Inverse Problem

Flow matching approaches to imaging inverse problems commonly incorporate measurements in two ways. Conditioning-based approaches supply measurement-derived information as a network input, often through concatenation, while inference-guided approaches combine an unconditional velocity field with a separate data-consistency update. In these common formulations, the forward model is not explicitly enforced within the learned conditional velocity field. We propose a principled parametrization of the measurement-conditional velocity field to solve inverse problems. Under linear interpolation, we express the conditional velocity $v(x_t,t,y)$ in terms of the posterior mean $E[x_1 | x_t,y]$, and characterize that mean as the unique minimizer of a variational objective whose data-consistency term is explicit. We further prove that the velocity field defines a probability flow from the source distribution to the measurement-conditioned posterior. Splitting the variational objective yields a conditional velocity parameterization with operator-dependent data-consistency updates, which we train end-to-end under the flow-matching objective, with no additional guidance at inference. Our method achieves state-of-the-art PSNR with $50\times$ fewer function evaluations than the strongest flow baseline. Varying the sampling steps provides test-time control over the distortion-perception trade-off without retraining.

This paper presents a novel approach for data-driven self-learning control of highly flexible, modular manufacturing systems. Specifically, we employ a novel framework for model-based reinforcement learning which introduces approximate inverse process models within the training of reinforcement policies. This approach disentangles the learning of actuation dynamics and the dynamics in state space, resulting in RL-based training solely within the task space. We propose a lightweight feedforward architecture for approximate inverse models and integrate them within the policy network of standard RL algorithms. We apply the approach to a laboratory modular production testbed with heterogeneous production modules. The results underline the efficiency improvements for modular manufacturing units in terms of both performance and training speed, particularly for off-policy algorithms.

Data is important in many deep learning-based inverse problem solvers. However, obtaining sufficient paired data in many scenarios remains highly challenging, while unpaired data is cheap. To maximize data utilization, this paper proposes LUD-DIF, a diffusion-based approach for solving inverse problems with unpaired data. Starting from the evidence lower bound (ELBO) of the joint distribution, we decouple it into two independent diffusion processes under the weak-coupling assumption. The method provides theoretical support from a variational inference perspective, derives the loss function, quantitatively analyzes the error bound introduced by the assumption, and offers a theorem-motivated heuristic for hyperparameter selection. Experimental results demonstrate that LUD-DIF achieves outstanding performance on multiple image inverse problems, validating its effectiveness and generalization capability in unpaired inverse problem settings.

Neural operators provide fast surrogates for partial differential equation (PDE) solvers, but their reliability can degrade for high-dimensional spatial inputs and inverse or repeated inference. State-only training constrains solution values but not the learned input--output response. We study sensitivity-constrained neural operators (SC-NOs), which augment standard training with sampled solver-derived Jacobian supervision. Selected sensitivities from differentiable solvers or discrete adjoints are matched during training, allowing response information to be amortized across minibatches without imposing the full Jacobian at every update. We evaluate SC-NO on advection--diffusion and RANS--Spalart--Allmaras benchmarks, input-dimensionality scaling tests, long-horizon autoregressive rollout, and a shallow-water Tohoku tsunami source-inversion case. Sensitivity supervision improves forward prediction and yields larger gains in gradient-based inverse reconstruction of distributed fields. Scaling experiments show an improved accuracy--cost tradeoff for high-dimensional gridded inputs, while ablations indicate that state values and Jacobian information provide complementary supervision. In the tsunami case, SC-FNO reconstructs gridded seafloor deformation from sparse early gauge observations and forecasts subsequent wave propagation in a near-real-time proof-of-concept workflow. These results support sampled sensitivity supervision as a practical way to improve neural PDE surrogates when forward accuracy, inverse stability, robustness, and computational cost must be considered together.

Modern recommender systems treat observed actions as reliable proxies for user preferences, yet interactions often reflect exploration or comparison rather than stable preference expression. As interfaces evolve from static layouts toward generative UIs and immersive extended reality (XR), the need for deeper, modality-agnostic user understanding grows: these adaptive environments must decide not only what to present but where, when, how prominently, and most importantly why a user acts. We propose an Inverse Theory of Mind (IToM) pipeline that reasons backward from observed interactions to infer the beliefs, preferences, and decision-making traits that explain behavior. The pipeline reconstructs each user's decision context, including what was chosen and what alternatives were available, applies LLM-driven counterfactual reasoning to produce evidence-grounded natural-language belief statements, and synthesizes these beliefs through multi-hypothesis abductive inference into a structured user persona. We evaluate on the OPeRA dataset against ground-truth personality assessments, attitudinal surveys, and interview-based personas across four tasks: next action prediction, shopping attitude alignment, Big Five personality inference, and held-out category prediction. Results show that inferred personas match or exceed ground-truth personas and that multi-hypothesis reasoning is essential for accurate personality prediction. We further demonstrate cross-modal transferability with a persona-driven spatial banking application on VisionOS.

Inverse decision modeling infers latent properties of decision processes from observed behavior, but existing formulations rely primarily on action trajectories. In verbalized cognitive tasks, task execution also produces response dynamics that action-only formulations leave unmodeled, such as verbal production, interaction, and hesitation. We propose a factorized inverse decision model (FIDM) that decomposes each individual's task-execution likelihood into an action factor and an effort factor, governed by separate individual-specific parameters. From raw verbal transcripts, a language model produces structured task-execution traces for factorized inference. On data from 400 older adults performing a grocery-shopping dialog task for cognitive screening, controlled recovery shows selective estimation of the intended factors, while matched semi-synthetic conditions show that FIDM preserves action-execution distinctions even when aggregate behavioral summaries are matched. Action evidence further localizes task-defined deviations across participants. In cognitive-status classification, FIDM provides information complementary to clinical scores, trajectory summaries, and frozen language representations, with consistent gains across all evaluated baselines in the binary setting.

We study what can be recovered about the transition probabilities of a Markov decision process from optimal actions alone. This is closely related to the inverse problem considered by Letcher et al., who ask when the dynamics can be recovered from numerical $Q$-values. Here the numerical values themselves are not observed; only the optimal actions are known, for every reward in a given class. For state-action rewards $r(s,a)$, knowing the optimal actions for every reward also tells us how much better one action is than another when each is followed by the same fixed policy. This is still not enough to determine the transition probabilities uniquely. We prove that two kernels give the same optimal actions for every reward exactly when $$Q_{s,a} = \Bigl(P_{s,a}+\tfrac1γe_s^{\mathsf T}(L-I)\Bigr)L^{-1}$$ for one invertible matrix $L$ satisfying $L\mathbf 1=\mathbf 1$. Near a kernel with strictly positive entries, there is an $n(n-1)$-dimensional family of different kernels with this property. The result is unchanged if we consider only rewards having a unique optimal action at every state. We then compare this with rewards of the forms $r(s)$ and $r(s,a,s')$. Rewards that depend on the next state can usually recover the transition kernel itself: every row at a state with at least two actions is determined, and we describe exactly when a row at a state with one action can remain hidden. State rewards reveal less: two kernels give the same optimal actions exactly when every deterministic policy is optimal for the same set of rewards. The results show how the form of the reward affects what can be learned about the dynamics from optimal actions alone.

