Diffusion Posterior Sampling

Deep generative models have emerged as state-of-the-art for solving inverse problems, but applying them to inverse problems for PDEs, like electrical impedance tomography (EIT) remains challenging. Because physical domains are naturally discretized as unstructured meshes rather than regular grids, standard convolutional architectures are often inadequate. In this paper, we propose a novel framework that extends diffusion posterior sampling (DPS) to graph-structured data. We develop an unconditional score-based diffusion model directly on a 2D triangular mesh to learn an accurate prior over the physical solution space. Furthermore, we introduce a regularized variant, RDPS, which incorporates explicit regularization terms, such as total variation and generalized Tikhonov, to complement the implicit diffusion prior and mitigate severe ill-posedness. Extensive experiments on synthetic and real 2D EIT datasets demonstrate that RDPS produces stable, physically plausible reconstructions. Our approach generalizes well to out-of-distribution inclusion geometries, is highly robust to measurement noise, and outperforms current state-of-the-art solvers (e.g., GPnP-BM3D, DP-SGS) in reconstruction accuracy and artifact reduction.

Generative models trained on public databases of protein structures, most of which have been determined by X-ray crystallography, now provide powerful priors for structure prediction. However, they are not readily conditioned on the measurements from a new crystallographic experiment, limiting their use for X-ray structure determination. In crystallography, the measured structure-factor amplitudes do not by themselves determine an electron density map or atomic structure because the associated phases are unobserved and must be inferred. Structure determination therefore remains an inverse problem in which candidate models must be both structurally plausible and consistent with measured diffraction data, often requiring substantial manual refinement by human experts. Emerging methods aim to incorporate experimental information more directly into predictive and refinement workflows. We present CrystalBoltz, a generative framework that casts crystallographic refinement as Bayesian inference over atomic structures and operates directly on structure-factor amplitudes. CrystalBoltz moves from unguided generation with a pre-trained prior over protein structures to experiment-guided posterior sampling, followed by atomic coordinate and B-factor refinement. Across multiple protein crystallography datasets, CrystalBoltz attains lower coordinate RMSD and lower R-factors than the strongest baselines considered, while reducing runtime by a factor of 33 relative to existing experimentally guided refinement.

Generative diffusion models can provide powerful prior probability models for inverse problems in imaging, but existing implementations suffer from two key limitations: $(i)$ the prior density is represented implicitly, and $(ii)$ they rely on likelihood approximations that introduce sampling biases. We address these challenges by introducing a new energy-based model trained for denoising with a covariance-based regularization term that enforces consistency across different measurement conditions. The trained model can compute normalized posterior densities for diverse linear inverse problems, without additional retraining or fine tuning. In addition to preserving the sampling capabilities of diffusion models, this enables previously unavailable capabilities: energy-guided adaptive sampling that adjusts schedules on-the-fly, unbiased Metropolis-Hastings correction steps, and blind estimation of the degradation operator via Bayes rule. We validate the method on multiple datasets (ImageNet, CelebA, AFHQ) and tasks (inpainting, deblurring), demonstrating competitive or superior performance to established baselines.

Diffusion models (DMs) have exhibited remarkable efficacy in various image restoration tasks. However, existing approaches typically operate within the high-dimensional pixel space, resulting in high computational overhead. While methods based on latent DMs seek to alleviate this issue by utilizing the compressed latent space of a variational autoencoder, they require repeated encoder-decoder inference. This introduces significant additional computational burdens, often resulting in runtime performance that is even inferior to that of their pixel-space counterparts. To mitigate the computational inefficiency, this work proposes projecting data into lower-dimensional subspaces using dynamic resolution DMs to accelerate the inference process. We first fine-tune pre-trained DMs for dynamic resolution priors and adapt DPS and DAPS, which are two widely used pixel-space methods for general image restoration tasks, into the proposed framework, yielding methods we refer to as SubDPS and SubDAPS, respectively. Given the favorable inference speed and reconstruction fidelity of SubDAPS, we introduce an enhanced variant termed SubDAPS++ to further boost both reconstruction efficiency and quality. Empirical evaluations across diverse image datasets and various restoration tasks demonstrate that the proposed methods outperform recent DM-based approaches in the majority of experimental scenarios. The code is available at https://github.com/StarNextDay/SubDAPS.git.

