Ill-Posed Inverse Problem

Plug-and-Play (PnP) methods have become standard tools for solving imaging inverse problems by replacing the intractable maximum a posteriori (MAP) denoiser with the MMSE one. While this mismatch has been widely treated as unavoidable, recent works have sought to close this gap by targeting the MAP with diffusion-model scores. We show this is problematic in practice: learned scores do not match the true ones, so MAP-targeting iterations converge to cartoon-like images rather than realistic ones, and better results are obtained by stopping short of convergence. We turn this observation into a design principle and introduce ProxiMAP, an iterative MAP approximation whose noise schedule keeps the iterate's residual noise matched to the denoiser's training noise. This keeps the denoiser in-distribution where its score is reliable, and yields implicit early stopping that avoids the failure mode above. ProxiMAP is a modular drop-in replacement for MMSE denoisers in standard PnP algorithms and consistently sharpens reconstructions across deblurring, inpainting, super-resolution, and phase retrieval. Building on the same principle, we propose a hybrid variant that applies ProxiMAP only in the late iterations of PnP, where the denoiser is most reliable -- matching or exceeding the full-replacement variant at a fraction of the cost.

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.

Training-free camera control for pretrained flow-matching video generators is a partial-observation inverse problem: a depth-warped guidance video supplies noisy evidence on a subset of latent sites, which the sampler must reconcile with the pretrained prior. Existing methods struggle to balance the trade-off between trajectory adherence and visual quality and the heuristic guidance-strength tuning lacks robustness. We propose \textbf{$h$-control}, which resolves this dilemma through a structural change to the sampler: each outer hard-replacement guidance step is augmented with an inner-loop \emph{block-conditional pseudo-Gibbs refinement} on the unobserved complement at the same noise level, with provable convergence to the partial-observation conditional data law. To accelerate convergence on high-dimensional video latents, we exploit their conditional locality, partitioning the unobserved complement into 3D patches, each tracked by a custom mixing indicator that adaptively freezes converged patches. On RealEstate10K and DAVIS, \textbf{$h$-control} attains the best FVD against all seven training-free and training-based competitors, outperforming every training-free baseline on every reported metric.

Data-free continual learning (DFCIL) relies on model inversion to synthesize pseudo-samples and mitigate catastrophic forgetting. However, existing inversion methods are fundamentally limited by a simplifying assumption: they model feature distributions using diagonal covariance, effectively ignoring correlations that define the geometry of learned representations. As a result, synthesized samples often lack fidelity, limiting knowledge retention. In this work, we show that modeling feature dependencies is a key ingredient for effective DFCIL. We introduce REMIX, a structured covariance modeling framework that enables scalable full-covariance modeling without the prohibitive cost of dense matrix inversion and log-determinant computation. By leveraging a Laplace kernel parameterization, REMIX captures structured feature dependencies using memory that scales linearly with the feature dimensionality, while requiring only an additional logarithmic factor in computation. Modeling these correlations produces more coherent synthetic samples and consistently improves performance across standard DFCIL benchmarks. Our results demonstrate that moving beyond diagonal assumptions is essential for effective and scalable data-free continual learning. Our code is available at https://github. com/pkrukowski1/REMIX-Model-Inversion-via-Laplace-Kernel.

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.

Generative (diffusion) priors demonstrate remarkable performance in addressing inverse problems in imaging. Yet, for scientific and medical imaging, it is crucial that reconstruction techniques remain stable and reliable under imperfect settings. Typical definitions of stability encompass the notion of ''convergent regularization'', robustness to out-of-distribution data, and to inaccuracies in the forward operator or noise model. We evaluate these properties numerically. Furthermore, we benchmark generative approaches against modern optimization-based methods inspired by the widely used variational techniques. Our results give insights for which settings and applications generative priors can deliver state-of-the-art reconstructions, and on those in which they fall short or may even be problematic.

Loss reweighting is a widely used strategy for long-tailed classification, but existing reweighting strategies often rely on heuristics and rarely define a well-specified target. Inspired by Neural Collapse (NC), the ideal simplex Equiangular Tight Frame (ETF) terminal geometry suggests equal per-class average loss as a reasonable target for reweighting. Based on the ideal equal loss objective, we consider loss reweighting as an inverse problem and propose an inverse-view reweighting strategy that infers class weights dynamically to match this ideal objective. Empirically, NC metrics suggest our method can effectively reduce the loss imbalance coefficient and closer alignment with NC geometry while consistently outperforming strong long-tailed baselines on different datasets.

Machine learning-based simulators offer the potential to model the dynamics of complex systems more efficiently than classical approaches, while retaining differentiability, a key property for materials design. Graph neural network (GNN)-based simulators have shown strong performance across a range of physical domains, including molecular dynamics. However, their reliance on temporal context for accurate prediction limits their use in inverse design settings, where simulations must be initialized from a single static configuration. Moreover, inverse design requires robust out-of-distribution (OOD) generalization, as candidate structures typically lie outside the training domain. Here, we address both challenges by introducing two complementary strategies that enable stable and accurate structure-only initialization of GNN-based simulations. To directly target OOD generalization, we propose an inference-time physics-based optimization framework that constrains model predictions to remain physically consistent during rollout. In addition, we introduce a differentiable, GNN-based barostat that enables accurate tracking of system dimensions and pressure, critical for capturing macroscopic responses and supporting OOD generalization. We evaluate these approaches in the context of uniaxial compression of disordered elastic networks spanning a broad range of geometries, Poisson ratios, and microscopic behaviors. We find that, together, these methods substantially improve rollout stability and enable reliable OOD generalization, including regimes with distinct, more complex dynamics than those in the training data. These results show that, when properly initialized and constrained, GNN-based simulators can serve as efficient and generalizable tools for materials discovery and structural optimization, advancing their use in materials, molecular, and dynamical system design.

