Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems

• URL: http://arxiv.org/abs/2606.26592v1
• Date: Thu, 25 Jun 2026 04:33:44 GMT
• Title: Latent Diffusion Posterior Sampling with Surrogate Likelihood Guidance for PDE Inverse Problems
• Abstract summary: latent-space diffusion posterior sampling (L-DPS) is an approximate Bayesian framework for high-dimensional inverse problems governed by partial differential equations (PDEs)<n>L-DPS produces accurate and robust inverse solutions, reduces inference cost relative to full-space DPS, and outperforms amortized inverse baselines in sparse and noisy regimes.
• Score: 48.15157016272698
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

Related papers

• Decoupled Diffusion Sampling for Inverse Problems on Function Spaces [73.52103661482242]
  Existing plug-and-play diffusion posterior samplers represent physics implicitly through coefficient joint-solution modeling.<n>We propose a physics-aware generative framework in function space for inverse PDE problems.<n>Our Decoupled Diffusion Inverse Solver (DDIS) employs a decoupled design: an unconditional diffusion learns the coefficient prior, while a neural operator explicitly models the forward PDE for guidance.
  arXiv  Detail & Related papers  (2026-01-30T18:54:49Z)
• Differentiable Inverse Modeling with Physics-Constrained Latent Diffusion for Heterogeneous Subsurface Parameter Fields [9.42765150299276]
  We present a latent diffusion-based differentiable inversion method (LD-DIM) for PDE-constrained inverse problems involving high-dimensional spatially distributed coefficients.<n>LD-DIM couples a pretrained latent diffusion prior with an end-to-end differentiable numerical solver to reconstruct unknown heterogeneous parameter fields in a low-dimensional nonlinear manifold.
  arXiv  Detail & Related papers  (2025-12-27T01:01:19Z)
• Solving Inverse Problems via Diffusion-Based Priors: An Approximation-Free Ensemble Sampling Approach [19.860268382547357]
  Current DM-based posterior sampling methods rely on approximations to the generative process.<n>We propose an ensemble-based algorithm that performs posterior sampling without the use of approximations.<n>Our algorithm is motivated by existing works that combine DM-based methods with the sequential Monte Carlo method.
  arXiv  Detail & Related papers  (2025-06-04T14:09:25Z)
• Guided Diffusion Sampling on Function Spaces with Applications to PDEs [112.09025802445329]
  We propose a general framework for conditional sampling in PDE-based inverse problems.<n>This is accomplished by a function-space diffusion model and plug-and-play guidance for conditioning.<n>Our method achieves an average 32% accuracy improvement over state-of-the-art fixed-resolution diffusion baselines.
  arXiv  Detail & Related papers  (2025-05-22T17:58:12Z)
• Geophysical inverse problems with measurement-guided diffusion models [0.4532517021515834]
  I consider two sampling algorithms recently proposed under the name of Diffusion Posterior Sampling (DPS) and Pseudo-inverse Guided Diffusion Model (PGDM)<n>In DPS, the guidance term is obtained by applying the adjoint of the modeling operator to the residual obtained from a one-step denoising estimate of the solution.<n>On the other hand, PGDM utilizes a pseudo-inverse operator that originates from the fact that the one-step denoised solution is not assumed to be deterministic.
  arXiv  Detail & Related papers  (2025-01-08T23:33:50Z)
• Enhancing Diffusion Models for Inverse Problems with Covariance-Aware Posterior Sampling [3.866047645663101]
  In computer vision, for example, tasks such as inpainting, deblurring, and super resolution can be effectively modeled as inverse problems.<n>DDPMs are shown to provide a promising solution to noisy linear inverse problems without the need for additional task specific training.
  arXiv  Detail & Related papers  (2024-12-28T06:17:44Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• Amortized Posterior Sampling with Diffusion Prior Distillation [55.03585818289934]
  Amortized Posterior Sampling is a novel variational inference approach for efficient posterior sampling in inverse problems.<n>Our method trains a conditional flow model to minimize the divergence between the variational distribution and the posterior distribution implicitly defined by the diffusion model.<n>Unlike existing methods, our approach is unsupervised, requires no paired training data, and is applicable to both Euclidean and non-Euclidean domains.
  arXiv  Detail & Related papers  (2024-07-25T09:53:12Z)
• Diffusion Prior-Based Amortized Variational Inference for Noisy Inverse Problems [12.482127049881026]
  We propose a novel approach to solve inverse problems with a diffusion prior from an amortized variational inference perspective. Our amortized inference learns a function that directly maps measurements to the implicit posterior distributions of corresponding clean data, enabling a single-step posterior sampling even for unseen measurements.
  arXiv  Detail & Related papers  (2024-07-23T02:14:18Z)
• Decomposed Diffusion Sampler for Accelerating Large-Scale Inverse Problems [64.29491112653905]
  We propose a novel and efficient diffusion sampling strategy that synergistically combines the diffusion sampling and Krylov subspace methods. Specifically, we prove that if tangent space at a denoised sample by Tweedie's formula forms a Krylov subspace, then the CG with the denoised data ensures the data consistency update to remain in the tangent space. Our proposed method achieves more than 80 times faster inference time than the previous state-of-the-art method.
  arXiv  Detail & Related papers  (2023-03-10T07:42:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.