Diffusion-based posterior sampling (PS) is a leading framework for imaging inverse problems, combining learned priors with measurement constraints. Yet, its standard formulations rely on instantaneous data-consistent estimates, which induce temporal variability in the reverse dynamics. We reinterpret PS from a dynamical perspective, showing that the standard PS update corresponds to a first-order discretization of the diffusion dynamics plus a residual correction capturing the mismatch between the denoised prediction and the data-consistent estimate. A second-order discretization, however, naturally introduces a temporal correction based on the variation of consecutive estimates. Building on this, we propose LAMP, combining the second-order update with the residual correction characterizing a PS technique. LAMP thus inherits a lagged temporal correction, and it can be implemented as a modular plug-in over the PS backbone. We show that LAMP preserves the structure of a posterior sampler, and we perform a one-step risk analysis to characterize when LAMP improves the reverse transition via a bias-variance trade-off. Experiments across multiple imaging tasks demonstrate consistent improvements over strong baselines such as DiffPIR and DDRM, without increasing the number of denoising evaluations.

Score-based diffusion models achieve state-of-the-art performance for inverse problems, but their practical deployment is hindered by long inference times and cumbersome hyperparameter tuning. While pretrained diffusion models can be reused across tasks without retraining, inference-time hyperparameters such as the noise schedule and posterior sampling weights typically require ad-hoc adjustment for each problem setup. We propose principled reparameterizations that induce invariances, allowing the same hyperparameters to be reused across multiple problems without re-tuning. In addition, building on the RED-diff framework, which reformulates posterior sampling as an optimization problem, we further develop the OptDiff pipeline. OptDiff provides a simplified tuning framework that facilitates the integration of convex optimization tools to accelerate inference. Experiments on image reconstruction, deblurring, and super-resolution show substantial speedups and improved image quality.

The remarkable achievements of both generative models of 2D images and neural field representations for 3D scenes present a compelling opportunity to integrate the strengths of both approaches. In this work, we propose a methodology that combines a NeRF-based representation of 3D scenes with probabilistic modeling and reasoning using diffusion models. We view 3D reconstruction as a perception problem with inherent uncertainty that can thereby benefit from probabilistic inference methods. The core idea is to represent the 3D scene as a stochastic latent variable for which we can learn a prior and use it to perform posterior inference given a set of observations. We formulate posterior sampling using the score-based inference method of diffusion models in conjunction with a likelihood term computed from a reconstruction model that includes volumetric rendering. We train the model using a two-stage process: first we train the reconstruction model while auto-decoding the latent representations for a dataset of 3D scenes, and then we train the prior over the latents using a diffusion model. By using the model to generate samples from the posterior we demonstrate that various 3D reconstruction tasks can be performed, differing by the type of observation used as inputs. We showcase reconstruction from single-view, multi-view, noisy images, sparse pixels, and sparse depth data. These observations vary in the amount of information they provide for the scene and we show that our method can model the varying levels of inherent uncertainty associated with each task. Our experiments illustrate that this approach yields a comprehensive method capable of accurately predicting 3D structure from diverse types of observations.

Pretrained diffusion models provide powerful learned priors, but in scientific sampling the target distribution often depends on physical context that is not fully represented by one generative model. We introduce Generative Gibbs for Physics-Aware Sampling (GG-PA), a training-free framework that formulates the composition of learned partial priors and explicit physical context as inference over a joint target distribution in an augmented state space. We derive a Gibbs sampler for this joint target, show that it is asymptotically exact as the diffusion time approaches zero, and prove that in settings with quadratic interactions it remains exact at finite diffusion times. We further introduce replica exchange over diffusion time to accelerate mixing. Experiments on a double-well system, a $φ^4$ lattice model, and atomistic peptide systems show that GG-PA recovers context-induced distribution shifts and emergent collective behavior in interacting systems using partial priors without retraining. These results demonstrate GG-PA as a practical approach for combining pretrained generative priors with explicit physical context.