Programming the static friction of mechanical interfaces is critical for soft robotics, haptics, and precision gripping. Static friction is governed by the real contact area, and standard rough surfaces exhibit a linear area-load scaling inherent to classical Archard and Greenwood-Williamson models, severely restricting their functional range. Here, we propose a framework for the inverse design of tribological metainterfaces engineered for programmable contact behaviors. By utilizing general axisymmetric asperities, we unlock nonlinear macroscopic responses unattainable by standard Hertzian contacts. To solve the inverse problem, we embed a fully differentiable contact mechanics engine within a neural network and a quadratic optimizer. We leverage regularized physical gradients to automatically discover non-standard topographies that reproduce complex target friction laws, with only a few asperities in unit cells. The predicted designs are strictly validated against high-fidelity Boundary Element Method (BEM) simulations. This framework bridges data-driven optimization and rigorous physics, offering a scale-invariant pathway for discovering functional tribological surfaces.

Methods based on diffusion models (DMs) for solving inverse problems (IPs) have recently achieved remarkable performance. However, DM-based methods typically struggle against outliers, which are common in real-world measurements. In this work, to tackle IPs with outliers, we first refine the measurement via explicit noise estimation to mitigate the effect of noise. Subsequently, we formulate an iteratively reweighted least squares objective based on the Huber loss to address the outliers. We propose a method utilizing gradient descent to approximately solve the corresponding optimization problem for the robust objective. To avoid delicate tuning of the learning rate required by the gradient descent method, we further employ the conjugate gradient method with an efficient strategy for updating. Extensive experiments on multiple image datasets for linear and nonlinear tasks under various conditions demonstrate that our proposed methods exhibit robustness to outliers and outperform recent DM-based methods in most cases.

Generative models are powerful tools for sampling from a learned distribution $\mathcal{P}(Y \mid X)$, and inverse-design methods invert this map to find an input $x$ that produces a desired point output $y^*$. However, many design goals are naturally distributional rather than pointwise, incorporating the inherent uncertainty of $Y$ and targeting a specific form for it, a task not addressed by standard inverse design. To address this issue we introduce Conditional Distribution Matching (CDM), a new inverse-design problem class in generative modeling: given a joint distribution $\mathcal{P}(X, Y)$ and a target distribution $\mathcal{G}(Y)$, find an input $x^*$ whose induced conditional distribution $\mathcal{P}(Y \mid X = x^*)$ matches $\mathcal{G}$. We formally define two variants: Conditional Distribution Matching Sampling (CDMS) and Conditional Distribution Matching Optimization (CDMO). To solve these problems, we propose MLGD-F (Matching-Loss Guided Diffusion with a Fast inner sampler), a plug-and-play inference-time algorithm that combines a pretrained score-based diffusion model with a pretrained fast conditional sampler, requiring no additional training or fine-tuning. By leveraging single-step conditional sampling, MLGD-F enables tractable gradient computation, making the estimation of $\mathcal{P}(Y \mid X)$ both memory-efficient and computationally lightweight. We validate MLGD-F on synthetic benchmarks, structured image transformations, and generative editing optimization, demonstrating reliable recovery of inputs whose conditional distributions match diverse user-specified targets, including discrete mixtures and continuous low-rank supports.

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.

Plug-and-play (PnP) methods for solving inverse problems have recently achieved strong performance by leveraging denoising priors based on powerful generative diffusion and flow models. However, existing diffusion- and flow-based PnP methods typically rely on stochastic renoise-denoise operations, which complicate the analysis of their convergence behavior. In this work, we identify and formalize the deterministic renoise-denoise operator underlying flow-based plug-and-play methods. This perspective reveals that these methods implicitly define a deterministic operator given by the expectation of a denoiser over the latent noise distribution. Building on this insight, we propose FlowADMM, a PnP algorithm that integrates the renoise-denoise operator into the classical alternating direction method of multiplier (ADMM) framework. We establish convergence guarantees for FlowADMM under weak Lipschitz conditions on the underlying flow network, and extend the analysis to non-stationary time schedules. Empirically, FlowADMM achieves state-of-the-art performance among flow-based PnP methods on a range of inverse problems, including denoising, deblurring, super-resolution, and inpainting, while requiring fewer data consistency evaluations than prior approaches.

Generative models based on flow matching have emerged as a powerful paradigm for inverse problems, offering straighter trajectories and faster sampling compared to diffusion models. However, existing approaches often necessitate differentiating through unrolled paths, leading to numerical instability and prohibitive computational overhead. To address this, we propose P-Flow, a framework that stabilizes the reconstruction process by leveraging a proxy gradient to update the source point. This approach effectively circumvents the numerical instability and memory overhead of long-chain differentiation. To ensure consistency with the prior distribution, we employ a Gaussian spherical projection motivated by the concentration of measure phenomenon in high-dimensional spaces. We further provide a theoretical analysis for P-Flow based on Bayesian theory and Lipschitz continuity. Experiments across diverse restoration tasks demonstrate that P-Flow delivers competitive performance, especially under extreme degradations such as severely ill-posed conditions and high measurement noise.

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.

Commercial Microwave Links (CMLs) offer dense spatial coverage for rainfall sensing but produce path-integrated measurements that make accurate ground-level reconstruction challenging. Existing methods typically oversimplify CMLs as point sensors and neglect line integration relating rainfall to signal attenuation, resulting in degraded performance under heterogeneous precipitation. In this work, we view rain field reconstruction as a Bayesian inverse problem with Diffusion Models (DMs) as high-fidelity spatial priors. We show that diffusion models better preserve key rainfall statistics compared to censored Gaussian processes. Framing rainfall estimation as a Bayesian inverse problem with a DM prior enables training-free posterior sampling using a broad family of methods, including Plug-and-Play, Sequential Monte Carlo, and Replica Exchange methods. Experiments on synthetic and real-world datasets demonstrate consistent improvements over established CML-based reconstruction baselines.

Inverse design of RF passive components from S-parameters is a high-dimensional, ill-posed problem, and prior generative approaches are limited to single-layer binary-metallization structures. This paper presents an inverse design approach that generates passive components from partial S-parameter inputs on an $8\times8$ mm board discretized at $64\times64$ pixels with sub-pixel grayscale metallization across 1-20 GHz. The framework generates two-layer copper layouts with vias, with hard physical constraints on feed locations enforced through annealed Langevin projection, flexible multi-modal conditioning on partial S-parameter specifications, port locations, dielectric properties, reference topology, and variable port placement. Candidate designs are generated in seconds, with surrogate-predicted S-parameters matching targets to within $0.77 \pm 1.28$ dB weighted mean absolute error. We validate the approach with two fabricated designs on RO4003C: a manufacturable alternative to a hairpin filter whose coupling gaps violate fabrication rules, and a combline bandpass filter designed from scratch given only target S-parameters.