Interpretability methods for large language models (LLMs) describe internal state but do not directly test whether that state is causally sufficient to produce the observed behavior. In earlier work, we lesioned LLMs to produce error profiles in picture naming, a central task for assessing aphasia, and found that specific lesions produced errors resembling those of individual stroke survivors. Here we ask the inverse question: given an error profile, can the lesion parameters that produced it be recovered, and what does this inverse problem reveal about transformer computation? Lesions in LLaVA-Vicuna 13B were parameterized by layer index, modification percentage, and noise sigma across 4,840 configurations, and error profiles were characterized by a seven-category clinical taxonomy (correct, semantic, unrelated, formal, mixed, neologism, no-response). We trained a multi-task neural network to map error profiles back to perturbation parameters. The problem admitted a partial solution: across 10 independently trained inverse models, modification percentage and noise sigma were recoverable, whereas layer index was recoverable only within a neighborhood. In counterfactual validation, a fresh model instance perturbed with the recovered parameters reproduced the target behavior in 81.4% of cases. This dissociation between low layer recovery and high counterfactual fidelity is consistent with functional redundancy across transformer layers, a property not captured by standard interpretability methods. As an out-of-distribution test, we applied the trained model to picture-naming error profiles from 278 stroke survivors; recovered parameters were syndrome-discriminative, most strongly for perturbation intensity, indicating generalization beyond the training distribution. Counterfactual validation provides a general framework for LLM interpretability claims beyond inverse mapping.

Latent Flow Models have revolutionized compressed-space image synthesis, yet their application to high-fidelity inverse problems remains bottlenecked. In this paper, we trace this dilemma to a fundamental geometric limitation of pre-trained autoencoders, which we term \emph{First-Order Manifold Blindness}. Severe decoder compression (e.g., retaining only $\sim\!2\%$ of the original degrees of freedom) produces a rank-deficient Jacobian, rendering high-frequency measurement residuals in its orthogonal complement invisible to latent gradients even when the decoder can represent the target image. To overcome this bottleneck, we propose Hybrid-Domain Posterior Sampling (HDPS), a decoupled inference framework that disentangles physical measurement consistency from semantic prior modeling. HDPS diverges into the pixel space, leveraging Langevin dynamics to absorb precise orthogonal measurement gradients, and subsequently projects these structural corrections back onto the generative manifold. An optimization-based latent alignment is introduced to filter pixel-space artifacts while avoiding the semantic drift of direct encoding. Extensive experiments on diverse inverse problems demonstrate that HDPS establishes a new state-of-the-art, successfully recovering the high-frequency structural precision that latent-only solvers inherently discard. The code is available at \href{https://github.com/74587887/HDPS}{https://github.com/74587887/HDPS}.

Inversion-based watermarking is a promising approach to authenticate diffusion-generated images, yet practical use is bottlenecked by inversion that is both slow and error-prone. While the primary challenge in the watermarking setting is robustness against external distortions, existing approaches over-optimize internal truncation error, and because that error scales with the sampler step size, they are inherently confined to high-NFE (number of function evaluations) regimes that cannot meet the dual demands of speed and robustness. In this work, we have two key observations: (i) the inversion trajectory has markedly lower curvature than the forward generation path does, making it highly compressible and amenable to low-NFE approximation; and (ii) in inversion for watermark verification, the trade-off between speed and truncation error is less critical, since external distortions dominate the error. A faster inverter provides a dual benefit: it is not only more efficient, but it also enables end-to-end adversarial training to directly target robustness, a task that is computationally prohibitive for the original, lengthy inversion trajectories. Building on this, we propose \textbf{FARI} (\textbf{F}ast \textbf{A}symmetric \textbf{R}obust \textbf{I}nversion), a one-step inversion framework paired with lightweight adversarial LoRA fine-tuning of the denoiser for watermark extraction. While consolidation slightly increases internal error, FARI delivers large gains in both speed and robustness: with approximately 20 minutes of fine-tuning on a single NVIDIA RTX A6000 GPU, it surpasses 50-step DDIM inversion on watermark-verification robustness while dramatically reducing inference time. Code and pretrained models are available at https://github.com/0xD009/FARI.

Simulation-based inference (SBI) with machine learning is an increasingly important tool for solving inverse problems in science and engineering, including parameter inference and the inversion of detector effects. We provide an overview of the Bayesian and frequentist statistical frameworks, describe how machine-learning-based SBI methods, such as neural posterior estimation and neural likelihood estimation, can be used for parameter estimation within these frameworks, and show that the same methods can also be applied to Empirical Bayes or unfolding tasks. We also discuss how to validate inference results and the limitations of SBI with machine learning.

Diffusion-based methods have achieved remarkable empirical success in solving inverse problems. However, many existing posterior samplers either lack rigorous theoretical guarantees or incur substantial computational overhead. We propose a simple and efficient algorithm, called \pddim, for solving linear inverse problems with diffusion priors via a DDIM-type sampler. Our method requires only lightweight, coordinate-wise modifications to the standard DDIM update, while explicitly incorporating the measurement model. The key idea is to perform posterior sampling separately along each singular direction of the measurement operator: for each direction, the sampler follows the learned diffusion prior when the observation signal-to-noise ratio (SNR) is below the corresponding diffusion SNR, and switches to a calibrated measurement-based predictor otherwise. We prove that the proposed sampler converges to the Bayesian posterior conditioned on the measurements. Empirical results show that the proposed sampler performs favorably against existing diffusion-based posterior samplers across a range of image restoration tasks, achieving the best performance on the majority of evaluation metrics considered. Overall, our results convert posterior sampling for noisy linear inverse problems to simple coordinate-wise DDIM updates, yielding an efficient, easy-to-implement algorithm with provable posterior consistency.

We develop a convergent scheme to train neural networks involving analytic activation functions based on gradient flows. Convergence properties are guaranteed by Lojasiewicz theory. The main advantage of this approach is its simplicity of implementation. The coefficients of the network are approximated by solving a system of ordinary differential equations. We test the method by constructing residual neural network approximations of solutions of parametric problems. The dependence of the solutions of simple ordinary differential equations on a few parameters is correctly reproduced. The solutions of inverse problems involving wave constraints which depend on a few parameters can be reasonably approximated, even in regions in which the problem is severely ill posed.