We study posterior sampling for inverse problems in discrete state spaces using discrete diffusion models as generative priors. While continuous diffusion models have become widely used for inverse problems, their discrete counterparts remain comparatively underexplored. Existing discrete posterior samplers often rely on continuous relaxations of discrete variables, Gibbs-style updates, or mechanisms specialized to particular corruption processes, which can limit scalability or generality. We propose $Δ$LPS, a Discrete Langevin-Inspired Posterior Sampler that uses gradient information to identify promising discrete moves without leaving the discrete state space. The resulting approach enables efficient parallel updates across all token dimensions and is agnostic to the training paradigm of the discrete diffusion prior, including masked and uniform-state diffusion. We evaluate our method on image restoration tasks across MNIST, CIFAR, and FFHQ, as well as spatial mapping, covering linear, nonlinear, and blind inverse problems. Across these settings, we improve over recent discrete diffusion posterior samplers and are competitive with strong continuous diffusion-based inverse solvers. Our results suggest that fully discrete, gradient-informed posterior samplers offer a scalable and general path toward solving inverse problems over discrete representations.

Vision-Language Latent Diffusion Models (LDMs) (Rombach et al., 2022) provide powerful generative priors for inverse problems. However, existing LDM-based inverse solvers typically require a large number of neural function evaluations (NFEs) and backpropagation through large pretrained components, leading to substantial computational costs and, in some cases, degraded reconstruction quality. We propose a unified Euclidean-Wasserstein-2 gradient-flow framework that jointly performs posterior sampling and prompt optimization in the latent space through a single flow that aligns the prior and posterior with the observed data. Combined with few-step latent text-to-image models, this formulation enables low-NFE inference without backpropagation through autoencoders. Experiments across several canonical imaging inverse problems show that our method achieves state-of-the-art performance with significantly reduced computational cost.

State estimation is a critical task in scientific, engineering and control applications. Since the reliability of reconstructions depends on the number and position of sensors, optimal sensor placement (OSP) is essential in scenarios where measurements are sparse and expensive. Classical OSP approaches rely on Gaussian assumptions and are consequently unable to account for the complex distributions encountered in many real-world systems. Generative-model-based reconstruction using sensor guided diffusion posterior sampling (DPS) has emerged as a promising technique for reconstructing states from highly complex distributions. However, existing sensor-selection methods either require unrealistically many sensors or emulate classical OSP, creating a mismatch between modern recovery models with classical OSP tools motivating the need for fundamentally new ideas towards OSP that match the recent advances made in powerful recovery models. We introduce a distribution-free sensor placement framework based on the Christoffel function: a mathematical formulation of optimal sampling and recovery guarantees for posterior sampling with arbitrary sensors and signal distributions, from which we derive a new OSP strategy with non-asymptotic bounds on the number of sensors needed for recovery. We develop Christoffel-DPS, with offline and online variants, instantiating Christoffel sampling for generative models. Christoffel-DPS outperforms Gaussian OSP baselines and existing generative-model placement methods, validating that distribution-free sensing is both theoretically principled and practically superior. The framework is model-agnostic; we demonstrate its application to a range of unconditional DPS and flow-matching models on structurally non-Gaussian benchmarks, showing the efficacy of Christoffel-DPS in low sensor budget regimes.

Diffusion-based posterior samplers use pretrained diffusion priors to sample from measurement- or reward-conditioned posteriors, and are widely used for inverse problems. Yet their theoretical behavior remains poorly understood: even with exact prior scores, their outputs are biased, and in low-temperature regimes their discretizations can become unstable. We characterize this bias by introducing a tractable surrogate path connecting the true posterior to a standard Gaussian and comparing it to the sampler's path. Their density ratio satisfies a parabolic PDE whose reaction term measures the accumulated bias. A Feynman-Kac representation then expresses the Radon-Nikodym correction as an explicit path expectation, identifying which posterior regions are over- or under-sampled. We apply this framework to DPS and STSL, a related sampler. For DPS, the correction is an Ornstein-Uhlenbeck path expectation coupling the data conditional covariance with the reward curvature, revealing where DPS over- or under-samples. Next, we reinterpret STSL as an auxiliary drift that steers trajectories toward low-uncertainty regions, flattening the spatially varying part of the DPS reaction term. Finally, we characterize early guidance-stopping, a common mitigation for low-temperature instabilities caused by forward-Euler integration of the vector field. Together, these results clarify sampler bias, explain existing correctives, and guide stable variant designs.