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.

This work presents a deeper analysis of the "irreversibility" property of PolyProtect, a biometric template protection method initially proposed for securing face embeddings. PolyProtect transforms embeddings into protected templates via multivariate polynomials, whose coefficients and exponents are distinct for each subject enrolled in the face recognition system. A polynomial is applied to consecutive sets of elements from a given embedding, where the amount of overlap between the sets is a tunable parameter. We begin our irreversibility analysis by demonstrating that PolyProtected templates are easier to invert using a numerical solver based on cosine distance, as opposed to Euclidean distance (used in the earlier PolyProtect work). To make this inversion more difficult, we then propose a "key selection algorithm", which tries to choose "keys" (coefficients and exponents of the PolyProtect polynomial) that enhance the irreversibility of PolyProtected templates, compared to when the keys are purely random. Our experiments show that this algorithm is effective at generating PolyProtected templates that are significantly more difficult to invert, and that it approximately equalises the irreversibility of PolyProtected templates generated using different "overlap" parameters. This allows for better control of the irreversibility versus accuracy trade-off, known to exist across different overlaps. We also show that accuracy in the PolyProtected domain can be affected by the range in which the embedding elements lie, but that this can be improved by normalizing the embeddings prior to applying PolyProtect. This work is reproducible using our open-source code.

Propagation-based X-ray phase-contrast imaging (PBI) enables high-contrast visualization of lung structures and holds strong medical potential. However, safe translation to the clinic will require a substantial radiation dose reduction, which inevitably increases image noise. Supervised convolutional-neural-network-based denoising can restore image quality but depends on paired low- and high-dose datasets, which are rarely available in practice. Self-supervised methods avoid this limitation, yet most are not well adapted to the inverse problem of PBI computed tomography (CT). We introduce Neighbor2Inverse, a self-supervised denoising framework designed for low-dose PBI-CT that generalizes to clinical CT. Building on the Neighbor2Neighbor principle, each noisy projection is subsampled into two variants that preserve structural information but contain independent noise realizations. These are reconstructed separately, and the resulting pairs are used to train a denoising network directly in the image domain. We benchmark the proposed method against established analytical and self-supervised denoising approaches. In region-of-interest PBI CT experiments, Neighbor2Inverse achieves superior noise suppression while preserving fine structural details, as demonstrated by improved contrast-to-noise ratio, spatial resolution, and composite image quality metrics. Competitive performance is also observed on clinical CT data under simulated low-dose conditions. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible. Code, data, and interactive figures are available at https://github.com/J-3TO/Neighbor2Inverse.

Nonlinear dynamical systems with regime transitions are typically described by ordinary differential equations with jumping parameters parameters. Traditional methods often treat change-point detection and parameter estimation as separate tasks, ignoring the inherent coupling between them. To address this, we propose residual-loss anomaly analysis of physics-informed neural networks, a unified framework that leverages dynamical consistency within the physics-informed learning paradigm. This approach jointly infers piecewise parameters and transition points under a single set of constraints. The method follows a two-stage strategy: First, local physical residuals are analyzed through overlapping subinterval decomposition. When a subinterval spans a true transition point, the residual exhibits a distinct structural elevation in noise-free conditions, which has a non-zero lower bound, enabling effective localization of potential transition intervals. Second, within our framework, change-point locations and piecewise parameters are integrated into a unified physical loss function for joint optimization, enabling simultaneous identification. Experiments on benchmark nonlinear dynamical systems, including Malthusian and logistic growth models, Van der Pol oscillator, Lotka-Volterra model and Lorenz system, demonstrate that the proposed method outperforms traditional decoupled approaches in both change-point localization and parameter estimation accuracy. This study provides an efficient, unified solution for structurally coupled inverse problems in nonlinear dynamical systems with regime switching.

Large (vision-)language models exhibit remarkable capability but remain highly susceptible to jailbreaking. Existing safety training approaches aim to have the model learn a refusal boundary between safe and unsafe, based on the user's intent. It has been found that this binary training regime often leads to brittleness, since the user intent cannot reliably be evaluated, especially if the attacker obfuscates their intent, and also makes the system seem unhelpful. In response, frontier models, such as GPT-5, have shifted from refusal-based safeguards to safe completion, that aims to maximize helpfulness while obeying safety constraints. However, safe completion could be exploited when a user pretends their intention is benign. Specifically, this intent inversion would be effective in multi-turn conversation, where the attacker has multiple opportunities to reinforce their deceptively benign intent. In this work, we introduce a novel multi-turn jailbreaking method that exploits this vulnerability. Our approach gradually builds conversational trust by simulating benign-seeming intentions and by exploiting the consistency property of the model, ultimately guiding the target model toward harmful, detailed outputs. Most crucially, our approach also uncovered an additional class of model vulnerability that we call para-jailbreaking that has been unnoticed up to now. Para-jailbreaking describes the situation where the model may not reveal harmful direct reply to the attack query, however the information that it reveals is nevertheless harmful. Our contributions are threefold. First, it achieves high success rates against frontier models including GPT-5-thinking and Claude-Sonnet-4.5. Second, our approach revealed and addressed para-jailbreaking harmful output. Third, experiments on multimodal VLM models showed that our approach outperformed state-of-the-art models.

Improving the effectiveness of human-robot interaction requires social robots to accurately infer human goals through robust intention understanding. This challenge is particularly critical in multimodal settings, where agents must integrate heterogeneous signals including text, visual cues to form a coherent interpretation of user intent. This paper presents IntentVLM, a novel two-stage video-language framework designed for open-vocabulary human intention recognition. The approach is inspired by forward-inverse modeling in cognitive science by decomposing intention understanding into goal candidate generation followed by structured inference through selection, effectively reducing hallucinations in latent reasoning. Evaluated on the IntentQA and Inst-IT Bench datasets, IntentVLM achieves state-of-the-art results with up to 80% accuracy, notably surpassing the baseline performance by 30% and matches human performance. Our findings demonstrate that this structured reasoning approach enhances open-vocabulary intention understanding without catastrophic forgetting, offering a robust foundation for human-centered robotics.

