Ill-Posed Inverse Problem

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

Interatomic potentials (IPs) enable large-scale atomistic simulations beyond the reach of first-principles methods, but their predictive reliability depends critically on the selection of training data, quantified uncertainty, and model expressiveness. Active learning (AL) provides a principled framework for constructing efficient and accurate IPs, yet most strategies reduce parameter uncertainty without explicitly accounting for the specific material properties being predicted. The information-matching (IM) approach addresses this limitation by requiring that the selected training data provide at least as much parameter space information as needed to achieve prescribed uncertainty targets for selected quantities of interest (QoIs). Here, we apply IM to develop bespoke IPs specifically tailored for predicting plastic strength in metals. Due to the high computational cost of simulating plastic strength, we employ an indirect IM strategy that targets inexpensive intermediate QoIs that correlate with strength. The IM method enables precise parameter constraints with minimal training data, yielding precise predictions for both the intermediate QoIs and plastic strength. Yet, model error remains a key limitation, and a post hoc uncertainty inflation correction provides a viable means to mitigate this limitation. These findings illustrate both the promise and limits of uncertainty-aware AL for predicting complex material properties.

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.

Physics-informed learning is increasingly used for partial differential equation (PDE)-governed inverse problems, but its reliability remains difficult to certify. This paper develops a no-harm certification-and-selection framework for physics-informed inverse learning. A learned reconstruction is accepted only when its residual-calibrated radius is no worse than the baseline radius, namely when $R_{\mathrm{learn}}\le R_{\mathrm{base}}+\varepsilon_{\mathrm{safe}};$otherwise, the method returns the baseline. The certificate combines data, physics, boundary or initial-condition, and optimization residuals. Under a conditional stability estimate, these residuals yield an a posteriori reconstruction-error bound and a deterministic uncertainty radius. A high-probability certificate is also derived for physics residuals estimated from independent random collocation points. Numerical tests on Poisson source recovery, inverse heat reconstruction, limited-angle tomography, elliptic coefficient identification, and stochastic residual validation show that the selector accepts certified improvements, rejects shifted, hallucinated, or unfinished candidates, and becomes conservative in strongly ill-posed regimes. The framework is therefore a certification-and-selection layer, not another reconstruction architecture.

Text-guided diffusion models have become effective tools for real-image visual editing, where the edited image must follow a target instruction while preserving editing-irrelevant structure. Most training-free editors rely on inversion: a source image is mapped to a noisy latent trajectory and the terminal latent is reused for target-prompt denoising. This reuse is useful for preservation, but it also couples source reconstruction and target editing. The resulting trajectory mismatch may either damage background/layout details or over-constrain the intended edit. This paper presents Consistent-Inversion, a training-free reverse consistency guidance framework for structure-preserving visual editing. Instead of treating the inverted source latent as a fixed initialization, Consistent-Inversion checks whether an intermediate target trajectory can be reversed toward the source inversion trajectory under the source prompt. To make this check well-defined, we construct an auxiliary target-side noise representation, perform source-guided reverse denoising, and use the resulting reverse consistency discrepancy as a correction signal for selected early target denoising steps. The method does not update model parameters, is compatible with inversion-based editors, and introduces only a small inference overhead when applied sparsely. Experiments on PIE-Bench show that Consistent-Inversion improves background and structural fidelity under a unified SD3.5 protocol while maintaining target-prompt alignment, and compatibility experiments further verify the same correction principle on classical Stable-Diffusion inversion pipelines.

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.

Aggressive distillation of the diffusion U-Net inverts the per-frame bottleneck of real-time text-to-image pipelines: once the denoiser is a 4-step or 1-step distilled student, the text encoder becomes the critical path. This inversion is most acute in vision-aware edit diffusion, where the encoder is a multimodal large language model (MLLM). We study a 0.39B distilled edit U-Net paired with a 2.13B MLLM text encoder (Qwen3-VL) and present a streaming pipeline for this regime built on three mechanisms that keep the encoder off the denoiser's critical path rather than shrinking the encoder: asymmetric side-stream / main-stream CUDA pipelining with batched text-encoder amortisation, a compile-friendly ControlNet-LLLite reformulation that folds the whole U-Net + adapter stack into one fused graph, and a periodic conditioning-refresh schedule with a hook subset that amortises the per-frame conditioning cost. On a single consumer RTX 3090 Ti at 512x512 this sustains 27-30 fps over a 480-frame run; at the same operating point steady-state throughput scales to 55 fps on RTX 4090 and 74 fps on RTX 5090. This shows that once distillation is aggressive enough, further gains come from encoder-side systems work rather than further denoiser compression -- the opposite lever from the one the streaming-diffusion literature has optimised to date. We report video-rate streaming throughput, not interactive low latency, and locate our numbers against same-stack StreamDiffusion re-runs as systems context, not a superiority claim. The released oil-painting adapter generalises within in-clip noise to 19 unused DAVIS-2017 sequences and 15 non-DAVIS clips from seven sources; prompt-level generalisation to unseen styles is bounded and reported separately.