We study zero-shot conditional sampling with pretrained diffusion models for linear inverse problems, including inpainting and super-resolution. In these problems, the observation determines only part of the unknown signal. The remaining degrees of freedom must be sampled according to the correct conditional data distribution. Existing projection-based samplers enforce measurement consistency by correcting the observed component during reverse diffusion. However, measurement consistency alone does not determine how probability mass should be distributed along the feasible set, and this can lead to biased conditional samples. We analyze this issue through a normal--tangent decomposition of the score function. For Gaussian noising, the observed-direction score is exactly determined by the measurement; only the tangent conditional score is unknown. We prove that the error from replacing this score by the unconditional tangent score is upper bounded by a dimension-free conditional mutual information between observed and unobserved components. This gives an information-theoretic decomposition into initialization and pathwise score-mismatch errors. Motivated by the theory, we propose a projected-Langevin initialization followed by guided reverse denoising, which outperforms a strong projection-based baseline in inpainting and super-resolution experiments.

Training-free conditional diffusion provides a flexible alternative to task-specific conditional model training, but existing samplers often allocate computation inefficiently: independent guided trajectories can vary widely in quality, and additional function evaluations along a single trajectory may not recover from poor early decisions. We propose Tempered Guided Diffusion (TGD), an annealed sequential Monte Carlo framework for training-free conditional sampling with diffusion priors. TGD targets tempered posterior distributions over the clean signal, using noisy diffusion states only as auxiliary variables for proposing reconstructions and propagating particles. Particles are reweighted by incremental likelihood ratios, resampled, and propagated across noise levels, concentrating computation on trajectories plausible under both the prior and observation. Under idealized exact-reconstruction assumptions, full TGD yields a consistent particle approximation to the posterior as the number of particles grows. For expensive reconstruction tasks, Accelerated TGD (A-TGD) retains early particle exploration but prunes to a single high-likelihood trajectory partway through sampling. Experiments on a controlled two-dimensional inverse problem and image inverse problems show improved posterior approximation and favorable wall-clock speed-quality tradeoffs over independent multi-trajectory baselines.

Computed Tomography (CT) is a widely used imaging modality in medical and industrial applications. To limit radiation exposure and measurement time, there is a growing interest in sparse-view CT, where the number of projection views is significantly reduced. Deep neural networks have shown great promise in improving reconstruction quality in sparse-view CT, especially generative diffusion models. However, these methods struggle to scale to large 3D volumes due to several reasons: (i) the high memory and computational requirements of 3D models, (ii) the lack of large 3D training datasets, and (iii) the inconsistencies across slices when using 2D models independently on each slice. We overcome these limitations and scale diffusion-based sparse-view CT reconstruction to large 3D volumes by combining conditional diffusion with explicit data consistency. We propose Conditional Diffusion Posterior Alignment (CDPA) to enable scalable 3D sparse-view CT reconstruction. A 2D U-Net diffusion model is conditioned on an initial 3D reconstruction to improve inter-slice consistency, combined with data-consistency alignment to match measured projections. Experiments on synthetic and real Cone Beam CT (CBCT) data show state-of-the-art performance, with ablations that confirm the synergistic effects of the proposed pipeline. Finally, we show that the same principles also strengthen fast denoising U-Nets, yielding near-diffusion quality at a fraction of the computational cost.