Preference-based argumentation frameworks (PAFs) extend Dung's approach to abstract argumentation (AAFs) by encoding preferences over arguments. Such preferences control the transformation of attacks into defeats, and different approaches to doing so result in different reductions from a PAF to an AAF. In this paper we consider a PAF inverse problem which takes an argumentation graph, a labelling and a semantics as an input, and outputs a yes" or no" as to whether there is a preference relation between the arguments which can yield the desired labelling. This inverse problem has applications in areas including preference elicitation and explainability. We consider this problem in the context of the four most widely-used preference based reductions under the complete semantics. We show that in most cases, the problem can be answered in polynomial time.

Current model training approaches incorporate user information directly into shared weights, making individual data removal computationally infeasible without retraining. This paper presents a three-layer architecture that decouples personal data from shared weights by combining a static base model, composable domain-expert LoRA adapters that shape behavior without imparting user data, and per-user proxy artefacts whose deletion constitutes deterministic unlearning. Evaluation on Phi-3.5-mini and Llama-3.1-8B confirms per-user differentiation in which personal data influences outputs while remaining isolated, verified by a return to baseline after proxy removal (KL divergence of approximately 0.21 nats, 82-89% verification pass rate) and near-zero cross-user contamination. Because user-specific information never enters shared weights, the architecture mitigates model inversion, membership inference, and training-data extraction against shared model components by construction. The approach converts machine unlearning from an intractable weight-editing problem into a deterministic deletion operation that preserves personalization alongside privacy-enhancing guarantees and is compatible with differentially private stochastic gradient descent (DP-SGD) for privacy-preserving shared model improvement.

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.

We study the deep image prior (DIP) framework applied to photoacoustic tomography (PAT) as an unsupervised reconstruction approach to mitigate limited-view artifacts and noise commonly encountered in experimental settings. Efficient implementation is achieved by employing recently published fast forward and adjoint algorithms for circular measurement geometries. Initialization via a fast inverse and total variation (TV) regularization are applied to further suppress noise and mitigate overfitting. For comparison, we compute a classical TV reconstruction. Our experiments comprise simulated PAT measurements under limited-view geometries and varying levels of added noise as well as experimental measurements together with using a digital twin for quality assessment. Our findings suggest that DIP framework provides an effective unsupervised strategy for robust PAT reconstruction even in the challenging case of a limited view geometry providing improvement in several quantitative measures over total variation reconstructions.

Diffusion models have achieved outstanding performance in unsupervised industrial anomaly detection (uIAD) by learning a manifold of normal data under the common assumption that off-manifold anomalies are harder to generate, resulting in larger reconstruction errors in data space or lower probability densities in the tractable latent space. However, their iterative denoising and noising nature leads to slow inference. In this paper, we propose OSD-IRF, a novel one-step diffusion with inverse residual fields, to address this limitation for uIAD task. We first train a deep diffusion probabilistic model (DDPM) on normal data without any conditioning. Then, for a test sample, we predict its inverse residual fields (IRF) based on the noise estimated by the well-trained parametric noise function of the DDPM. Finally, uIAD is performed by evaluating the probability density of the IRF under a Gaussian distribution and comparing it with a threshold. Our key observation is that anomalies become distinguishable in this IRF space, a finding that has seldom been reported in prior works. Moreover, OSD-IRF requires only single step diffusion for uIAD, thanks to the property that IRF holds for any neighboring time step in the denoising process. Extensive experiments on three widely used uIAD benchmarks show that our model achieves SOTA or competitive performance across six metrics, along with roughly a 2X inference speedup without distillation.

Digital marketplaces processing billions of dollars annually represent critical infrastructure in sociotechnical ecosystems, yet their performance optimization lacks principled measurement frameworks that can inform algorithmic governance decisions regarding market efficiency and fairness from complex market data. By looking at orderbook data from double auction markets alone, because bids and asks do not represent true maximum willingnesses to buy and true minimum willingnesses to sell, there is little an economist can say about the market's actual performance in terms of allocative efficiency. We turn to experimental data to address this issue, `inverting' the standard induced value approach of double auction experiments. Our aim is to predict key market features relevant to market efficiency, particularly allocative efficiency, using orderbook data only -- specifically bids, asks and price realizations, but not the induced reservation values -- as early as possible. Since there is no established model of strategically optimal behavior in these markets, and because orderbook data is highly unstructured, non-stationary and non-linear, we propose quantile-based normalization techniques that help us build general predictive models. We develop and train several models, including linear regressions and gradient boosting trees, leveraging quantile-based input from the underlying supply-demand model. Our models can predict allocative efficiency with reasonable accuracy from the earliest bids and asks, and these predictions improve with additional realized price data. The performance of the prediction techniques varies by target and market type. Our framework holds significant potential for application to real-world market data, offering valuable insights into market efficiency and performance, even prior to any trade realizations.

Physics-informed neural networks (PINNs) have emerged as a promising framework for solving inverse problems governed by partial differential equations (PDEs), including the reconstruction of turbulent flow fields from sparse data. However, most existing PINN formulations are deterministic and do not provide reliable quantification of epistemic uncertainty, which is critical for ill-posed problems such as data-driven Reynolds-averaged Navier-Stokes (RANS) modeling. In this work, we develop and systematically evaluate a set of probabilistic extensions of PINNs for uncertainty quantification in turbulence modeling. The proposed framework combines (i) Bayesian PINNs with Hamiltonian Monte Carlo sampling and a tempered multi-component likelihood, (ii) Monte Carlo dropout, and (iii) repulsive deep ensembles that enforce diversity in function space. Particular emphasis is placed on the role of ensemble diversity and likelihood tempering in improving uncertainty calibration for PDE-constrained inverse problems. The methods are assessed on a hierarchy of test cases, including the Van der Pol oscillator and turbulent flow past a circular cylinder at Reynolds numbers Re=3,900 (direct numerical simulation data) and Re = 10,000 (experimental particle image velocimetry data). The results demonstrate that Bayesian PINNs provide the most consistent uncertainty estimates across all inferred quantities, while function-space repulsive ensembles offer a computationally efficient approximation with competitive accuracy for primary flow variables. These findings provide quantitative insight into the trade-offs between accuracy, computational cost, and uncertainty calibration in physics-informed learning, and offer practical guidance for uncertainty quantification in data-driven turbulence modeling.