Metasurfaces enable precise manipulation of electromagnetic waves for applications such as beam steering, sensing, and stealth technology. However, inverse design of metasurfaces with targeted EM responses remains challenging due to the computational expense of iterative full wave simulation driven optimization and the limited conditioning fidelity and diversity of existing generative approaches. To address these challenges, this paper presents a generative inverse design framework for controllable and physically consistent metasurface synthesis under continuous spectral constraints. The proposed approach employs a progressively growing Wasserstein generative adversarial network with gradient penalty integrated with feature wise linear modulation based conditioning for stable propagation of continuous spectral and fabrication constraints. EM consistency is embedded directly into the generative learning process through a surrogate assisted spectral alignment loss, enabling physics constrained generation during training. Further, a determinantal point process based diversity regularization strategy is incorporated to generate geometrically diverse yet spectrally consistent realizations for the same target response. The effectiveness of the proposed framework is demonstrated through the generation of practically realizable metasurface absorbers exhibiting diverse reflection characteristics in the frequency range of 2 to 18 GHz. EM simulations validate that the generated designs meet the target specifications with high accuracy. The final proposed framework achieved an average mean squared error of 0.0052, diversity score of 0.8730, band alignment accuracy of 0.8533, and a valid EM design generation percentage of 89.57, clearly demonstrating its capability to generate highly accurate, diverse, electromagnetically consistent and fabrication realizable metasurface configurations.

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.

Inverting a robotic task requires more than reversing symbolic state transitions or rewinding motor trajectories. In robot manipulation tasks, symbolic inverse plans often fail to fully restore the effects of forward executions under continuous interaction dynamics. We present a hybrid framework for inverse manipulation that derives inverse-skill objectives from STRIPS-like operators automatically extracted from demonstrations through soft geometric predicates. For each extracted operator, we construct an inverse restoration objective that preserves preconditions, restores delete effects, and negates add effects. A task planner first attempts to satisfy this objective using available action primitives. Unresolved symbolic predicates then induce a residual operator learning problem solved through Reinforcement Learning (RL). We evaluate the framework on the ManiSkill3 PushCube task. For a forward pushing skill, the symbolic inverse performs a coarse pick-and-place restoration, while a residual Soft Actor-Critic policy refines the cube pose to satisfy the remaining inverse predicates. Our results show that predicate-derived residual control can turn an approximate symbolic inverse into a physically grounded inverse skill.

Automatic-differentiation-based inverse analysis methods, including physics-informed neural networks (PINNs) and differentiable programming, have recently shown great promise due to their ability to compute accurate gradients and convergence efficiency. However, their applicability to falling weight deflectometer (FWD) backcalculation remains unexplored. This study critically evaluates PINN-based inverse analysis for a multilayer pavement system and investigates differentiable finite element method (DiffFEM) as an alternative based on a synthetic benchmark. The standard PINN does not recover layer moduli because of the sharp domain discontinuities inherent to layered pavement systems. Although we use an extended PINN with domain decomposition (XPINN), which shows better performance on discontinuous domains, its performance remains highly sensitive to loss weighting and network architecture, and degrades under measurement noise. By contrast, DiffFEM consistently achieves more accurate, stable, and computationally efficient inversion results. These results indicate that DiffFEM, which enforces the governing physics as a hard constraint, yields better accuracy, robustness, and computational efficiency than PINN-based approaches, in which the governing physics is imposed as a soft constraint through the loss function. More broadly, the findings suggest that the choice between PINN- and DiffFEM-based inverse analysis needs careful consideration, with DiffFEM offering practical advantages when an efficient and robust differentiable forward solver is available.

This paper studies the problem of inverting the DDIM image generation process to recover latent variables, particularly the initial noise map, from a generated image. Existing methods often struggle with accuracy in this task. We propose a novel hybrid approach that combines direct inversion via gradient descent for the first step, followed by a fixed-point method for subsequent steps. Empirical evaluations across three datasets demonstrate that our method significantly improves the prediction of initial latent variables while achieving superior reconstruction accuracy. Additionally, we introduce a new evaluation, called the self-interpolation test, which assesses the quality of images generated from interpolated points between the true and predicted latent maps, offering deeper insights into performance. Our results reveal that while existing methods perform reasonably well in reconstruction, they consistently fail to accurately predict the initial latent variables, resulting in poor performance on the self-interpolation test. In contrast, our method outperforms all others across all metrics, providing valuable insights into diffusion models and enhancing their applications in image generation and editing.

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.

The transition from single-turn models to Multi-Agent Systems (MAS) promises enhanced problem-solving capabilities, yet the centralized orchestration topology remains a critical point of fragility. To analyze this, we propose a Mean-Field Entropy Dynamics framework, modeling the orchestration process as a system governed by the competing forces of task resolution and cumulative context loading. To facilitate validation, we introduce Inverse Workflow Generation (IWG), a multi-agent pipeline that synthesizes process-verifiable, high-complexity benchmarks with dense intermediate checkpoints. We demonstrate that our entropy dynamics model fits empirical trajectories, providing physically interpretable parameters that quantify system stability and performance collapse. Crucially, our analysis uncovers a ``Reasoning Trap": while reasoning-heavy models excel in isolated tasks, they frequently fail as orchestrators due to context squeezing. Elucidating the physical mechanisms underlying the Orchestrator and quantifying systemic uncertainty offers insights for the MASs' architectural design.

This paper presents a fast, recursive neural solver for the J2-perturbed Lambert problem based on Tiny Recursive Models (TRM), termed the TRM-Perturbed Lambert (TRM-PL) model. TRM is a weight-shared architecture whose effective capacity emerges from iteration depth rather than parameter count: a compact reasoning module is applied repeatedly within a two-level latent hierarchy, refining a candidate departure velocity by simulating the J2 trajectory and correcting it from the resulting tracking error. This unifies initial-guess generation and iterative correction in a single, end-to-end differentiable architecture. The recursive refinement loop is a learned alternative to the homotopy and continuation schemes of classical perturbed-Lambert solvers: rather than following a hand-designed path from the Keplerian to the perturbed solution, the network learns its own sequence of corrections. We evaluate TRM-PL on three test cases of increasing difficulty: single-revolution low-Earth-orbit (LEO) transfers, multi-revolution LEO transfers, and multi-revolution Jovian transfers. Three training paradigms are compared: jointly learning the Lambert solution and the J2 correction; refining the Lambert initial velocity with target-position and J2-corrected velocity supervision; and refining it with target-position supervision alone. Across all cases, the refinement-only approaches are the most reliable. The position-supervised variant reduces the median terminal-position error from 21.7 km to 0.027 km on single-revolution LEO, from 340.9 km to 0.31 km on multi-revolution LEO, all with the same 2.3M-parameter architecture. A single Newton corrector iteration on the TRM-PL output tightens the Jovian median to 0.063 km, yielding compact models accurate enough for embedded deployment.