This work studies the inverse problem of recovering the relative magnitudes of the tension, bending, and bearing loads acting on a crack from its stress-intensity-factor profile along the crack front, using the public SIFBench finite-element data. The central claim is not forensic load recovery on field cases, but a rigorous characterization of when the combined load is identifiable at all, together with an estimator that returns calibrated uncertainty precisely in the regimes where it is not. For a known geometry the forward map from loads to profile is exactly linear, and identifiability reduces to a single geometric question: whether the three elementary load profiles are linearly independent as functions along the front. When they are nearly dependent, many different load combinations produce almost the same profile and the inverse problem is illposed; the analysis shows that the degree of ill-posedness is controlled by an intrinsic stability margin, not by the conditioning number alone. A single crack-front operator serves both as a structured forward surrogate and as the differentiable map required by a simplex-constrained, set-valued inverse estimator. On the SIFBench corner-crack scenario the empirical behaviour matches the theory: the typical geometry is well posed while a sizable minority is genuinely ill-posed, so a point estimate is reliable on the majority and provably uninformative on the rest. Validation is on controlled synthetic noise; no real fracture cases are used or claimed.

Physics-informed neural networks (PINNs) often face ill-posed optimization, competing losses, and parameter compensation in partial differential equation (PDE) inverse problems. Transfer learning can reuse source-task representations, but direct fine-tuning may induce negative transfer when source and target physics differ, leading to low field error but inaccurate parameter recovery. To address this issue, we propose Target-Guided Selective Reweighting PINN (TGSR-PINN), a target-evidence-driven representation correction method for PINN inverse transfer learning. TGSR-PINN transfers source network weights and biases but initializes target physical parameters independently. After short target adaptation, it scores neurons using first-order Taylor sensitivity and pre-activation variance on fixed batches. These scores are converted into continuous weak-adaptation signals using a Gaussian mixture model with rank fallback. TGSR-PINN then applies bounded selective soft decay to the corresponding input weight rows and biases without pruning or resetting them. Experiments on a zero-source high-Péclet inflow--outflow problem with nonzero Dirichlet data and an outflow boundary layer, Allen--Cahn to Burgers cross-PDE transfer, and 5%-noise reaction--diffusion inverse problems show that TGSR-PINN improves parameter recovery while maintaining low field error. Ablation studies indicate that neuron target scoring, weak-adaptation estimation, layer protection, and selective soft decay jointly contribute to the observed benefits.

This paper presents a machine learning framework for data-driven inverse design of V-beam thermal sensors. The goal is to determine the optimal sensor geometry: beam inclination angle, beam length and beam width that achieves a target displacement under a given temperature. The design should also provide the geometry with minimum structure volume and minimum mechanical stress the sensor must support. This problem is ill-posed as for a given displacement there are multiple possible geometric configurations, causing direct regression methods to fail. We document a series of five exploratory trials that progressively revealed the nature of the problem culminating in a two-phase solution: a neural network forward model trained to map geometry and material constants to sensor responses, a gradient-descent inverse optimization over the frozen forward model, minimizing stress and volume simultaneously. The proposed pipeline utilizes a 3000-sample dataset and achieves a MAPE of 4.76% for predicting the displacement, more than 70% of predictions having MAPE of under 5%.

Meta-learning without labeled data is crucial for real-world applications, where obtaining labeled datasets can be expensive or restricted due to privacy concerns. Data-Free Meta-Learning (DFML) addresses this challenge by leveraging pre-trained models without access to training data. However, existing DFML methods rely on model inversion to generate training data, a process that is generally difficult and computationally expensive due to the need to generate high-dimensional data matching the original distribution. To address this limitation, we propose a novel meta-learning setting that avoids model inversion by jointly leveraging pre-trained models and unlabeled data. Our method generates meta-training tasks by assigning soft labels from pre-trained models to unlabeled data. Since the quality of these tasks can vary, we introduce a task-weighting mechanism based on task confidence and class distribution balance to ensure effective meta-learning. Extensive experiments demonstrate that our approach substantially reduces computational cost and improves generalization, achieving up to 104-fold speedup and 8.4 percent to 36.4 percent improvements in few-shot classification accuracy compared to state-of-the-art DFML methods.

Operator learning for partial differential equations (PDEs) on arbitrary geometries builds fast neural surrogates for large-scale simulation. Although recent geometry-adaptive neural operators have made substantial progress, they are mainly designed for forward problems in which inputs and outputs share the same spatial domain. This limits their applicability for boundary value problems (BVPs) and inverse problems, where inputs and outputs may live on different domains. We introduce the Geometry-Adaptive Integral Autoencoder (GAIA), an operator learning model that encodes the domain boundary and the interior field distribution into geometry tokens, and conditions integral transform layers on these tokens via cross-attention, allowing the kernel to adapt locally to geometric features. This yields a single architecture for forward (including BVPs) and inverse problems on arbitrary domains in one pass, without retraining, iterative optimization, or graph construction. We evaluate GAIA on seven 2D and 3D benchmarks, four of which are new or substantially extended benchmarks for inverse problems and BVP: electrical impedance tomography, optical tomography, 3D Darcy flow on varying geometries, and a modified setting of Poisson BVP on mechanical components benchmark (MCB). GAIA sets new state-of-the-art results on every inverse and BVP task, reducing median relative $L^2$ error by 64% on airfoil flow reconstruction and 27% on EIT relative to the next best amortized method, and outperforming all baselines on every shape category of MCB. On other forward problems, GAIA is competitive with specialized solvers while maintaining stable accuracy across point resolutions on which transformer-based baselines degrade.

This work investigates uncertainty-aware deep learning approaches for direction of arrival (DOA) estimation in automotive radar, focusing on probabilistic modeling and downstream integration. A circular-statistics-based von Mises (VM) ensemble (ENS) is compared with an evidential deep learning (EDL) framework based on a normal inverse gamma formulation, yielding a Student t predictive distribution in the Euclidean domain. The ENS framework produces angular predictions parameterized by (mu, kappa), enabling interpretable uncertainty aligned with directional geometry. Performance is evaluated under in distribution and multiple out-of-distribution conditions using risk coverage and ROC or AUROC analyses. Results indicate that ENS achieves lower uncertainty under nominal conditions and exhibits stronger sensitivity to severe perturbations, whereas EDL provides smoother uncertainty variation and slightly improved ranking consistency. Importantly, the ENS representation enables direct probabilistic integration into association modules via closed form VM likelihoods, facilitating a unified detection tracking pipeline. These findings highlight a trade-off between geometric consistency and statistical generality in uncertainty-aware DOA estimation.