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.

Machine learning has achieved impressive performance in tomographic reconstruction, but supervised training requires paired measurements and ground-truth images that are often unavailable. This has motivated self-supervised approaches, which have primarily addressed denoising and, more recently, linear inverse problems. We address nonlinear inverse problems and introduce SPLIT (Self-supervised Partitioning for Learned Inversion in Nonlinear Tomography), a self-supervised machine-learning framework for reconstructing images from nonlinear, incomplete, and noisy projection data without any samples of ground-truth images. SPLIT enforces cross-partition consistency and measurement-domain fidelity while exploiting complementary information across multiple partitions. Our main theoretical result shows that, under mild conditions, the proposed self-supervised objective is equivalent to its supervised counterpart in expectation. We regularize training with an automatic stopping rule that halts optimization when a no-reference image-quality surrogate saturates. As a concrete application, we derive SPLIT variants for multispectral computed tomography. Experiments on sparse-view acquisitions demonstrate high reconstruction quality and robustness to noise, surpassing classical iterative reconstruction and recent self-supervised baselines.

Proximal operators are fundamental across many applications in signal processing and machine learning, including solving ill-posed inverse problems. Recent work has introduced Learned Proximal Networks (LPNs), providing parametric functions that compute exact proximals for data-driven and potentially non-convex regularizers. However, in many settings it is important to include additional structure to these regularizers--and their corresponding proximals--such as shift and scale equivariance. In this work, we show how to obtain learned functions parametrized by neural networks that provably compute exact proximal operators while being equivariant to shifts and scaling, which we dub Affine-Equivariant Learned Proximal Networks (AE-LPNs). We demonstrate our results on synthetic, constructive examples, and then on real data via denoising in out-of-distribution settings. Our equivariant learned proximals enhance robustness to noise distributions and affine shifts far beyond training distributions, improving the practical utility of learned proximal operators

Text-driven inversion of generative models is a core paradigm for manipulating 2D or 3D content, unlocking numerous applications such as text-based editing, style transfer, or inverse problems. However, it relies on the assumption that generative models remain sensitive to natural language prompts. We demonstrate that for state-of-the-art native text-to-3D generative models, this assumption often collapses. We identify a critical failure mode where generation trajectories are drawn into latent ``sink traps'': regions where the model becomes insensitive to prompt modifications. In these regimes, changes to the input text fail to alter internal representations in a way that alters the output geometry. Crucially, we observe that this is not a limitation of the model's \textit{geometric} expressivity; the same generative models possess the ability to produce a vast diversity of shapes but, as we demonstrate, become insensitive to out-of-distribution \textit{text} guidance. We investigate this behavior by analyzing the sampling trajectories of the generative model, and find that complex geometries can still be represented and produced by leveraging the model's unconditional generative prior. This leads to a more robust framework for text-based 3D shape editing that bypasses latent sinks by decoupling a model's geometric representation power from its linguistic sensitivity. Our approach addresses the limitations of current 3D pipelines and enables high-fidelity semantic manipulation of out-of-distribution 3D shapes. Project webpage: https://daidedou.sorpi.fr/publication/beyondprompts

High-fidelity simulations and complex inverse problems, such as detector modeling and unfolding, are computationally intensive bottlenecks across subatomic physics, yet essential for accurate physical interpretation. While Conditional Flow Matching (CFM) offers a robust acceleration approach, we demonstrate its standard training loss is fundamentally misleading. Specifically, utilizing a Jefferson Lab Nuclear Physics (NP) kinematic dataset ($γp \to ρ^0 p \to π^+π^- p$), we expose that CFM loss plateaus prematurely, obscuring ongoing physical refinement. To verify this disconnect is a dataset-agnostic pathology, we introduce ScatterPrism, an efficient generative surrogate evaluated against both the NP data and synthetic stress tests modeling challenging 1D distribution topologies. Coupling these benchmarks, we establish that physics-informed metrics continue improving long after standard loss converges. Consequently, we propose a multi-metric diagnostic protocol to ensure true kinematic fidelity without data memorization. Driven by NP challenges relevant to the forthcoming Electron-Ion Collider (EIC), this unified machinery has strong potential to extend to High-Energy Physics (HEP) applications, such as jet modeling. Furthermore, the framework holds promise for broader domains requiring rigorous generative reliability, including medical imaging, astrophysics, and quantitative finance.

Inverse design aims to find design parameters $x$ achieving target performance $y^*$. Generative approaches learn bidirectional mappings between designs and labels, enabling diverse solution sampling. However, standard conditional flow matching (CFM), when adapted to inverse problems by pairing labels with design parameters, exhibits strong sensitivity to their arbitrary ordering and scaling, leading to unstable training. We introduce Diagonal Flow Matching (Diag--CFM), which resolves this through a zero-anchoring strategy that pairs design coordinates with noise and labels with zero, making the learning problem provably invariant to coordinate permutations. This yields substantially lower round-trip error than CFM and invertible neural network baselines across design dimensions up to $P{=}784$, including order-of-magnitude gains on several benchmarks. We develop two architecture-intrinsic uncertainty metrics, Zero-Deviation and Self-Consistency, that enable three practical capabilities: selecting the best candidate among multiple generations, abstaining from unreliable predictions, and detecting out-of-distribution targets; consistently outperforming ensemble and general-purpose alternatives across all tasks. We validate on airfoil, gas turbine combustor, scalable analytical benchmarks, a photonics inverse-design task, and an image-statistics benchmark.