While diffusion priors generate high-quality posterior samples across many inverse problems, they are often trained on limited training sets or purely simulated data, thus inheriting the errors and biases of these underlying sources. Current approaches to finetuning diffusion models rely on a large number of observations with varying forward operators, which can be difficult to collect for many applications, and thus lead to overfitting when the measurement set is small. We propose a method for tuning a prior from only a single observation by combining existing diffusion priors into a single product-of-experts prior and identifying the exponents that maximize the Bayesian evidence. We validate our method on real-world inverse problems, including black hole imaging, where the true prior is unknown a priori, and image deblurring with text-conditioned priors. We find that the evidence is often maximized by priors that extend beyond those trained on a single dataset. By generalizing the prior through exponent weighting, our approach enables posterior sampling from both tempered and combined diffusion models, yielding more flexible priors that improve the trustworthiness of the resulting posterior image distribution.

Diffusion models (DMs) have recently shown remarkable performance on inverse problems (IPs). Optimization-based methods can fast solve IPs using DMs as powerful regularizers, but they are susceptible to local minima and noise overfitting. Although DMs can provide strong priors for Bayesian approaches, enforcing measurement consistency during the denoising process leads to manifold infeasibility issues. We propose Noise-space Hamiltonian Monte Carlo (N-HMC), a posterior sampling method that treats reverse diffusion as a deterministic mapping from initial noise to clean images. N-HMC enables comprehensive exploration of the solution space, avoiding local optima. By moving inference entirely into the initial-noise space, N-HMC keeps proposals on the learned data manifold. We provide a comprehensive theoretical analysis of our approach and extend the framework to a noise-adaptive variant (NA-NHMC) that effectively handles IPs with unknown noise type and level. Extensive experiments across four linear and three nonlinear inverse problems demonstrate that NA-NHMC achieves superior reconstruction quality with robust performance across different hyperparameters and initializations, significantly outperforming recent state-of-the-art methods. The code is available at https://github.com/NA-HMC/NA-HMC.

Generative semantic communication (SemCom) harnesses pretrained generative priors to improve the perceptual quality of wireless image transmission. Existing generative SemCom receivers, however, rely on maximum a posteriori (MAP) estimation, which fundamentally cannot preserve the data distribution and thus limits achievable perceptual quality. Moreover, current diffusion-based approaches using single-domain guidance face significant limitations: latent-domain guidance is sensitive to channel noise, while image-domain guidance inherits decoder bias. Simply combining both domains simultaneously yields an overconfident pseudo-posterior. In this paper, we formulate semantic decoding as a Bayesian inverse problem and prove that posterior sampling achieves optimal perceptual quality by preserving the data distribution. Building on this insight, we propose alternating dual-domain posterior sampling (ADDPS), a diffusion-based SemCom receiver that alternately enforces latent-domain and image-domain consistency during the sampling process. This alternating strategy decomposes joint posterior sampling into simpler subproblems, avoiding gradient conflicts while retaining the complementary strengths of both domains. Experiments on FFHQ demonstrate that the proposed ADDPS achieves superior perceptual quality compared with existing methods.

Physics-guided sampling with diffusion priors has recently shown strong performance in solving complex systems of partial differential equations (PDEs) from sparse observations. However, these methods are typically evaluated on benchmark problems that do not fully demonstrate their ability to generate temporally consistent solutions of time-dependent PDEs, often focusing instead on reconstructing a single snapshot. In this work, we apply these methods to gas-phase reaction kinetics problems governed by the advection-reaction-diffusion (ARD) equation, providing a setting that more closely reflects realistic laboratory experiments. We demonstrate that guided sampling can be used to reconstruct full spatiotemporal trajectories, rather than isolated states. Furthermore, we show that these methods generalise to previously unseen parameter regimes, highlighting their potential for real-world applications.

Test-time guidance is a widely used mechanism for steering pretrained diffusion models toward outcomes specified by a reward function. Existing approaches, however, focus on maximizing reward rather than sampling from the true Bayesian posterior, leading to miscalibrated inference. In this work, we show that common test-time guidance methods do not recover the correct posterior distribution and identify the structural approximations responsible for this failure. We then propose consistent alternative estimators that enable calibrated sampling from the Bayesian posterior. We significantly outperform previous methods on a set of Bayesian inference tasks, and set a new state-of-the-art PSNR in black hole image reconstruction.

