measurement-guided diffusion posterior sampling within a unified formulation. Local structural slopes estimated through plane-wave destruction are used both to propagate well information along geological dip directions and to guide the diffusion sampling process through a joint velocity--slope generative prior. Numerical experiments on the Volve synthetic model and the Viking Graben field dataset demonstrate that the proposed framework improves structural continuity, lateral consistency, and geological realism compared with conventional structurally preconditioned inversion approaches while maintaining computationally practical inference through DDIM sampling.
Diffusion posterior samplers for accelerated MRI can reconstruct accurately yet still disagree on the acquired k-space across samples, placing posterior variability on coefficients the scanner has already measured. We identify this measured-subspace leakage as a physical-admissibility failure. Under a hard-constraint model it violates the measurement constraint and inflates the reported uncertainty with disagreement about coefficients the scanner has already determined. To quantify this leakage, we introduce complementary measured- and unmeasured-subspace k-space dispersion metrics (MSD/USD). We then present Measured-Subspace Consistency (MSC), a training-free terminal correction that wraps any compatible image-space posterior sampler with a standard multi-coil consistency lock. The ideal lock follows classical range/null-space data consistency. Our contribution is to repurpose it as a black-box posterior audit and correction rather than a new reconstructor or learned sampler. Theoretically, we prove that the ideal transform confines pairwise sample differences to the MRI null space and bound the residual cross-subspace coupling left by practical sensitivity-weighted implementations. Across six base samplers and two MRI anatomies, including out-of-distribution transfer where a knee prior reconstructs brain, MSC substantially reduces measured-subspace dispersion for Soft samplers (a median 16.5x reduction for DPS across five brain contrasts, up to ~29x), while preserving unmeasured-subspace diversity and acting as a near-identity map for Consistent ones. Furthermore, MSC maintains or modestly improves PSNR/SSIM, with no retraining, retuning, or significant computational overhead.
We propose latent-space diffusion posterior sampling (L-DPS), an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs). The method addresses three challenges in PDE-constrained inversion: implicit sample-based priors without tractable densities, high-dimensional spatially distributed parameters, and the high cost of repeated forward-model evaluations during posterior sampling. L-DPS combines a variational autoencoder, an unconditional latent diffusion model, diffusion posterior sampling, and a differentiable neural surrogate. The VAE maps the parameter field to a lower-dimensional latent space, the diffusion model learns an implicit prior score in this latent space, and DPS combines this learned prior with likelihood-based guidance. The likelihood gradient is evaluated through the decoder-surrogate composition, avoiding repeated calls to the full numerical PDE solver. We evaluate the method on an inverse Darcy flow problem with an unknown spatially distributed permeability field inferred from sparse and noisy pressure observations. L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines such as conditional latent diffusion and inverse FNO in sparse and noisy regimes. We further compare L-DPS with a KLE-MAP baseline and study mixed-prior generalization and the sensitivity of inversion accuracy to surrogate forward-model error.