Reconstructing spatiotemporal fields from partial observations is fundamental to scientific inference, from inferring atmospheric states from satellite data to recovering fluid states from imaging. When observations are incomplete, the inverse problem is fundamentally ill-posed: even when the underlying PDE dynamics are Markovian in the full state, partial observation operators induce a non-Markovian posterior that cannot be resolved from a single timestep. We propose a history-bootstrapped autoregressive flow matching (HB-ARFM) for spatiotemporal inverse reconstruction under partial observability. Observation history bootstraps the initial reconstruction via conditional flow matching, reducing ambiguities. The same conditional transport model is then applied autoregressively, conditioning on both new observations and past predictions to propagate the reconstruction forward in time. We evaluate the method on boiling dynamics reconstruction, recovering full velocity and temperature fields from interface geometry and motion. Across two inverse tasks with varying observation sparsity, HB-ARFM produces physically and temporally valid reconstructions where other models fail.

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.

Nonlinear conservation laws are at the heart of many of the most important dynamical systems in science and engineering. In practical applications, such systems are often subject to various sources of uncertainty, e.g. due to sparse or noisy measurements. Inferring physical quantities and fields of interest then becomes an ill-posed problem which both classical numerical methods and modern deep learning-based methods struggle to treat appropriately. Recent work has framed classical numerical methods as Bayesian inference under Gaussian process priors, resulting in a physics-aware treatment of uncertainties. Following this line of work, we develop a novel numerically conservative method for uncertainty-aware simulations of nonlinear conservation laws. We use recent sparse approximation techniques to scale up to large-scale forward and inverse problems. For forward simulation, we inherit the accuracy of classical solvers while providing structured uncertainty quantification. On inverse problems, we recover posteriors over nonparametric source fields in seconds -- outperforming neural baselines that take minutes to produce a less accurate point estimate.

Inverse reinforcement learning (IRL) typically assumes demonstrations from a single optimal demonstrator, but in many applications data come from multiple imperfect demonstrators with heterogeneous suboptimality levels. We study reward learning in this setting through a feasible-reward-set framework: for each demonstrator, we encode its declared suboptimality level as a linear constraint and intersect the resulting feasible sets across demonstrators. Our theoretical analysis shows that the joint feasible set shrinks monotonically as data are added, and we give an exact characterization of when a new demonstrator strictly tightens it. We further establish two recovery guarantees for the feasible reward set of the ground-truth optimal demonstrator: one bound depends on closeness to the optimal occupancy, while the other requires only sufficient coverage and no near-optimal demonstrator. On the practical side, we introduce strategies to address the inherent reward ambiguity in the obtained reward set and provide an offline algorithm with function approximation for high-dimensional environments. Experiments in tabular grid-world and large language model (LLM) fine-tuning settings are consistent with the theoretical predictions and demonstrate the effectiveness of the proposed framework over baselines.

Sampling from high-dimensional, non-log-concave distributions with unnormalized densities remains a fundamental challenge in machine learning, particularly in black-box settings where gradient information is inaccessible or computationally prohibitive. While Langevin dynamics provides a principled framework for sampling when gradients are accessible, its extension to the black-box settings suffers from high variance and lacks non-asymptotic convergence guarantees for non-log-concave sampling. To address these limitations, we propose a variance-reduced zeroth-order Langevin sampling method. Our method employs a gradient estimator that substantially reduces the variance of the classical batched zeroth-order estimator and eliminates the unfavorable dimensional dependence of the batch size required for accurate estimation, enabling practical and stable sampling. We establish the first non-asymptotic convergence guarantees for zeroth-order non-log-concave sampling in terms of $\varepsilon$-relative Fisher information, and, under a Poincaré inequality assumption, squared total variation distance. We further propose ZO-APMC, a posterior sampling algorithm for black-box inverse problems with pre-trained score-based generative priors, establishing the first non-asymptotic convergence guarantees for such methods. We validate our theory through synthetic experiments and demonstrate strong empirical performance on practical linear and nonlinear inverse problems.

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.

Photonic crystal fiber (PCF) inverse design remains challenging because candidate geometries must satisfy coupled optical targets under expensive electromagnetic simulation. Existing pipelines improve surrogate prediction or one-shot parameter recommendation, but they do not accumulate reusable design knowledge across iterative trials. We formulate PCF inverse design as a memory-policy learning problem and propose SkillPCF, a closed-loop agent framework that combines a physics-guided memory skill bank, reinforcement-learned skill selection, and simulator-grounded skill evolution. We further construct a real-world dataset with 479 expert interaction traces (2,507 spans) and 553 memory-dependent evaluation queries covering dispersion engineering, loss optimization, and multi-objective design. Experiments across multiple LLM backbones and classical baselines show that SkillPCF achieves stronger design-quality and efficiency trade-offs under practical simulation budgets, demonstrating the effectiveness of our proposed memory-skill learning paradigm for physics-aware PCF inverse design.

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.