Constructing reduced-order models (ROMs) capable of efficiently predicting the evolution of parameter-dependent high-dimensional dynamical systems is crucial in many applications in engineering and applied sciences. A popular class of projection-based ROMs projects the high-dimensional full-order model (FOM) dynamics onto a low-dimensional manifold. These projection-based ROMs approaches often rely on classical model reduction techniques such as proper orthogonal decomposition (POD) or, more recently, on neural network architectures such as autoencoders (AEs). In the case that the ROM is constructed by the POD, one has approximation guaranteed based based on the singular values of the problem at hand. However, POD-based techniques can suffer from slow decay of the singular values in transport- and advection-dominated problems. In contrast to that, AEs allow for better reduction capabilities than the POD, often with the first few modes, but at the price of theoretical considerations. In addition, it is often observed, that AEs exhibits a plateau of the projection error with the increment of the dimension of the trial manifold. In this work, we propose an invertible AE architecture, named inv-AE, that computationally improves upon the stagnation of the reconstruction error typical of traditional AE architectures. Inv-AE is composed of several invertible neural network layers that allows for gradually recovering more information about the FOM solutions the more we increase the dimension of the reduced manifold. Through the application of inv-AE to a 1-dimensional Burgers' equation, a 2-dimensional fluid flow around an obstacle with variable geometry, and a 3-dimensional Korteweg-de Vries, we show that (i) inv-AE mitigates the issue of the characteristic plateau of AEs and (ii) inv-AE can be combined with popular autoencoder-based ROM approaches, e.g., DL-ROM, to improve their accuracy.

Subsurface ore detection is of paramount importance given the rising depletion of shallow mineral resources in recent years. It is crucial to explore approaches that go beyond the limitations of traditional geological exploration methods. Due to readily available surface readings, joint magnetic and gravitational inversion is a promising new method - given magnetic and gravitational data on a surface, jointly reconstructing the underlying densities that generate them. However, this is ill-posed and has non-unique solutions. Deterministic methods often require handcrafted priors and converge to a single solution and do not capture the distribution, which is often of interest. We introduce a novel framework that reframes 3D gravity and magnetic joint inversion as a rectified flow on the Noddyverse dataset, the largest physics-based dataset for inversion. We introduce a Ginzburg-Landau (GL) regularizer, a generalized version of the Ising model that aids in ore identification, enabling physics-aware training. We also propose a guidance methodology based on GL theory that can be used as a plug-and-play module with existing unconditional denoisers. Lastly, we also train and release a VAE for the 3D densities, which facilitates downstream work in the field.

Generalizing skill policies to novel conditions remains a key challenge in robot learning. Imitation learning methods, while data-efficient, are largely confined to the training region and consistently fail on input data outside it, leading to unpredictable policy failures. Alternatively, transfer learning approaches offer methods for trajectory generation robust to both changes in environment and tasks, but they remain data-hungry and lack accuracy in zero-shot generalization. We address these challenges in the context of task inversion learning and propose a novel joint learning approach to achieve accurate and efficient knowledge transfer. Our method constructs a common representation of the forward and inverse tasks, and leverages auxiliary forward demonstrations from novel configurations to successfully execute the corresponding inverse tasks, without any direct supervision. We demonstrate the extrapolation capabilities of our framework through ablation studies and experiments in simulated and real-world environments that require complex manipulation skills with a diverse set of objects and tools, where we outperform diffusion-based and multimodal VAE alternatives.

Volume denoising is a foundational problem in computational imaging, as many 3D imaging inverse problems face high levels of measurement noise. Inspired by the strong 2D image denoising properties of Field of Junctions (ICCV 2021), we propose a novel, fully volumetric 3D Field of Junctions (3D FoJ) representation that optimizes a junction of 3D wedges that best explain each 3D patch of a full volume, while encouraging consistency between overlapping patches. In addition to direct volume denoising, we leverage our 3D FoJ representation as a structural prior that: (i) requires no training data, and thus precludes the risk of hallucination, (ii) preserves and enhances sharp edge and corner structures in 3D, even under low signal to noise ratio (SNR), and (iii) can be used as a drop-in denoising representation via projected or proximal gradient descent for any volumetric inverse problem with low SNR. We demonstrate successful volume reconstruction and denoising with 3D FoJ across three diverse 3D imaging tasks with low-SNR measurements: low-dose X-ray computed tomography (CT), cryogenic electron tomography (cryo-ET), and denoising point clouds such as those from lidar in adverse weather. Across these challenging low-SNR volumetric imaging problems, 3D FoJ outperforms the evaluated classical denoisers, untrained neural denoisers, and denoisers trained only on noisy examples. Code is available at https://github.com/voilalab/3D-Field-of-Junctions.

This article addresses the issue of estimating observation parameters (response and error parameters) in inverse problems. The focus is on cases where regularization is introduced in a Bayesian framework and the prior is modeled by a diffusion process. In this context, the issue of posterior sampling is known to be thorny, and a recent paper proposes a notably simple and effective solution. Additionally, it opens an remarkable flexibility when it comes to estimating observation parameters. The proposed strategy enables to define an optimal estimator for both observation parameters and image of interest. Furthermore, the strategy provides a means for uncertainty quantification. In addition, MCMC algorithms allow for the computation of estimates and properties of posteriors, while offering some guarantees. The paper presents several numerical experiments that clearly confirm the computational efficiency and the quality of both estimates and uncertainty quantification.

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.

We develop an operator-theoretic framework for stability and statistical concentration in nonlinear inverse problems with block-structured parameters. Under a unified set of assumptions combining blockwise Lipschitz geometry, local identifiability, and sub-Gaussian noise, we establish deterministic stability inequalities, global Lipschitz bounds for least-squares misfit functionals, and nonasymptotic concentration estimates. These results yield high-probability parameter error bounds that are intrinsic to the forward operator and independent of any specific reconstruction algorithm. As a concrete instantiation, we verify that the Gaussian Splatting rendering operator satisfies the proposed assumptions and derive explicit constants governing its Lipschitz continuity and resolution-dependent observability. This leads to a fundamental stability--resolution tradeoff, showing that estimation error is inherently constrained by the ratio between image resolution and model complexity. Overall, the analysis characterizes operator-level limits for a broad class of high-dimensional nonlinear inverse problems arising in modern imaging and differentiable rendering.

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.

Inverse problems are fundamental to many scientific and engineering disciplines; they arise when one seeks to reconstruct hidden, underlying quantities from noisy measurements. Many applications demand not just point estimates but interpretable uncertainty. Providing fast inference alongside uncertainty estimates remains challenging yet desirable in numerous applications. We propose the Variational Sparse Paired Autoencoder (vsPAIR) to address this challenge. The architecture pairs a standard VAE encoding observations with a sparse VAE encoding quantities of interest (QoI), connected through a learned latent mapping. The variational structure enables uncertainty estimation, the paired architecture encourages interpretability by anchoring QoI representations to clean data, and sparse encodings provide structure by concentrating information into identifiable factors rather than diffusing across all dimensions. To validate the effectiveness of our proposed architecture, we conduct experiments on blind inpainting, computed tomography (CT), and initial-condition inference for the heat equation, demonstrating that vsPAIR is a capable inverse problem solver that can provide interpretable and structured uncertainty estimates.