This paper addresses the issue of inversion in cases where (1) the observation system is modeled by a linear transformation and additive error, (2) the problem is ill-posed and regularization relies on a Bayesian strategy, (3)~the prior is modeled by a diffusion process adjusted on an available large set of examples. In this context, it is known that the issue of posterior sampling is a thorny one and the paper introduces a Gibbs algorithm. It appears that this avenue has not been explored, and we show that it is particularly effective and remarkably simple. In addition, it provides clear elements regarding convergence guarantees in a specific case and arguments supporting such guarantees in practical cases. The results are clearly confirmed by numerical simulations based on a toy example.

Score-based diffusion models are a recently developed framework for posterior sampling in Bayesian inverse problems, enabling high-quality reconstructions in inverse problems by leveraging expressive prior distributions learned from empirical data. Despite their strong empirical performance and growing interest within the machine learning community, their behaviour in realistic, severely ill-posed inverse problems with experimental measurement data remains under-explored. Diffuse optical tomography (DOT) is an inverse boundary value problem that uses boundary measurements of near-infrared light to recover spatially varying absorption and scattering parameters in biological tissue. The problem is highly ill-posed and particularly sensitive to both measurement noise and modelling errors. We introduce a regularization strategy by constructing a mixed score consisting of a learned component and a model-based component. We show that the resulting mixed score approximates the score of a corresponding mixture distribution locally and in the small diffusion-time regime, providing a theoretical justification for the approach. We compare four approaches for difference imaging in DOT: a classical model-based method, an approximate score-based diffusion method (DPS), an exact posterior sampling method (UCoS) and a novel, regularized version of UCoS. We show that both the model-based approach and approximate diffusion-based sampling degrade significantly in the presence of limited-view geometry and real experimental data, whereas UCoS yields more accurate reconstructions.

Single-photon avalanche diodes (SPADs) record light as a discrete stream of individual detections. The signal is stochastic. Its statistical structure depends on the sensor's operation mode: binary detection in fixed bins, timestamped detection in fixed bins, or free-running timestamped detection. We derive the likelihood score function for each of these three passive modes. From this single object, stem both fundamental limits of recovery (Cramer-Rao bounds) and practical recovery algorithms based on diffusion posterior sampling. The paper further generalizes fundamental limits to Bayesian Cramer-Rao lower bounds. This generalization makes use of a learned approximation of the score function of signal priors. In prior art, analyses and diffusion-based reconstruction for SPAD data have treated individual modes in isolation. Our unified treatment shows a qualitative high-flux gap between modes: binary counts saturate exponentially, while timestamped modes degrade only linearly. We further extend diffusion posterior sampling, previously restricted to binary SPAD data, to a full timestamped case using the suitable score function. We demonstrate experimentally that matching the score to the operation mode is beneficial for high-fidelity reconstruction. By tying the recovery bounds and diffusion to the score function, this work aims to establish a common foundation for both asking what is recoverable in single-photon sensing, and building methods that approach the bound.

Diffusion models provide powerful generative priors for solving inverse problems by sampling from a posterior distribution conditioned on corrupted measurements. Existing methods primarily follow two paradigms: direct methods, which approximate the likelihood term, and proximal methods, which incorporate intermediate solutions satisfying measurement constraints into the sampling process. Under standard Gaussian approximations and locally-linear measurements, we demonstrate that these approaches differ fundamentally in their treatment of the diffusion denoiser's Jacobian within the likelihood term. While this Jacobian encodes critical prior knowledge of the data distribution, training-induced non-idealities can degrade performance in zero-shot settings. In this work, we bridge direct and proximal approaches by proposing a principled Jacobian-Aware Posterior Sampler (JAPS). JAPS leverages the Jacobian's prior knowledge while mitigating its detrimental effects through a corresponding proximal solution, requiring no additional computational cost. Additionally, we integrate our guidance into DDIM sampling, with a corrected conditional factor that has been missing in previous works. Our method enhances reconstruction quality across diverse linear and nonlinear noisy imaging tasks, outperforming existing diffusion-based baselines in perceptual quality while maintaining or improving distortion metrics.