The distortion-perception (D-P) tradeoff is a fundamental phenomenon of Bayesian inverse problems, which characterizes the inherent tension between distortion performance and perceptual quality. Enabling flexible traversal of the D-P tradeoff at inference time is crucial for practical applications. Despite the recent success of diffusion models in zero-shot inverse problem solving, efficient and principled strategies for D-P traversal in diffusion-based inverse algorithms remain inadequately characterized. In this paper, we propose a stage-wise framework for realizing D-P traversal using a single diffusion model in zero-shot inverse problems. Our proposed method, termed MAP-RPS, starts with an MAP estimation stage that approximates the MMSE solution and provides a low-distortion initialization, followed by a re-noised posterior sampling stage that progressively improves perceptual quality. We provide theoretical analyses for both stages, establishing the validity and effectiveness of the proposed design. Furthermore, we extend MAP-RPS to the latent space, yielding LMAP-RPS, which enjoys broader applicability by leveraging large-scale pre-trained latent diffusion backbones. Extensive experiments demonstrate that MAP-RPS and LMAP-RPS enable more effective D-P traversal on various tasks, while also exhibiting strong performance as efficient solvers for real-world inverse problems.

Mathematical reasoning benchmarks are vital for evaluating large language models (LLMs), but many are static and repeatedly exposed through public evaluation and training pipelines, making it difficult to separate genuine reasoning from memorization. Meanwhile, manually constructing new math problems with reliable answers remains costly. We introduce ReverseMath, a scalable method for generating new math problems through answer inversion. Given a problem and its answer, ReverseMath masks a numerical value in the original problem, treats the original answer as a known condition, and rewrites the problem so that the masked value becomes the new answer. The generated problem reverses the original input-output relation, making its answer known by construction. We study ReverseMath for both evaluation and training. For evaluation, paired original/reversed problems reveal substantial behavioral shifts: models sometimes fail on reversed problems and even incorrectly output the original answer, suggesting memorization-like behavior. For training, ReverseMath provides automatically labeled reversed problems as data augmentation for reinforcement learning (RL). Experiments show that including ReverseMath-generated data improves mathematical reasoning performance across multiple benchmarks, demonstrating its value as both an analysis tool and a scalable source of verifiable training data.

Polymer discovery is central to fields ranging from energy storage to biomedicine, but it is hindered by an astronomically large chemical design space and fragmented representations of structure, properties, and prior knowledge. This fragmentation leaves many AI models disconnected from physical and experimental reality, restricting their ability to support directly actionable design decisions. Here we introduce PolyFusionAgent, an interactive framework coupling a multimodal polymer foundation model (PolyFusion) with a tool-augmented, literature-grounded design agent (PolyAgent). PolyFusion aligns complementary polymer views including sequence, topology, 3D geometry, and fingerprints across millions of polymers to learn a shared latent space transferable across chemistries and data regimes, improving thermophysical property prediction and enabling property-conditioned generation of chemically valid, structurally novel polymers beyond the reference design space. PolyAgent closes the design loop by linking prediction and inverse design with evidence retrieval from the polymer literature, proposing, evaluating, and contextualizing hypotheses with explicit precedent in one workflow. Together, PolyFusionAgent enables interactive, evidence-linked polymer discovery combining large-scale representation learning, multimodal chemical knowledge, and verifiable scientific reasoning.

The inverse problem of multilayer thin-film optical coatings design represents a complex combinatorial-continuous optimization challenge. We present PRISM (Position-encoded Regressive Inverse Spectral Model), a unified decoder-only autoregressive transformer that streamlines this process by jointly predicting discrete material selection and continuous thickness regression within a single backbone. PRISM introduces two primary architectural innovations: (1) spectrum prefix conditioning, which utilizes standard prefix tokens for in-context target injection, and (2) cumulative-depth Rotary Position Embeddings, which encode continuous thickness directly into the positional representation to preserve the physical spatial relationships of the stack. Our benchmarks demonstrate that a PRISM-13M model reduces MAE by over 50% compared to other transformer baselines while utilizing only one-fifth of the parameters. Furthermore, a 44M-parameter variant achieves state-of-the-art performance (MAE = 0.010) on our in-distribution validation benchmark and operates significantly faster than simulated annealing, offering a highly efficient alternative to classical optimization methods.

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.

Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.

Reconstructing PDE solutions from sparse observations is a core challenge in scientific computing. We present FM4PDE, a flow-matching generative framework that learns the joint distribution of PDE coefficients (or initial states) and solutions (or final states), enabling both forward simulation and inverse recovery with limited paired data. At inference, sampling is guided by a composite loss that enforces agreement with sparse measurements and reduces the PDE residual; we support deterministic, stochastic, and hybrid samplers. We provide error guarantees for these guided procedures. For the deterministic optimizer, a coercivity condition ensures trajectory boundedness and a phase-wise contraction yields logarithmic complexity in the target accuracy. For the stochastic sampler, we introduce adaptive guidance and assume dissipativity of the velocity field to obtain uniform moment bounds independent of the noise-floor parameter. This leads to polynomial-time error bounds, and a matching lower bound shows constant guidance induces an unavoidable positive bias, motivating adaptivity. A hybrid deterministic-stochastic analysis is also provided. Experiments on static and time-dependent benchmark PDEs demonstrate competitive accuracy and faster inference than diffusion-based generative models.