Image correction and rectangling are valuable tasks in practical photography systems such as smartphones. Recent remarkable advancements in deep learning have undeniably brought about substantial performance improvements in these fields. Nevertheless, existing methods mainly rely on task-specific architectures. This significantly restricts their generalization ability and effective application across a wide range of different tasks. In this paper, we introduce the Unified Rectification Framework (UniRect), a comprehensive approach that addresses these practical tasks from a consistent distortion rectification perspective. Our approach incorporates various task-specific inverse problems into a general distortion model by simulating different types of lenses. To handle diverse distortions, UniRect adopts one task-agnostic rectification framework with a dual-component structure: a {Deformation Module}, which utilizes a novel Residual Progressive Thin-Plate Spline (RP-TPS) model to address complex geometric deformations, and a subsequent Restoration Module, which employs Residual Mamba Blocks (RMBs) to counteract the degradation caused by the deformation process and enhance the fidelity of the output image. Moreover, a Sparse Mixture-of-Experts (SMoEs) structure is designed to circumvent heavy task competition in multi-task learning due to varying distortions. Extensive experiments demonstrate that our models have achieved state-of-the-art performance compared with other up-to-date methods.

Learned image reconstruction has become a pillar in computational imaging and inverse problems. Among the most successful approaches are learned iterative networks, which are formulated by unrolling classical iterative optimisation algorithms for solving variational problems. While the underlying algorithm is usually formulated in the functional analytic setting, learned approaches are often viewed as purely discrete. In this survey we present a unified operator view for learned iterative networks. Specifically, we formulate a learned reconstruction operator, defining how to compute, and separately the learning problem, which defines what to compute. In this setting we present common approaches and show that many approaches are closely related in their core. We review linear as well as non-linear inverse problems in this framework and present a short numerical study to conclude.

Recent approaches in controllable novel view video generation often rely on fine-tuning pre-trained Video Diffusion Models (VDMs). This dominant paradigm is computationally expensive and frequently suffers from catastrophic forgetting of the model's original generative priors. To address this challenge, here we propose InverseCrafter, a VDM training-free framework that reformulates novel view video generation as an inpainting-based inverse problem in the latent space, eliminating the need for any annotated 4D training data. The core of our method is to establish operator equivalence by employing a lightweight latent mask encoder to define a latent-domain masking operation via a continuous, multi-channel representation. This principled representation faithfully models the forward process in the latent domain, enabling efficient, backpropagation-free solvers while bypassing the costly bottleneck of repeated VAE operations. InverseCrafter achieves high-fidelity, spatio-temporally coherent novel view synthesis with near-zero additional inference overhead and excels at general-purpose video inpainting and editing by fully preserving the pre-trained VDM's generative capabilities.

How does irrelevant information (i.e., distractors) affect test-time scaling in vision-language models (VLMs)? Prior work on text-only language models has shown that textual distractors can intensify inverse scaling, causing models to reason longer but less effective reasoning traces. In this work, we investigate whether similar phenomena arise in multimodal settings. We introduce Idis (Images with distractors), a visual question-answering dataset that systematically varies distractors along semantic and numerical dimensions. Our analyses reveal that visual distractors affect reasoning VLMs in a fundamentally different way from textual distractors: although inverse scaling still emerges, visual distractors reduce accuracy without increasing reasoning length. We further show that attribute counts extracted from reasoning traces provide key insights into how distractors interact with reasoning length and accuracy. As a sanity check, we propose a simple prompting strategy that mitigates distractor-driven predictions in reasoning vision-language models.

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.

This work proposes a two-stage physics-informed deep learning framework that combines neural-network-based sampling with statistical inference and constrained parameter refinement. In the first stage, a dual-network physics-informed architecture is used, where a main-network approximates the PDE solution and an auxiliary coefficient sub-network provides a relaxed continuous soft approximation of the true discontinuous coefficient field. A gradient-adaptive weighting strategy is incorporated to improve residual training and enhance sampling reliability near possible discontinuity regions. The sampled coefficient values are then analyzed using Bayesian learning for Gaussian mixture models and birth-death Markov chain model selection, which identify the number of coefficient regimes and provide candidate intervals for coefficient values and transition regions. In the second stage, the inverse problem is reformulated as a constrained physics-informed estimator, in which the coefficient is replaced by a form-consistent hard approximation explicitly represented as a piecewise-constant function over the spatiotemporal domain. Comprehensive numerical experiments on PDEs with jump-discontinuous coefficients demonstrate that the proposed framework achieves adaptability and accurate parameter identification with acceptable computational costs compared to existing methods. Applications to solution reconstruction further illustrate its practical potential. This work provides a generalizable computational approach for inverse problems governed by PDEs with discontinuous parameter structures, particularly in non-stationary and heterogeneous systems.

Diffusion models have emerged as powerful generative priors for solving inverse imaging problems. However, their practical deployment is hindered by the substantial computational cost of slow, multi-step sampling. Although Consistency Models (CMs) address this limitation by enabling high-quality generation in only one or a few steps, their direct application to inverse problems has remained largely unexplored. This paper introduces a modified consistency sampling framework specifically designed for inverse problems. The proposed approach regulates the sampler's stochasticity through a measurement-consistency mechanism that leverages the degradation operator, thereby enforcing fidelity to the observed data while preserving the computational efficiency of consistency-based generation. Comprehensive experiments on the Fashion-MNIST and LSUN Bedroom datasets demonstrate consistent improvements across both perceptual and pixel-level metrics, including the Fréchet Inception Distance (FID), Kernel Inception Distance (KID), peak signal-to-noise ratio (PSNR), and structural similarity index measure (SSIM), compared with baseline consistency and diffusion-based sampling methods. The proposed method achieves competitive or superior reconstruction quality with only a small number of sampling steps.

