生成模型理论与方法

In many industrial applications, 1) tabular data is scarce and imbalanced and thus requires synthetic expansion; 2) input distributions drift between training and deployment (covariate shift); 3) validation sets often diverge from unseen test environments; or 4) standard generative models simply mimic outdated source distributions. This learning setting limits the stability of standard augmentation and adaptation pipelines. We generalize the task under such setting as the Augmented and Weighted Learning under Covariate Shift problem (AWL-CS). AWL-CS imposes two critical challenges on existing methods: 1) misleading generative guidance where models optimize for source similarity rather than downstream task relevance, and 2) structural instability of distributional density where reweighting mechanisms overfit to noisy validation signals. To tackle these challenges, we propose IGDPR (Invariant-Guided Diffusion with Prototype Reweighting), a unified framework that synergizes stable synthesis and structural adaptation: i) To achieve task-relevant generation, we steer the diffusion sampling process using invariant potentials to ensure synthetic samples align with stable decision boundaries rather than outdated correlations. ii) To ensure stable adaptation, we develop a prototype-based reweighting strategy that assesses sample reliability through structural clusters instead of isolated points, effectively filtering validation noise. Extensive experiments on real data demonstrate our method improves data quality by augmenting the most beneficial data for robust learning.

Bayesian inference increasingly uses informative but implicit priors represented only by samples, such as historical ensembles, simulator outputs, and pretrained generative models. The same computational problem appears in the test-time guidance task (generalized Bayes), where an explicit positive weight, e.g., an exponentiated reward, tilts an implicit prior. We develop a framework for source-space generalized Bayesian inference that combines inexpensive few-step prior transports with posterior stability guarantees. Specifically, we represent the prior using a one- or few-step improved MeanFlow (iMF) map and perform posterior sampling in its Gaussian source space. We establish Wasserstein error bounds between the exact and learned posteriors in terms of the joint population iMF and auxiliary-velocity loss, decomposed into training suboptimality and model-class approximation error. In the iMF source space, we adopt parallel tempering with preconditioned Crank-Nicolson updates and introduce a hybrid variant that incorporates split Hamiltonian Monte Carlo to improve sampling efficiency. Synthetic experiments show that the proposed framework can approximate posterior distributions accurately and efficiently, while CLIP-guided ImageNet experiments demonstrate its ability to steer a pretrained iMF image prior toward text-specified preferences.

Diffusion models are a powerful generative paradigm used across multimedia and scientific applications. Guided diffusion methods impose requirements on the generation by adding the gradient of a differentiable loss (the guidance function) as a drift term during inference. The weight of this drift (the guidance scale) is critical for the trade-off between data quality and requirement satisfaction. To achieve both of these goals, guided diffusion must resort to small guidance scales and lengthy sampling, incurring high computational costs. This work proposes learned end-to-end guidance schedules (LEEGS) to achieve these objectives with fewer sampling steps. LEEGS trains a time-dependent schedule by minimizing the guidance function over a small set of examples using stochastic gradient descent. Backpropagating through guided sampling is computationally expensive, so LEEGS uses an approximation of the gradient that cuts training time by a factor of 4. We evaluate LEEGS on diverse guidance tasks, including (a) image inpainting, (b) noisy image inverse problems, (c) face-ID-guided generation, and (d) forward and inverse PDE problems, outperforming baselines at equal budget (50 or 100 NFEs), or matching constant guidance with only 10% of the steps.

Topology is, by its nature and design, suited to structure that is nonlinear, multiscale, and nonstationary - however, within machine learning, its use remains largely confined to topological data analysis. We advocate that tools from low-dimensional topology which have remained almost exclusively contained within the domain of pure mathematics (such as Morse theory) offer a strong, complementary, and yet virtually unexplored perspective on the hidden structure of data-generating processes and learning tasks built upon them. Here we introduce concepts from cobordism theory and harness tools from discrete Morse theory to improve the performance of graph diffusion models through our pipeline MG-Diff. Further, we derive theoretical guarantees and sufficient conditions so that under a positive decision-gap, the Morse-theoretic tools and their application for induced diffusion guidance are stable under small perturbations. Finally, we illustrate the utility of discrete Morse theory in application to graph diffusion models for spatio-temporal graph forecasting and graph regeneration, and argue that these applications are only a small window into the part of what low-dimensional topology can offer to the field of machine learning.