Learned reconstruction operators for inverse problems are typically trained under a fixed noise model, and generalize poorly when the distribution during testing differs from the one assumed during training. Distributionally robust optimization (DRO) addresses this by optimizing against the worst-case distribution within a prescribed ambiguity set, but standard Wasserstein DRO perturbs the full joint distribution uniformly, which can be overly conservative and ignores the physics of the measurement process. We develop a structured DRO framework in which the ambiguity set is restricted to structured perturbations aligned with the data-acquisition process. This allows us to learn data-driven reconstruction operators that remain robust to distributional shifts. By constraining perturbations to subsets such as $P(Y|X)$, our framework models uncertainty in the forward operator and noise model more faithfully, accommodating any noise model expressible as a stochastic forward operator. We establish strong duality for this general formulation and derive explicit finite-dimensional dual representations for perturbations in the joint, marginal, and conditional distributions. A central result is an explicit worst-case risk bound that induces Tikhonov regularization on the Lipschitz constant of the reconstruction operator, and is less conservative relative to standard DRO for well-posed problems. Numerical experiments on deblurring and sinogram-to-CT reconstruction demonstrate improved robustness, stability, and interpretability over standard DRO and MSE baselines. In the linear setting, the learned operator becomes effectively low-rank, truncating at the intrinsic dimension of the data and recovering a data-driven analogue of truncated-SVD regularization.

Modeling chaotic systems is crucial yet challenging. Inverse problems in chaotic dynamics, namely inferring initial conditions from final states, remain largely unsolved because of ill-posedness, non-uniqueness, instability, and potentially chaotic time-reverse dynamics. We address this open problem with Bidirectional Conditional Flow Matching (Bi-CFM), which learns bidirectional mappings between distributions of initial and final states to capture the stochasticity of chaotic evolution and mitigate exponential error accumulation over time. Furthermore, for systems with conservation laws, we extend it to Conservation-constrained Bi-CFM (CBi-CFM). Across the classic Lorenz, Circuit, and high-dimensional Lorenz 96 systems, Bi-CFM improves five distribution-level metrics over baselines while achieving a speedup of more than two orders of magnitude. In the three-body planet-planet scattering problem in planetary dynamics, CBi-CFM better respects conservation laws, with conservation errors comparable to those of the ground truth. Finally, on real observations of globular clusters, collisional million-body systems shaped by $\sim 10^{10}$ years (10 Gyr) of evolution, our method represents an advance in accuracy, establishing a scalable route to solving inverse problems of long-timescale real-world chaotic dynamics.

Principled regression for stochastic processes is a long-standing challenge with deep connections to scientific inverse problems. We introduce Flow Annealing Posterior Sampling (FAPS), to our knowledge the first function-space posterior sampling framework that unifies stochastic-process regression and PDE inverse problems. Built on pretrained function-space flow-matching priors, FAPS enables likelihood-guided posterior inference from sparse and noisy observations, supports variable query discretizations, and avoids explicit prior-density evaluation. Its Langevin correction uses a low-rank covariance preconditioner to exploit dominant function-space correlations across discretizations. Across Gaussian and non-Gaussian stochastic-process regression benchmarks and diverse PDE inverse problems, FAPS produces coherent posterior samples with accurate uncertainty quantification, significantly outperforming existing functional regression baselines and achieving competitive or better PDE noisy inverse performance than diffusion-based posterior samplers while reducing test-time sampling cost.

Inverse problems for differential equations arise throughout science and engineering, where one seeks to infer unknown model parameters from noisy or incomplete observations. Traditional numerical methods for these problems are often computationally expensive, particularly in Bayesian settings where evaluating the likelihood becomes costly for complex forward models and high-dimensional parameter spaces. To address this challenge, we introduce DeepGaLA, a neural-network surrogate for differential equation solvers that provides uncertainty-aware predictions, reducing overconfident inference when training data are limited. To evaluate the fidelity of the surrogate-induced posterior approximations in practice, we show that a short run of delayed-acceptance Markov chain Monte Carlo can serve as an effective diagnostic. Across a range of numerical experiments, DeepGaLA delivers forward-model approximations with accuracy comparable to established Gaussian-process surrogates, while better maintaining efficiency as parameter dimension grows. Moreover, it can incorporate differential-equation constraints, including in nonlinear settings. Overall, these results indicate that uncertainty-quantified neural surrogates can enable scalable and reliable Bayesian inference for inverse problems in complex systems.

Image restoration faces a fundamental tradeoff: methods that minimize error produce blurry reconstructions, while those that maximize perceptual quality yield sharp but less faithful images. Existing approaches either commit to a single operating point on this distortion perception (DP) frontier or require paired-data supervision, auxiliary models, or hyperparameter tuning of the sampler to access different points. We show that flow map models, a recent extension of flow matching for few-step sampling that learns an average field, implicitly define a one-parameter family of denoisers that continuously spans the DP frontier. The lookahead parameter t acts as a control knob between the MMSE and perceptual regimes. For Gaussian targets, we prove that varying t exactly recovers the optimal DP frontier; for natural images, we observe similar behavior empirically. Within a Plug-and-Play solver, the same mechanism extends to general inverse problems, where it controls a tradeoff between perceptual alignment and data consistency. Despite the lack of exact optimality guarantees in this setting, a single trained flow map spans the DP tradeoff, matching or exceeding specialized baselines at both extremes. Extensive experiments on CelebA ($128\times 128$) and AFHQ ($256\times 256$) across several linear and nonlinear inverse tasks validate our findings.

Realistic visual simulation of food manipulation requires accurate material parameters, yet these are difficult to measure directly and vary across the heterogeneous regions of a single food item. We address the inverse problem of estimating material parameters from a target description of fracture behavior in a non-differentiable continuum damage mechanics simulator. Using orange peeling as a test case, we train a neural surrogate on 2,000 forward simulations and compare Covariance Matrix Adaptation Evolution Strategy (CMA-ES, a gradient-free evolutionary optimizer) with Proximal Policy Optimization (PPO, a reinforcement learning algorithm) across the original 9-dimensional parameter space and two learned 4-dimensional latent representations. Since different oranges have different material properties, a practical inverse system must handle arbitrary targets without retraining. We train a goal-conditioned PPO policy that learns a general inverse mapping: given any target description of peeling behavior, the policy produces a material parameter estimate in a single forward pass (8 surrogate evaluations, approximately 10ms). Operating in a normalizing flow latent space with a shared surrogate evaluator, the goal-conditioned policy achieves 0.642 actual recovery when validated through the simulator, outperforming the original parameter space by 23%. A warm-start extension that initializes CMA-ES refinement from the policy's output further improves recovery to 0.828 with 540 evaluations. These findings provide a practical framework for inverse food physics and lay groundwork for vision-driven material identification from video observations of food manipulation.