Reconstructing a three-dimensional hyperspectral cube from a two-dimensional compressed measurement is a severely ill-posed inverse problem. Existing deep unfolding networks (DUNs) retain fidelity to the imaging model, but regression-trained denoisers can suppress spatial detail and smooth spectral structure under strong modulation. This paper proposes \emph{FMU}, a deep unfolding framework that couples a measurement-conditioned flow-matching prior with a sensing-model-guided measurement update. A two-phase scheme first learns a compact clean-HSI latent target and then trains a conditional velocity field to generate this target from Gaussian noise. A mean-velocity regularizer additionally penalizes the first-moment error of the learned field. On the KAIST 10-scene benchmark under the optical-filter setting, FMU obtains 42.13,dB PSNR and 0.9900 SSIM, outperforming LADE-DUN by 1.16,dB in PSNR under the same training data, sensing mask, and evaluation protocol. Under the same optical-filter operator, FMU is further evaluated on held-out KAIST and ICVL scenes without fine-tuning. We also report quantitative simulated-CASSI results and qualitative reconstructions of real CASSI measurements.

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.

We seek to recover an unknown signal from nonlinear amplitude-only measurements, a challenging inverse problem. Strong theoretical guarantees have been established for idealized random measurements, defining the sampling ratio required for signal recovery. However, these results neglect signal priors, which can fundamentally shift these limits, potentially enabling reconstruction with far fewer measurements and simpler models. We evaluate a variety of image priors in the context of severe undersampling with physically-grounded random measurement models. Our results show that these priors enable accurate recovery well below the weak recovery limit, the theoretical threshold required for recovery better than a random guess.

We present a framework for fine-tuning flow-matching generative models to enforce physical constraints and solve inverse problems in scientific systems. Starting from a model trained on low-fidelity or observational data, we apply a differentiable post-training procedure that minimizes weak-form residuals of governing partial differential equations (PDEs), promoting physical consistency and adherence to boundary conditions without distorting the underlying learned distribution. To infer unknown physical inputs, such as source terms, material parameters, or boundary data, we augment the generative process with a learnable latent parameter predictor and propose a joint optimization strategy. The resulting model produces physically valid field solutions alongside plausible estimates of hidden parameters, effectively addressing ill-posed inverse problems in a data-driven yet physicsaware manner. We validate our method on canonical PDE benchmarks, demonstrating improved satisfaction of PDE constraints and accurate recovery of latent coefficients. Our approach bridges generative modelling and scientific inference, opening new avenues for simulation-augmented discovery and data-efficient modelling of physical systems.

Maximum A Posteriori (MAP) estimation is a cornerstone framework for blind inverse problems, where an image and a forward operator are jointly estimated as the maximizers of a posterior distribution. In applications such as blind deblurring, this principle is used to recover sharp images from degraded observations. In this paper, we analyze the recovery guarantees of MAP-based methods by adopting a \emph{Morse--Bott framework}. We model the image potential as a Morse--Bott function, where natural images are modeled as residing locally on a critical submanifold. This means that while the potential is locally flat along the natural'' directions of the image manifold, it is strictly convex in the directions normal to it. We demonstrate that this Morse--Bott hypothesis aligns with the structural properties of state-of-the-art learned priors, a finding we validate through an experimental analysis of the potential landscape and its Hessian spectrum. Our theoretical results show that, in a neighborhood of the ground-truth image and operator, the posterior admits local minimizers that are stable both with respect to initialization (gradient descents converge to the same minimizer) and to small perturbations of the data (solutions vary smoothly with the observations). This local stability potentially provides a theoretical justification for the empirical success of well designed gradient-based optimization in these settings. However, we also demonstrate that this local stability is a \textbf{local} property: the blurry trap'', well-known for sparse priors in blind deconvolution, persists even with state-of-the-art learned priors. Our findings demonstrate that the failure of MAP in blind deconvolution is not a limitation of prior quality, but an intrinsic characteristic of the landscape. We conclude that successful recovery depends on strategic initialization around favorable local minima.

Protein generative models have shown remarkable promise in protein design, yet their success rates remain constrained by reliance on curated sequence-structure datasets and by misalignment between supervised objectives and real design goals. We present ProteinZero, an online reinforcement learning framework for inverse folding models that enables scalable, automated, and continuous self-improvement with computationally efficient feedback. ProteinZero employs a reward pipeline that combines structural guidance from ESMFold with a novel self-derived ddG predictor, providing stable multi-objective signals while avoiding the prohibitive cost of physics-based methods. To ensure robustness in online RL, we further introduce a novel embedding-level diversity regularizer that mitigates mode collapse and promotes sequence-level diversity among generated designs. Within a general RL formulation balancing multi-reward optimization, KL-divergence from a reference model, and diversity regularization, ProteinZero achieves robust improvements across designability, predicted stability, recovery, and diversity. On the CATH-4.3 benchmark, it consistently outperforms state-of-the-art baselines including ProteinMPNN, ESM-IF, and InstructPLM, reducing design failure rates by 36-48% and achieving success rates above 90% across diverse folds. Importantly, a complete RL run can be executed on a single 8xGPU node within three days, including reward computation and data generation. These results indicate that efficient online RL fine-tuning can complement supervised pretraining by allowing protein generative models to evolve continuously from their own outputs and optimize multiple design objectives without labeled data, opening new possibilities for exploring the vast protein design space. Code and model checkpoints are available at https://github.com/ziwenwang28/ProteinZero.

Understanding the internal representations of large language models (LLMs) is a central challenge in interpretability research. Existing feature interpretability methods often rely on strong structural assumptions--such as linearity or sparsity--that may not hold in practice. In this work, we introduce InverseScope, an assumption-light and scalable framework for interpreting neural activations via input inversion. Given a target activation, InverseScope characterizes its encoded information by generating natural-language inputs that produce nearby activations, grounding abstract internal states in concrete language. To overcome the prohibitive cost of sampling in high-dimensional activation spaces, we propose a novel control-layer conditioning architecture that substantially improves sample efficiency compared to prior token-prepending approaches. We demonstrate that InverseScope reveals rich geometric structure in LLM representation spaces, including sentence-level linear analogies. The framework scales to state-of-the-art open-source models of up to 14B parameters and generalizes to out-of-distribution inputs, enabling systematic analysis of activation neighborhoods.