Inverse problems in partial differential equations (PDEs) involve estimating the physical parameters of a system from observed spatiotemporal solution fields. Neural networks are well-suited for PDE parameter estimation due to their capability to model function-to-function space transformations. While existing benchmarks of machine learning methods for PDEs primarily focus on the forward problem, there are no similar comprehensive studies and benchmark datasets on PDE inverse problems, i.e., mapping solution fields to underlying physical parameters. We fill this gap by introducing PDEInvBench, a comprehensive benchmark dataset consisting of numerical simulations for both time-dependent and time-independent PDEs across a wide range of physical behaviors and parameters. Our dataset includes evaluation splits that assess performance in both in-distribution and various out-of-distribution settings. Using our benchmark dataset, we comprehensively explore the design space of neural networks for PDE inverse problems along three key dimensions: (1) optimization procedures, analyzing the role of supervised, self-supervised, and test-time training objectives on performance, (2) problem representations, where we study the value of architectural choices with different inductive biases and various conditioning strategies, and (3) scaling, which we perform with respect to both model and data size. Our experiments reveal several practical insights: 1) neural networks perform best with a two-stage training procedure: initial supervision with PDE parameters followed by test-time fine-tuning using the PDE residual, 2) incorporating PDE derivatives as input features consistently improves accuracy, and 3) increasing the diversity of initial conditions in the training data yields greater performance gains than expanding the range of PDE parameters. We make our dataset and codebase publicly available.

Recently, Deep Image Prior (DIP) has demonstrated strong capabilities for solving inverse imaging problems (IIPs) by optimizing a randomly initialized convolutional neural network in a training-data-free regime. However, DIP suffers from overfitting to noisy measurements due to network over-parameterization, making early stopping (ES) essential. The most successful ES method tracks fluctuations in the running variance of the network output to detect overfitting. However, in many applications, these fluctuations may appear prematurely, leading to unstable reconstructions. In this paper, we first show that nearly optimal DIP early stopping can be achieved when two independent noisy copies of the degraded image are available. Motivated by this observation, and since obtaining two fully independent copies is infeasible, we propose an overfitting detection framework based on constructing pseudo self-referenced images, resulting in three IIP-specific algorithms. Our approach is further supported by theoretical results on single-reference validation, pseudo-validation estimation, and the impact of shared noise. Across different IIPs, ranging from natural image restoration to medical image reconstruction, and under varying noise levels and noise types, our methods consistently outperform existing DIP early stopping approaches, all without requiring an accurate estimate of the noise level.

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.

The widespread integration of Large Language Models (LLMs) necessitates rigorous and systematic safety evaluation. Existing paradigms either rely on constructed benchmarks to assess safety from predefined perspectives, or employ dynamic red-teaming to probe potential vulnerabilities. While effective, these approaches face challenges, as they depend heavily on expert domain knowledge, offer limited systematic guarantees, and are vulnerable to rapid obsolescence. To address these limitations, we introduce a novel framework POLARIS that brings the rigor of specification-based software testing to AI safety. POLARIS first compiles unstructured natural-language policies into First-Order Logic (FOL) representations, establishing a traceable link between high-level rules and concrete test cases. This formalization enables the construction of a Semantic Policy Graph, where complex policy violation scenarios are encoded as traversable paths. By systematically exploring this graph, POLARIS uncovers compositional violation patterns, which are then instantiated into executable natural-language test queries, enabling coverage-driven and reproducible safety testing. Experiments demonstrate that POLARIS achieves higher policy coverage and attack success counts compared to established baselines. Crucially, by bridging formal methods and AI safety, POLARIS provides a principled, automated approach to ensuring LLMs adhere to safety-critical policies with verifiable traceability. We release our code at https://github.com/huac-lxy/POLARIS.

We present a neuro-inspired framework for embodied planning and control. Building on three principles that enable fast and highly effective goal-directed behavior in the mammalian brain - paired forward/inverse internal models, open-loop multi-step motor commands, and sequential, hierarchical organization of action - our Inverter framework uses learned components, trained end-to-end through Inverse Learning (IL) and supplemented where natural by analytic or algorithmic modules; we formalize IL and delineate it from supervised, reinforcement, and imitation learning. IL bridges Reinforcement Learning (RL)-style amortization, which runs in a single forward pass but emits only one action at a time, and Optimal Control (OC)-style sequence planning over whole trajectories, but with iterative test-time computation. Single Inverters or hierarchical n=2 Inverter stacks match or improve on offline-RL and diffusion-planner baselines on all 3 maze2d and 6 antmaze D4RL variants by an average of +24.2% (range -1.9% to +78.2%), at one-to-two orders of magnitude less inference compute time. Distinctively, optimizing through the Forward Model (FoM) over the entire T-step action sequence - rather than per step - lets Inverters produce smooth, goal-coherent, trajectory-wide structure and reach control policies closer to the analytic optimum than the policy underlying the training data itself. We also identify a failure mode of IL: FoM hacking under narrow training-data coverage, which we mitigate by using random training data with broader coverage. As an application example, a Pulse Inverter synthesizes arbitrary single-qubit quantum gates with fidelity matching the standard iterative numerical baseline (GRAPE), at more than 1000x lower per-gate compute time. In summary, we conclude that IL enables a versatile class of world-interfaces, especially for latency- and resource-critical embodied AI.