From a Bayesian perspective, score-based diffusion solves inverse problems through joint inference, embedding the likelihood with the prior to guide the sampling process. However, this formulation fails to explain its practical behavior: the prior offers limited guidance, while reconstruction is largely driven by the measurement-consistency term, leading to an inference process that is effectively decoupled from the diffusion dynamics. We show that the diffusion prior in these solvers functions primarily as a warm initializer that places estimates near the data manifold, while reconstruction is driven almost entirely by measurement consistency. Based on this observation, we introduce \textbf{DAPS++}, which fully decouples diffusion-based initialization from likelihood-driven refinement, allowing the likelihood term to guide inference more directly while maintaining numerical stability and providing insight into why unified diffusion trajectories remain effective in practice. By requiring fewer function evaluations (NFEs) and measurement-optimization steps, \textbf{DAPS++} achieves high computational efficiency and robust reconstruction performance across diverse image restoration tasks.

Solving ill-posed inverse problems requires powerful and flexible priors. We propose leveraging pretrained latent diffusion models for this task through a new training-free approach, termed Diffusion-regularized Wasserstein Gradient Flow (DWGF). Specifically, we formulate the posterior sampling problem as a Wasserstein gradient flow in the latent space of an expected negative log posterior objective, regularized by a Kullback-Leibler divergence to the diffusion prior. We demonstrate the performance of our method on standard benchmarks using StableDiffusion (Rombach et al., 2022) as the prior.

Typically, numerical simulations of Earth systems are coarse, and Earth observations are sparse and gappy. We apply four generative diffusion modeling approaches to super-resolution and inference of forced two-dimensional quasi-geostrophic turbulence on the beta-plane from coarse, sparse, and gappy observations. Two guided approaches minimally adapt a pre-trained unconditional model: SDEdit modifies the initial condition, and Diffusion Posterior Sampling (DPS) modifies the reverse diffusion process score. Two conditional approaches, a vanilla variant and classifier-free guidance, require training with paired high-resolution and observation data. We consider multiple test cases spanning: two regimes, eddy and anisotropic-jet turbulence; two Reynolds numbers, 10^3 and 10^4; and two observation types, 4x coarse-resolution fields and coarse, sparse and gappy observations. Our comprehensive skill metrics include norms of the reconstructed vorticity fields, turbulence statistical quantities, and quantifications of the super-resolved probabilistic ensembles and their errors. We also study the sensitivity to tuning parameters such as guidance strength. Results show that the generated super-resolution fields of SDEdit are unphysical, while those of DPS are reasonable but with smoothed fine-scale features; however, neither of these lower-cost models propagates observational information effectively to unobserved regions. The two conditional models require re-training, but reconstruct missing fine-scale features, are cycle-consistent with observations, and predict correct turbulence statistics, including the tails. Further, their mean errors are highly correlated with and predictable from their ensemble standard deviations. Results highlight the tradeoffs between ease of implementation, fidelity (sharpness), and cycle-consistency of the diffusion models, and offer practical guidance for deployment.

Diffusion models represent the state-of-the-art for solving inverse problems such as image restoration tasks. Diffusion-based inverse solvers incorporate a likelihood term to guide prior sampling, generating data consistent with the posterior distribution. However, due to the intractability of the likelihood, most methods rely on isotropic Gaussian approximations, which can push estimates off the data manifold and produce inconsistent, poor reconstructions. We propose Equivariance Regularized (EquiReg) diffusion, a general plug-in framework that improves posterior sampling by penalizing trajectories that deviate from the data manifold. EquiReg formalizes manifold-preferential equivariant functions that exhibit low equivariance error for on-manifold samples and high error for off-manifold ones, thereby guiding sampling toward symmetry-preserving regions of the solution space. We highlight that such functions naturally emerge when training non-equivariant models with augmentation or on data with symmetries. EquiReg's largest gains are under reduced sampling and measurement consistency steps, where many methods suffer severe quality degradation. By regularizing trajectories toward the manifold, EquiReg implicitly accelerates convergence and enables high-quality reconstructions. EquiReg consistently improves performance in linear and nonlinear image restoration tasks and solving partial differential equations. Our code is available at https://github.com/Anima-Lab/EquiReg