Driven by their remarkable success in computer vision and inverse problem solving, score-based models are increasingly applied to wireless communications, where they show promise across a range of physical-layer tasks. However, despite this growing interest, the current literature often lacks a rigorous analysis of when score-matching offers a tangible advantage over traditional discriminative learning. This paper aims to address this gap through the use-case of channel estimation, a fundamental inverse problem in wireless systems. We present a theoretically grounded interpretation of score-based channel estimation through the lens of the perception-distortion tradeoff, identifying the conditions where score matching excels as well as its key limitations. In particular, by modeling downstream wireless tasks (e.g., capacity maximization) as functionals of the channel estimation process, we quantify the excess risk incurred by standard distortion-minimization approaches. Extensive numerical results show that under high predictive uncertainty, the large excess risk gap can be offset by score-based estimation, enabling near Bayesian-optimal precoding via the learned posterior, whereas in the low predictive uncertainty regime, discriminative distortion-minimization approaches are preferable due to lower complexity and more efficient use of model capacity.

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities is a fundamental challenge in machine learning, particularly when the exact gradient of the potential is unavailable and must be approximated via stochastic gradients that exhibit high variance under a fixed budget of gradient computations per iteration. Although variance reduction techniques such as SGD with momentum, STORM, and PAGE have demonstrated improved convergence properties in non-convex optimization, their implications for sampling from non-log-concave distributions remain largely unexplored. In this work, we develop the first unified analysis of these estimators for sampling from non-log-concave distributions. We establish improved non-asymptotic convergence rates in $\varepsilon$-relative Fisher information and, under a Poincaré inequality assumption, in squared total variation distance, and further prove weak convergence to the target distribution. We extend our analysis to solving inverse problems with score-based generative priors. We empirically validate our theory and demonstrate that, under a fixed gradient computations per iteration, variance-reduction techniques consistently improve sample quality in two standard imaging applications.

Large language models (LLMs) are increasingly deployed in privacy-sensitive domains, where users must balance the risk of data exposure through external APIs against the high computational cost of local deployment. Split learning has therefore emerged as a promising paradigm for LLM fine-tuning and inference under limited local resources. However, it introduces new privacy risks. Prior work primarily studies leakage of private input prompts, typically via inversion attacks on intermediate representations, while the potential for sensitive information leakage through generative response outputs remains largely unexplored. In this work, we unveil novel vulnerabilities of Split-LLM by presenting Patched Model Inversion with Dual-Sided Initialization (PIDI), a two-stage attack that simultaneously targets both private input prompts and output responses in Split-LLM settings. It combines dual-sided initialization with a patched inversion strategy to tackle long sequences, substantially outperforming prior inversion methods. To counter threats from both sides, we further propose the Adapter-based DualGuard with Mutual Information Defense (ADMI), which integrates an adapter-based local warmup strategy and mutual information regularization to provide a strong empirical privacy protection with minimal impact on task performance. Extensive experiments across diverse tasks and models demonstrate that ADMI effectively defends against PIDI and other state-of-the-art inversion attacks. Our code is publicly available at https://github.com/FLAIR-THU/VFLAIR-LLM.

We present Full-Self Diagnostics (FSD), a unified mathematical framework for recovering latent physiological states from unconstrained 9-second facial videos captured by consumer smartphones. The approach integrates five mutually reinforcing components: (1) a physics-based forward model derived from the radiative transfer equation and chromophore absorption that maps camera observables to biomarker concentrations; (2) an information-theoretic observability theory proving that multi-channel visual signals (spectral, pulse, respiratory, micro-expression, and oculomotor) contain strictly increasing mutual information with physiological state; (3) a stable, Tikhonov-regularized inverse problem with domain-uniform identifiability guarantees; (4) an operator-learning formulation that enables generalization across devices, resolutions, and populations; and (5) a supervised learning procedure, interpretable as stochastic variational inference, that continuously refines the model from paired biosensor ground truth with performance improving proportionally to one over the square root of the number of paired observations. Empirical validation on 38812 real-world paired scans across 59 subjects demonstrates practical performance. Self-collected data from the lead author (glucose range 35-550 mg/dL) yields MARD of 29.86 percent with 97.57 percent of predictions in Clarke Error Grid Zones A+B and only 0.27 percent in the dangerous Zone E. A well-managed diabetic participant achieves MARD of 17 percent in the narrower 70-180 mg/dL band. These results confirm that consumer-grade facial video encodes sufficient structured information for clinically relevant, non-invasive biomarker inference under fully unconstrained conditions, with performance scaling predictably as more paired data becomes available.

Physics-Informed Neural Networks (PINNs) are a machine learning method for solving forward and inverse Partial Differential Equations (PDEs). When applied to PDEs with Dirac delta functions in the forcing terms, boundary conditions, or initial conditions, PINNs require approximating them with smooth surrogate functions, a practice that can introduce significant modeling errors. In this work, we exploit the interpretation of PINNs as Residual Least Squares (RLS) methods and show that this perspective enables direct treatment of Dirac delta terms by integrating the weak-form equation. Among RLS formulations other than PINN, we focus on the Radial Basis Function (RBF) expansion (also known as a single-layer RBF Network). We show that while integrating out the Dirac delta in PINNs causes residuals to fail to converge to zero, RBF-RLS consistently provides good forward and inverse solutions to transport problems. We explain this finding using the Neural Tangent Kernel (NTK) theory. We test both approaches on linear PDEs that represent groundwater flow and transport in porous media and rivers. We solve inverse problems to fit synthetic data, noisy synthetic data, and real-world measurements.

Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-informed neural networks (PINNs) have emerged as a flexible alternative. Their relative performance remains difficult to assess because the two approaches are often compared under different formulations, parameterizations, optimizers, and regularization choices. We present a fair comparison of adjoint optimization and PINNs for PDE-constrained inverse problems. From a common abstract formulation, we instantiate both methods on identical domains, governing equations, observation models, and regularization terms, while matching the optimizer, unknown parameterization, and arithmetic precision wherever applicable. The benchmarks include unsteady Burgers, noisy Darcy permeability inversion, three-dimensional Allen--Cahn reaction identification, and unsteady Navier--Stokes viscosity identification. The results show that the representation of the unknown largely determines the preferred method: grid-based fields favor the discrete adjoint, whereas neural representations are native to PINNs and relevant for closure and constitutive modeling. For time-dependent problems, adjoint inversion can be dominated by trajectory storage and differentiation, while PINNs provide satisfactory reconstructions at lower cost. A PINN-warm-started adjoint strategy then recovers adjoint-level accuracy at substantially reduced cost.