The rapid evolution of generative video foundation models has propelled the field toward professional-grade cinematic synthesis. To achieve such demanding quality, the community transitions towards Reinforcement Learning (RL) and agentic workflows. However, reliable evaluation has emerged as a critical bottleneck. Existing benchmarks predominantly evaluate ''whether it is right'' (basic prompt-following) while fundamentally neglecting ''whether it is good'' (cinematic quality, acting, and aesthetics). Furthermore, current automated metrics lack the domain-specific rigor required to provide trustworthy signals, creating a severe credibility gap between human aesthetic perception and machine scoring. To bridge this gap, we introduce EvalVerse, a comprehensive, pipeline-aware, and expert-calibrated evaluation framework. We treat video generation assessment not merely as an engineering task, but as a core scientific problem: the systematic digitization of subjective cinematic expertise. First, we organize domain knowledge into an evaluation taxonomy aligned with the professional filmmaking workflow (pre-production, production, and post-production). Second, we distill human expert judgments into a curated dataset with large-scale human annotations. Third, we inject this knowledge into Vision-Language Models (VLMs) through an expert-calibrated fine-tuning strategy, enabling the VLM to perform explicit Chain-of-Thought reasoning. Compared to previous works, EvalVerse not only retains compatibility with foundational ''rightness'' metrics, but also significantly expands the criteria to ''goodness'' and broaden the task coverage to complex multi-shot sequencing and audio-visual integration. Consequently, by providing granular diagnostic signals, EvalVerse transcends a static leaderboard and establishes a fundamental infrastructure for future work, such as reward models and evaluator agent.

We document inverse scaling in LLMs on forecasting problems whose underlying time series exhibit superlinear growth and tail risk of regime change, a structure common in finance and epidemiology. On these tasks, more capable models produce worse distributional forecasts. The pattern appears on ForecastBench-Sim (FBSim), a contamination-free, simulated-world benchmark we release, in forecasting synthetic SIR epidemics with a matched linear control, and replicates in real-world datasets on COVID-19, measles, housing markets, and hyperinflation. A per-quantile decomposition shows the failure concentrates at the upper tail, which more capable models shift upward to track aggressive extrapolations of growth, while the lower tail stays put. A within-family study of Llama-3.1 shows that both model scale and post-training independently contribute to this effect. Domain knowledge does not reliably rescue calibration. This inverse scaling does not appear on single-threshold metrics common in LLM forecasting benchmarks, reversing the sign of the capability--accuracy relationship on identical outputs. Single-threshold scoring at conventional cutoffs misses the upper-tail cost; tail-inclusive scoring reverses the sign of the capability--accuracy relationship on the same outputs. We recommend that LLM forecasting evaluations use continuous (and unbounded) measures of accuracy alongside bounded binary threshold metrics.

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.

Reconstructing continuous physical fields from sparse measurements is a central inverse problem, but data-driven generative models can produce states that violate governing dynamics. We introduce a physics-informed generative solver that separates stable prior learning from inference-time enforcement of conservation laws. Martingale-Regularized Score Matching regularizes score pretraining with a Score Fokker-Planck constraint, yielding a dynamically stable prior. Physics-Informed Implicit Score Sampling then guides denoising trajectories by gradients of physical residuals, projecting samples toward admissible manifolds without retraining. In acoustics, the method co-generates pressure and particle velocity from sparse sensors, enabling dense virtual arrays that suppress spatial aliasing. The same framework generalizes to real-world ERA5 meteorological fields under extreme sparsity. Together, this work establishes a rigorous and generalizable paradigm for solving high-dimensional inverse problems, bridging the gap between generative artificial intelligence and first-principles science.

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.

This work presents an end-to-end strategy for solving inverse problems constrained by Partial Differential Equations within a fully differentiable Machine Learning framework. The proposed formulation provides a unified and user-friendly methodology applicable to a wide range of problems, from data assimilation to closure modeling. Our approach combines a baseline differentiable PDE solver, which predicts the state w from the nonlinear system $R(w) = 0$, with a generic additive, parametrized, and differentiable correction $f_φ(w)$, with trainable parameters $φ$. We show how to optimize phi within a fully differentiable Python workflow by reformulating the PDE as an implicit layer, enabling its integration into arbitrary objective functions, while leveraging PyTorch's automatic differentiation graph. The method is demonstrated on the Reynolds-Averaged Navier-Stokes equations for compressible flows, where the closure term, or a portion of it, is modeled using trainable parameters or a Neural Network. The first application considers the 2D NASA Wall-Mounted Hump test case, where a production-term parameter is optimized against time-averaged LES data. A second application is carried out on the VKI LS-59 turbine blade, where the Spalart-Allmaras eddy viscosity field is reconstructed through the optimization of a trainable spatial field. A dataset is generated starting from the VKI LS-59 turbine blade geometry using the differentiable BROADCAST solver with the Spalart-Allmaras turbulence model. The results highlight the flexibility of the framework, showing its applicability beyond turbulence modeling to a broader class of physics-informed PDE-constrained problems with data-driven components.

Inferring unknown initial states in shock-dominated compressible flows from sparse and noisy measurements is a challenging ill-posed inverse problem due to nonlinear wave interactions and limited sensing. In this work, we develop a non-intrusive reduced-order modeling framework for efficient Bayesian initial-state inversion with uncertainty quantification. The framework combines a convolutional autoencoder with a learned latent-space forward operator. The autoencoder compresses high-dimensional flow fields into a compact nonlinear latent representation, while the forward operator predicts final-time latent states from encoded initial conditions. This AE-ROM surrogate enables rapid forward evaluations and is embedded within a No-U-Turn Sampler (NUTS) for posterior exploration. The framework is demonstrated using 500 high-fidelity Sod shock tube simulations generated through Latin hypercube sampling and solved using a fifth-order WENO scheme. The inverse problem seeks to recover unknown left and right density and pressure states from sparse noisy observations of final-time density and pressure fields. Results show that the AE-ROM accurately reconstructs key shock-tube structures, including the rarefaction wave, contact discontinuity, and shock front. A latent dimension of 32 provides an effective balance between reconstruction accuracy and reduced-space compactness, while 250 training simulations are sufficient for accurate reconstruction. Increasing observation density significantly contracts posterior uncertainty, reducing the mean posterior standard deviation by approximately 78% for density and 76% for pressure. Overall, the proposed framework provides a computationally efficient and uncertainty-aware approach for inverse analysis of shock-dominated flows, with potential extensions to multidimensional compressible-flow and digital-twin applications.

Solving inverse partial differential equation (PDE) problems is a fundamental topic in scientific research due to its broad significance across a wide range of real-world applications. Inverse PDE problems arise across medical imaging, geophysics, materials science, and aerodynamics, where the goal is to infer hidden causes, design structures, or control physical states. In this paper, we provide a comprehensive review of recent advances in solving inverse PDE problems using artificial intelligence (AI). We first introduce the basic formulation, key challenges, and traditional numerical foundations of inverse PDE problems, and then organize it into three major categories: inverse problems, inverse design, and control problems. For each category, we further present a methodological paradigms, and review representative state-of-the-art approaches from recent years. We then summarize representative applications across scientific and industrial domains, including mechanical systems, aerodynamic problems, thermal systems, full-waveform inversion, system identification, and medical imaging. Finally, we discuss open challenges and future prospects, such as physics-informed architectures, limited real-world data, uncertainty quantification, and inverse foundation models. This survey aims to provide the first unified and systematic perspective on AI for inverse PDE problems, demonstrating how modern learning-based methods are reshaping inverse problems, inverse design, and control problems in PDE-governed systems.

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.

Solving imaging inverse problems has usually been addressed by designing proper prior models of the underlying signal. However, minimizing the data fidelity term poses significant challenges due to the ill-conditioned sensing matrix caused by physical constraints in the acquisition system. Thus, preconditioning techniques have been adopted in classical optimization theory to address ill-conditioned data-fidelity minimization by transforming the algorithm gradient step to achieve faster convergence and better numerical stability. We extend the preconditioning concept beyond convergence acceleration and use it to improve reconstruction quality. We introduce DIPA: Distilled Preconditioned Algorithms, where a preconditioning operator (PO) is optimized using teacher-guided distillation criteria. Unlike standard model-compression KD, the teacher and student differ by the sensing operators available during reconstruction: the teacher uses a simulated, better-conditioned, and more informative sensing matrix, whereas the student uses the physically feasible sensing matrix. We design different distillation loss functions to transfer different properties of the teacher algorithm to the preconditioned student. The PO can be linear (L-DIPA), allowing interpretability, or non-linear (N-DIPA), parametrized by a neural network, offering better scalability. We validate the proposed PO design across several imaging modalities, including magnetic resonance imaging, compressed sensing, and super-resolution imaging.

We consider amortized Bayesian inference for nonlinear inverse problems in settings where only samples from the joint distribution of parameters and observations are available. Classical methods such as Markov chain Monte Carlo require solving a new inference problem for each observation, which can be computationally prohibitive when inference must be repeated many times. We propose a transport-based approach that learns an observation-dependent map pushing forward a reference measure to approximate the posterior distribution. The map is trained by minimizing an averaged energy-distance objective between the true posterior and the learned pushforward. This formulation is likelihood-free, requiring only joint samples, and avoids density evaluation, invertibility constraints, and Jacobian determinant computations. For function-space inverse problems with Gaussian priors, we parameterize the transport map as the identity plus a perturbation in the Cameron-Martin space of the prior, preserving absolute continuity with respect to the prior. In infinite-dimensional settings, the map is represented using neural operators. We illustrate the method on a finite-dimensional nonlinear inverse problem and two PDE-constrained inverse problems arising in porous medium flow and seismic inversion. The results show that the learned transport captures posterior structure, including multimodality and dominant modes, while enabling fast posterior sampling for new observations.

Recently, deep generative models have been used for posterior inference in inverse problems, including high-stakes applications in medical imaging and scientific discovery, where the uncertainty of a prediction can matter as much as the prediction itself. However, posterior uncertainty is difficult to interpret because it can mix ambiguity inherent to the forward operator with uncertainty propagated through inference. We introduce a structural decomposition of posterior uncertainty that isolates intrinsic ambiguity. A cascade formulation makes this ambiguity accessible for calibration analysis, enabling qualitative diagnostics and simulation-based calibration tests that reveal failure modes that remain hidden when models are selected by reconstruction quality alone. We first validate the approach on a Gaussian example with analytical posterior structure, then illustrate the decomposition on accelerated magnetic resonance imaging (MRI), and finally apply the calibration diagnostics to electroencephalography (EEG) source imaging.

We present VectraYX-Nano, a 41.95M-parameter decoder-only language model trained from scratch in Spanish for cybersecurity, with a Latin-American regional focus and native tool invocation via the Model Context Protocol (MCP). The model has four contributions. (i) Corpus: VectraYX-Sec-ES, a 170M-token Spanish corpus assembled by an eight-VM distributed pipeline at ~$25 USD of cloud compute and split into three curriculum phases (conversational 42M, cybersecurity 118M, offensive tooling 10M). (ii) Architecture: a 42M Transformer decoder with GQA, QK-Norm, RMSNorm, SwiGLU, RoPE and z-loss, paired with a domain-balanced 16,384-token byte-fallback BPE. (iii) Curriculum with replay across the three phases yields a monotonic loss descent (9.80 -> 3.17 -> 3.00 -> 2.16); after SFT (loss 1.74) the v2 bootstrap-ablation reference attains a conversational gate of 0.775 +/- 0.043 on B5 over N=4 seeds, and a controlled Phase-2 replay sweep over {0,5,10,25,50}% saturates B5 at >=25% replay. (iv) Two empirical findings, both N=4. A controlled bootstrap-corpus ablation across v2 (OpenSubs), v4 (mC4-ES), and v6 (60/25/15 OpenSubs/mC4/Wiki) exposes a loss-versus-register inversion: lower-perplexity bootstraps yield measurably worse conversational behavior (v2 > v4 > v6 on B5 at every paired seed). The B4 (tool-selection) floor of 0.000 is a corpus-density artifact, not a capacity gate: rebalancing the SFT mixture to tool-use ratio 1:21 yields VectraYX-Nano v7, the released headline configuration, reaching B4 = 0.230 +/- 0.052 at 42M while retaining B1 = 0.332 +/- 0.005 and B5 = 0.725 +/- 0.130; a LoRA replication on a 260M from-scratch mid-tier reaches 0.445 +/- 0.201. The released GGUF is 96 MB in F16, runs sub-second TTFT on commodity hardware under llama.cpp, and is, to our knowledge, the first published Spanish-native cybersecurity LLM with end-to-end MCP integration.