Scientists have historically relied on mathematical models based on differential equations to relate system inputs -- forces, fluxes, or heat sources -- to outputs, such as displacement, velocity, concentration, and temperature. These models rely on deep domain knowledge to determine the form of the governing differential equation, which is then calibrated with data by solving an inverse problem. In recent years, the field of Scientific Machine Learning has introduced a variety of alternative modeling strategies for physical systems. A method called Sparse Identification of Nonlinear Dynamics learns the governing equation as a sparse linear combination of terms in a user-defined library. Neural Ordinary Differential Equations construct the governing equation by taking in the state and its derivatives at the input layer of a neural network. Entirely foregoing the modeling framework of differential equations, neural operators directly learn a non-linear mapping between the system inputs and outputs. From inverse problems to neural operators, all of these modeling strategies can be conceptualized as data-driven machinery to predict a system's response over a range of inputs. It is then natural to wonder how exactly these various strategies relate to each other, and whether they can be neatly taxonomized. Drawing from the philosophical literature on scientific models, we argue that many model types have a common structure, differing only in the assumed model class of the input-output relation they define. Connecting to philosophical ideas on mechanism, and arguing that data from physical systems arises from solutions to parsimonious differential equations, we propose that only certain models are capable of mechanism discovery, and thus generalization. Our analysis is intended to unite apparently disparate modeling strategies and provide insight into their appropriate use cases.

Designing 3D metamaterial microstructures that meet the intended functions remains a major challenge, as it typically requires domain expertise, iterative simulations, and extensive manual tuning. Existing work on inverse design that automatically generates microstructures based on desired target properties often suffers from limited design diversity and faces challenges in ensuring the physical feasibility of the generated structures. To address this issue, a property-informed diffusion-based network is proposed that enables the generation of 3D microstructures directly from textual descriptions. Unlike traditional property conditioning methods, our approach leverages rich guidance in terms of semantics and physical properties in the text input to support diverse structure synthesis. To enforce consistency between the generated structures and the target textual prompts, a dual alignment strategy is adopted, including contrastive text-structure alignment and test-time reward-guided alignment. Experimental results show that the model is capable of generating semantically meaningful and physically plausible structures across a wide range of material categories. Our approach has good potential for interactive microstructure design and opens up new directions for combining language-based interfaces with inverse material discovery. Code is available at: https://github.com/hongsong-wang/PropDiff-TMG

Graph neural networks (GNNs) are widely deployed on relational data, yet they can leak sensitive or proprietary information about the training graph adjacency, e.g., social ties, transactions, and interactions. This work studies graph reconstruction attacks (GRA), a form of model inversion that reconstructs the training adjacency from a trained GNN, given different levels of attacker-side information. We first provide a systematic characterization of when and why adjacency becomes recoverable through features, labels, embeddings, and predictions, with leakage modulated by graph homophily, heterophily, and the model's inductive bias. Motivated by these findings, we view GNN inference through a Markov chain approximation lens, treating the layered forward computation as a chain of topology-dependent representations. Building on this view, we develop complementary attack and defense methods. On the attack side, we propose MC-GRA (+), which reconstructs the adjacency by optimizing a surrogate adjacency whose GNN-induced representations align with those of the target model at each layer. On the defense side, we propose MC-GPB (+), which suppresses adjacency-dependent information throughout the representation chain while aiming to preserve classification accuracy under a privacy-utility trade-off. Experiments across homophilic/heterophilic graph benchmarks and GNNs show that our attacks improve reconstruction fidelity over prior methods, while our defenses reduce reconstruction success with only minor accuracy loss.

Pretrained diffusion models demonstrate impressive potential in solving highly ill-posed 3D computed tomography (CT) inverse problems, while the inference process suffers from significant computational overhead. Furthermore, existing uniform timestep schedules fail to capture the non-uniform evolution of the reverse conditional diffusion stochastic differential equation, thereby introducing substantial truncation errors. To overcome this limitation, we propose Tracing the Oracle (TrO), a plug-and-play framework for improved timestep scheduling. Specifically, we treat densely sampled numerical integration trajectories on a few samples as the reference oracle. The optimized schedule is extracted by leveraging dynamic programming to globally minimize the cumulative error between the few-step approximation and the oracle. This mechanism precisely allocates the limited sampling steps to critical evolution stages that are highly susceptible to truncation errors. Our extensive experiments on the AAPM dataset across multiple 3D CT reconstruction tasks demonstrate that, when combined with the state-of-the-art 3D CT reconstruction method DDS, our optimized timesteps significantly improve reconstruction fidelity and computational efficiency compared to existing heuristic schedules, especially under a strict budget of no more than 10 sampling steps.

Generative models -- diffusion and flow matching -- are increasingly used to solve partial differential equation (PDE) inverse problems, enforcing the governing physics as a \emph{hard constraint} (via projection or guidance) and reporting the resulting samples as a Bayesian posterior with calibrated uncertainty. We show that this widely adopted recipe samples the wrong distribution. Conditioning a generative prior on a hard PDE constraint is conditioning on a measure-zero manifold -- an operation that is intrinsically ambiguous (the Borel--Kolmogorov paradox) and whose physically correct resolution, the small-residual-noise limit, carries a co-area (Fixman) Jacobian factor $[det(JJ^{\top})]^{-1/2}$ that projection- and guidance-based methods silently omit. We make the bias precise, show that it grows with the heterogeneity of the constraint sensitivity, and validate it on controlled problems against an \emph{i.i.d.} ground-truth arbiter. The omitted factor is not a second-order detail: removing it inflates the posterior error to $20\times$ the sampling-noise floor; minimal-displacement projection (as in PCFM) is biased at $9\times$ the floor; and a naive scalar reweighting does not fix it. We introduce \textbf{CoCoS}, a measure-aware constrained sampler that targets the correct co-area posterior, and show that it matches the gold-standard posterior to within sampling noise. Our results imply that satisfying the physics'' is not the same as sampling the posterior,'' and give a principled correction for uncertainty-aware scientific inference.

Diffusion-based inverse problem solvers can produce realistic reconstructions, but realism alone does not ensure that the recovered details are supported by the measurement. We study this failure as measurement-conditioned hallucination: visually meaningful content that is either implausible or inconsistent with the measured instance. Our analysis separates Bayes-rule-based diffusion inverse solvers into a prior update and a measurement-conditioning step, showing that hallucinated content can enter through the prior-side proposal before the measurement correction is applied. Motivated by this view, we propose Robust Prior Update (RPU), a solver-level module that probes the local stability of the diffusion prior update, re-anchors the resulting displacement at the current iterate, and leaves the measurement update unchanged. We instantiate RPU in DPS and evaluate it on FFHQ and ImageNet inverse problems using automatic metrics and human faithfulness studies. On FFHQ, RPU improves PSNR and LPIPS over DPS across box inpainting, Gaussian deblurring, and motion deblurring. In human judgments, RPU receives 91.9% of blind non-tie majority preferences and 91.1% of ground-truth-assisted non-tie preferences on FFHQ box inpainting, while the ImageNet Gaussian reader study is tie-heavy but favors RPU among non-tie cases. These results support a targeted claim: robustifying the prior update can improve instance faithfulness in diffusion inverse solvers, especially when the prior shapes weakly constrained content.

Generative models are increasingly used as priors for inverse problems, but their ability to produce realistic images creates a basic trust problem: a plausible reconstruction may be supported by the measurements, or it may be filled in by the prior along unobserved directions. This distinction is especially important in medical imaging, where acquisition operators are designed under scan-time, dose, and calibration constraints. We study generative inverse problems from a measurement-geometry perspective. The central question is whether a fixed measurement operator can distinguish nearby images that are plausible under the generative prior, and whether this relationship can guide better measurements. We introduce a local measurement-manifold compatibility measure that quantifies how well the operator observes prior-relevant tangent directions. Under local regularity assumptions, we prove that this quantity controls the stable part of the reconstruction error, while the generative prior controls off-manifold drift. This worst-direction certificate motivates practical fixed and sequential acquisition rules based on overall local volume preservation, including a posterior-cloud design that adapts measurements at test time without training a sampling policy. Across row-sampling, tomographic, and MR acquisition settings, the proposed scores predict failure modes, explain measurement-induced hallucinations, and guide better sampling. In fastMRI Cartesian sampling, posterior-cloud measurement design improves over strong non-learned ACS-preserving baselines, including variable-density and Poisson-like masks.

Recently, partial differential equations (PDEs) have been used to directly model the measurement process in signal processing, although their evaluation is costly. In this paper, we propose a novel alternating direction method of multipliers (ADMM)-based algorithm called physics-aware linearized ADMM (PA-LADMM) for inverse problems from PDE-based measurement processes. The key idea is the linearization of the subproblem with PDEs, leading to a cost-efficient update rule that calls only a PDE solver and its gradient evaluation per iteration. The algorithm has a theoretical convergence guarantee under certain conditions. In addition, we combine it with deep unfolding to unroll the PA-LADMM and train its internal parameters using supervised data. Two distinct experiments, compressed sensing with optical fiber communication and image restoration from noisy anisotropic diffusion, demonstrated the effectiveness of the proposed algorithms.

Diffusion models have shown promising performance as data-driven priors for computational imaging, as well as some capacity to detect out-of-distribution (OOD) images. However, existing approaches to OOD detection often require some knowledge of the shifted distribution, fail to detect subtle or localized distribution shifts, and operate on full images, rather than the indirect measurements available in inverse problems. We propose an OOD detection metric based on the Kullback-Leibler divergence between the diffusion prior and the posterior distribution, that (i) does not require any calibration data or knowledge of the shifted distribution, and (ii) can detect whole images as OOD as well as localize OOD patches within an image. Experimentally, we show that this metric can detect subtle yet semantically meaningful distribution shifts, such as the shift from healthy liver CT scans to those with tumors, and generalizes across different types of diffusion models, datasets, and inverse problems. Our code can be found at https://github.com/voilalab/KLIP.

Diffusion models have excellent capacity to model complex distributions of natural data, which has made them a popular and effective choice for posterior sampling in imaging inverse problems. Existing methods can incorporate any measurement model at inference time but must use an inexact approximation for the likelihood at intermediate timesteps for computational tractability. Although these approximations can often work well empirically, their downstream effect on the sampled posterior is poorly understood and can result in unexplained failures. To understand when, why, and how these likelihood approximations propagate to erroneous posterior distributions, we introduce a finite-sample perspective on posterior sampling that approximates the posterior to arbitrary precision as training set size tends towards infinity, for any forward model and prior distribution. Using this finite-sample lens, we observe that popular posterior sampling approximations tend to under- or over-estimate the spread of the posterior at intermediate timesteps, causing downstream consequences including sensitivity to early stopping time, inaccurate relative weighting of posterior modes, and hallucination, both of prior modes that are not in the posterior and likelihood modes that are not supported by the prior. Moreover, we find that the cause of these posterior errors requires neither a nonlinear measurement model nor a multimodal posterior, but can arise solely due to a multimodal prior and inaccurate posterior spread at intermediate sampling times. Our finite-sample posterior sampling approach is agnostic to the type of likelihood approximation and the type of (linear or nonlinear) forward model, and can thus serve as a drop-in diagnostic to evaluate the accuracy and failure modes of existing and future posterior samplers.

Solving high-dimensional PDE-governed inverse problems is often challenging due to complex non-Gaussian posterior distributions, expensive forward model evaluations, and misspecified prior information. To address these issues, we propose a deep adaptive dimension-reduction Bayesian inference framework based on the Variational Flow (VF) model. Since standard normalizing flows are restricted by bijective mappings and cannot directly reduce dimensions, VF overcomes this limitation by integrating VAE-based nonlinear dimension reduction with dual normalizing flows for the latent prior and encoder. This design provides a strictly higher evidence lower bound than VAE and allows more flexible approximation of complex posterior distributions. We further introduce an iterative prior updating strategy that gradually moves the prior mean toward high-probability posterior regions, avoiding manual prior tuning. These components form a closed adaptive loop together with an adaptively fine-tuned Fourier Neural Operator (FNO) surrogate: VF generates posterior-concentrated samples to refine the surrogate, while the updated surrogate further improves posterior inference. Numerical experiments on a 100-dimensional Rosenbrock problem and three standard PDE-governed inverse problems show that our method delivers competitive or superior accuracy compared with MCMC, UKI, and SVGD baselines across all tested configurations, with the most pronounced advantages emerging in challenging scenarios such as high-noise observations and high-dimensional parameter spaces.

Diffusion-based and iterative methods have become effective tools for solving imaging inverse problems. Their reconstruction process naturally forms a trajectory of intermediate estimates. Although these intermediate estimates define a reconstruction trajectory, most methods do not explicitly regularize the transitions between consecutive states. To address this limitation, we introduce TRACE, a training-free TRAjectory-Constrained rEconstruction framework that stabilizes the reconstruction path by coupling adjacent states along the trajectory. This gives a trajectory-level model that can be interpreted as a sequence of proximal updates. Since the exact proximal update is generally intractable, we approximate it with a neural mapping. This yields a diffusion-like reconstruction process with an explicit coupling between neighboring states. We provide a stability analysis showing that temporal coupling bounds trajectory variation and that this control is preserved under untrained network updates. Experiments on linear and nonlinear image reconstruction tasks show that TRACE improves reconstruction quality. Trajectory-level analyses and ablations confirm that temporal coupling directly affects state transitions along the reconstruction path.