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.

Partial differential equations (PDEs) are fundamental for modeling complex natural and physical phenomena. In many real-world applications, however, observational data are extremely sparse, which severely limits the applicability of both classical numerical solvers and existing neural approaches. While neural methods have shown promising results under moderately sparse observations, their inference efficiency at high resolutions is limited, and their accuracy degrades substantially in the extremely sparse regime. In this work, we propose the Di-BiLPS, a unified neural framework that effectively handle both forward and inverse PDE problems under extremely sparse observations. Di-BiLPS combines a variational autoencoder to compress high-dimensional inputs into a compact latent space, a latent diffusion module to model uncertainty, and contrastive learning to align representations. Operating entirely in this latent space, the framework achieves efficient inference while retaining flexible input-output mapping. In addition, we introduce a PDE-informed denoising algorithm based on a variance-preserving diffusion process, which further improves inference efficiency. Extensive experiments on multiple PDE benchmarks demonstrate that Di-BiLPS consistently achieves SOTA performance under extremely sparse inputs (as low as 3%), while substantially reducing computational cost. Moreover, Di-BiLPS enables zero-shot super-resolution, as it allows predictions over continuous spatial-temporal domains.

Image restoration is an inherently ill posed inverse problem. Equivariant networks that embed geometric symmetry priors can mitigate this ill posedness and improve performance. However, current understanding of the relationship between network equivariance and data symmetry remains largely heuristic. Particularly for real world data with imperfect symmetry, existing research lacks a systematic theoretical framework to quantify symmetry, select transformation groups, or evaluate model data alignment. To bridge this gap, we conduct an analysis from an optimization perspective and formalize the intrinsic relationship among data symmetry priors, model equivariance, and generalization capability. Specifically, we propose for the first time a quantifiable definition of non strict symmetry at the dataset level (rather than sample level) and use it as a constraint to formulate the restoration inverse problem. We then show that the equivariance for restoration models can be naturally derived from this inverse problems incorporated the proposed symmetry constraints, and that the equivariance error of the optimal restoration operator is strictly bounded by the data symmetry error and the discretization mesh size. Furthermore, by analyzing the network's empirical risk, we demonstrate that aligning equivariance with data symmetry optimizes the bias variance trade off, minimizing the total expected risk. Guided by these insights, we propose a Sample Adaptive Equivariant Network that uses a hypernetwork and transformation learnable equivariant convolutions to dynamically align with each sample's inherent symmetry. Extensive experiments on super resolution, denoising, and deraining validate our theoretical findings and show significant superiority over standard baselines and traditional equivariant models. Our code and supplementary material are available at https://github.com/tanfy929/SA-Conv.

Recent advances in AI have been primarily driven by large-scale neural architectures that excel at function approximation, rather than by tailored inductive biases and inference or learning strategies that could be important for resource-efficient real-world perception and planning through the solution of inverse problems. In this work, we consider the possibility of enabling robust inversion of continuous forward processes $p \mapsto y$ by learning representations of $y$ that are bijectively aligned with $p$ while remaining insensitive to perturbations in $y$ caused by noise or model mismatch. We propose Twincher, a class of architectures based on stacks of structured diffeomorphic transformations and tailored adversarial training strategies that enable learning such bijective representations. We provide a public API for training and inference and empirically demonstrate the ability of the proposed architecture to efficiently learn bijective representations of synthetic systems, thereby enabling robust and efficient iterative inverse inference. Compared to a baseline inverse-modeling approach, the method exhibits improved data efficiency and robustness, providing initial evidence for the potential of bijective representation learning in robotics, vision, and physical AI.

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.

Nonlinear inverse problems often trade inexpensive but fragile first-order updates against curvature-aware methods such as Gauss-Newton and Levenberg-Marquardt, which obtain stronger directions by repeatedly solving Jacobian-based linearized systems. We propose a learned alternative: amortize local inverse geometry into a reusable reverse operator. Our framework learns a bidirectional surrogate, Deceptron, and deploys it through D-IPG (Deceptron Inverse-Preconditioned Gradient), an iterative solver that pulls residual-corrected measurement-space proposals back to latent space. The key mechanism is a Jacobian Composition Penalty (JCP), which trains the reverse Jacobian to act as a local left inverse of the forward Jacobian; its runtime counterpart, RJCP, measures the same inverse-consistency error along optimization trajectories. We prove that D-IPG is first-order equivalent to damped Gauss-Newton under local pseudoinverse consistency, with deviation controlled by composition error and conditioning. Across seven PDE inverse-problem benchmarks, D-IPG outperforms standard baselines, achieves 94.8% mean success across the six-problem reliability suite, and reaches comparable or better recovery quality at up to 77x lower inference-time solve cost on the main benchmarks.

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.