Generative posterior sampling using diffusion models has emerged as a dominant paradigm for solving inverse problems in imaging, which usually consists of three main components: data consistency (DC) guidance, classifier-free guidance (CFG) and stochasticity. While prior arts have focused on how to develop each or all components, less attention has given to how to schedule them, leading to heuristically fixed or partially adjusted suboptimal schedules. In this work, we argue that the interactions among all three components in terms of scheduling are crucial for significantly improved performance in solving inverse problems in imaging. Our analysis shows that aggressive CFG early in sampling conflict with DC guidance, while stochasticity brings the trajectory back to higher-probability regions. Based on these findings, we propose Triadic Dynamics Aware Posterior Sampling (TriPS), which reformulates posterior sampling as a time-varying control problem and optimizes schedules following a triadic trend of decreasing DC and stochasticity scales alongside increasing CFG scale. TriPS achieves this through two strategies: template-based search over functional priors for reliable baseline schedules, and Group Relative Policy Optimization (GRPO)-based reinforcement learning for more flexible temporal curves. Experiments demonstrate TriPS outperforms state-of-the-art baselines in data fidelity and perceptual realism.

Existing score-based methods for inverse problems often resort to approximate minimization of the KL divergence between the inversion distribution and the Bayesian posterior. Such an approximation leads to severe mode collapse and unreliable uncertainty quantification. In this paper, we propose Principled Posterior Matching (PPM), a framework that returns to the fundamentals of variational inference, rather than using tricky approximations. Instead of relying on heuristic approximations, we rigorously formulate the exact optimization of the KL divergence via the integration of Fisher divergence. We derive a tractable, equivalent gradient form of this integral, enabling precise optimization without the biases introduced by prior approximations. Our analysis clearly reveals that the mode collapse in previous methods stems directly from this approximation gap. Supported by our theoretical solution, PPM unifies two complementary paradigms: (1) In variational inference, PPM adopts mass-covering divergences that significantly improve the inversion diversity and uncertainty quantification; (2) In amortized inference, it enables the training of an efficient reconstruction network for rapid, single-step reconstruction. Furthermore, our formulation naturally extends to a broader family of divergence measures by generalizing the integral of the Fisher divergence. We validate PPM across challenging computational imaging tasks, including inpainting, super-resolution fluorescent microscopy, and radio interferometric black-hole imaging. In all experiments, PPM achieves superior reconstruction fidelity, faithful multimodal posterior recovery, and well-calibrated uncertainty estimates, establishing a robust framework for scientific imaging.

Evaluation in scientific reconstruction is dominated by pointwise metrics - RMSE, MAE, per-event resolution - under the implicit assumption that lower error means better reconstruction. We show that this assumption fails structurally for inverse problems with multimodal posteriors. By the law of total variance, point estimators trained to minimize MSE or MAE produce a marginal spectrum strictly narrower than the truth whenever the posterior has nonzero width. The resulting bias is independent of architecture, training, and dataset size, and it compresses precisely the spectral features - tails, modes, shapes - that downstream scientific measurements rely on. We propose a three-part evaluation protocol where each step targets a failure mode the others miss: per-event distributional accuracy via CRPS, population-level marginal accuracy via a spectrum-fidelity diagnostic, and uncertainty trustworthiness via coverage-based calibration. On a synthetic benchmark with an analytic posterior and on a realistic many-to-one inverse problem from particle physics, model rankings reverse between pointwise and distributional metrics, and calibration further separates architectures indistinguishable under CRPS. The evaluation protocol, not the model, determines the scientific conclusion.

We investigate unpaired image inverse problems, a challenging setting where only independent, non-paired sets of noisy measurements and clean target signals are available for training. We propose a novel inverse problem solver based on Unbalanced Optimal Transport, called Unbalanced Optimal Transport Map for Inverse Problems (UOTIP). Our method formulates the reconstruction task, predicting clean target signals from noisy measurements, as learning a UOT Map from noisy measurement distribution to clean signal distribution by incorporating a likelihood-based cost function. By relaxing the exact marginal constraint, the UOT framework provides key advantages to our model: robustness to multi-level observation noise, adaptability to class imbalance between noisy and clean datasets, and generalizability to diverse noise-type scenarios. Furthermore, we theoretically demonstrate that incorporating a quadratic cost term ensures the existence and uniqueness of the transport map by satisfying the twist condition, even for ill-posed inverse problems. Our experiments demonstrate that UOTIP achieves state-of-the-art performance on unpaired image inverse problem benchmarks, across linear and nonlinear inverse problems.

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 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.

Inverse problems in scientific sensing are often solved with either hand-designed regularizers or supervised networks trained on simulated labels, yet both can fail when the forward model is nonlinear, spectrally coupled, and physically delicate. We study this issue for noise sensing based on nitrogen-vacancy (NV) centers in diamond, where a quantum sensor measures magnetic-noise spectra generated by sparse spin sources. We show that replacing a common scalar/coherent forward approximation with a tensor power-summed dipolar operator changes the inverse landscape and exposes a center-collapse failure mode in free-density optimization. We propose NeTMY, an amortization-free coordinate neural field coupled to the differentiable NV forward model, with annealed positional encoding, multiscale optimization, sparsity/gating, and spectrum-fidelity losses. Across sparse synthetic reconstructions generated by the corrected operator, NeTMY achieves the best localization and distributional metrics in the tested benchmark. Mechanism experiments show that NeTMY does not directly execute the raw density-space gradient; its parameterization smooths and redistributes updates, mitigating the center-collapse pathology. These results position NV quantum sensing as a useful testbed for physics-faithful neural inverse problems.

Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.

Artificial intelligence (AI) has transformed imaging inverse problems, from medical diagnostics to Earth observation. Yet deep neural networks can produce hallucinations, realistic-looking but incorrect details, undermining their reliability, especially when ground truth data is unavailable. We develop a theoretical framework showing that such hallucinations are not merely artifacts of particular models, but can arise from the ill-posed nature of the inverse problem itself. We derive necessary and sufficient conditions for hallucinations, together with computable bounds on their magnitude that depend only on the forward model. Building on this theory, we introduce algorithms to: (1) estimate the minimum hallucination magnitude achievable by any reconstruction model for a given input; (2) assess the faithfulness of reconstructed details by a given reconstruction model. Experiments across three imaging tasks demonstrate that our approach applies broadly, including to modern generative models, and provides a principled way to quantify and evaluate AI hallucinations.

Length-dependent logit rescaling is widely used to stabilize long-context self-attention, but existing analyses and methods suggest conflicting inverse-temperature laws for the context length $n$, ranging from $(\log n)^{1/2}$ to $\log n$ and $(\log n)^2$. We provide a general theory showing that the desirable scale is determined by the gap-counting function $N_n$ of each attention row. Counting how many competitors lie within each gap from the maximum, we define an upper-tail accumulation scale and prove that it gives the critical inverse-temperature scale for softmax concentration: below this scale, the top competitors remain unseparated, whereas above it, the attention entropy collapses. This framework unifies prior scaling laws as different $N_n$ and yields a direct diagnostic for attention-score families, from idealized theoretical models to more practical transformers.

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.

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.

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.