Model collapse arises when generative models are trained on synthetic data produced by earlier models. The phenomenon has attracted considerable attention because of its societal and technical implications. However, previous studies have reached seemingly contradictory conclusions: replacing real data with synthetic data causes collapse (Shumailov et al.), yet accumulating real data alongside synthetic data can prevent it. For diffusion models, we study an intermediate regime typical of finite-budget pipelines: all past datasets and the real data are kept, but each new model is trained on a fixed-size sample from this growing pool, so the real fraction vanishes without any data being removed. Experiments on a 2D spiral dataset as well as the image benchmarks (MNIST, Fashion-MNIST, and CIFAR-10) show that replacement protocol degrades dataset rapidly as in the literature, whereas the fixed budget degrades only partially, sparing some features. A linear-response model of the multi-generational parameter dynamics, analyzed by stochastic recursion, confirms that the two protocols differ: some features will be fragile and lost within a few generations for both protocols, while some will be robust and preserved over practically unbounded horizons under the fixed budget protocol.

Inference-time scaling adapts pretrained diffusion models to new sampling tasks without additional training. Existing methods rely primarily on Monte Carlo sampling with more particles, yet are premised on the pretrained model being exact. In practice, data and training limitations make the model imperfect, and these methods inherit its error. More particles reduce Monte Carlo error but cannot remove the mismatch between the endpoint and the desired target or the error in tracking the prescribed probability path. We introduce the Energy-based Feynman-Kac Corrector (EBFKC), a framework for energy-based diffusion models that corrects these errors on the fly given a reference energy. We first derive Feynman-Kac dynamics that track a prescribed path exactly in the continuous-time population limit even when the model is imperfect, and approximate these dynamics using sequential Monte Carlo with variance-controlling guidance. To remove the endpoint mismatch, we use the pretrained energy as a surrogate along the diffusion path and progressively incorporate the discrepancy between the learned and target terminal energies. Experiments on Gaussian mixture models, particle systems, alanine dipeptide, and alanine tetrapeptide show that our method closely matches target distributions and molecular free-energy profiles under annealing and reward tilting, whereas standard inference-time scaling baselines retain substantial sampling errors.

Generating and predicting spatiotemporal physical fields from scarce measurements is challenging, as observations are insufficient to characterize a distribution over complete fields. This limits conventional data-driven diffusion models that rely on full-field datasets. We introduce PhysDEM, a physics-defined diffusion framework that combines governing equations with spatially sparse observations to generate multiple plausible fields. First, we construct a Gibbs target by reweighting a measurement-conditioned Gaussian reference with PDE residual energy. Second, we derive an exact conditional-mean identity that reduces denoising to supervised learning of the standardized energy-induced mean correction. Third, a physics-displacement probability flow cancels Gaussian reference terms and enables amortized sampling with changing measurements through Gaussian conditioning, without retraining. Experiments on synthetic PDE systems and real-world-informed applications demonstrate that PhysDEM supports coherent field recovery and efficient sampling while maintaining stable diagnostics under tested noise levels, illustrating its practical value for field assessment. To our knowledge, PhysDEM is the first physics-defined diffusion model enabling amortized spatiotemporal field inference without preassembled full-field datasets.

Uniform-state diffusion models (USDMs) can revise any token at any denoising step, which lets them correct their own mistakes, a key advantage over masked diffusion. Self-correction, however, requires both revising incorrect tokens and retaining correct ones, and we show that current USDMs lack the latter. Even under greedy-tail decoding, state-of-the-art USDMs (DUO, UDLM, and uniform-noise SEDD) keep revising 173--270 of 512 positions at every step, and these large, uncoordinated edits collapse sample diversity. A random-token corruption experiment traces this deficit to the models themselves: they reconstruct clean and corrupted tokens with nearly identical accuracy, even though clean tokens are easier targets. A decomposition of the validation NELBO shows that training barely rewards retention: incorrect predictions are heavily penalized at corrupted positions but almost free at clean ones. We propose Correct-Token Retention Regularization (CTR-Reg), a simple but effective auxiliary loss that trains the model to retain tokens left unperturbed by the forward process and requires no change to the sampler. CTR-Reg improves clean-token accuracy by 26.5 percentage points on average across six benchmarks, while leaving corrupted-token accuracy virtually unchanged, and its per-step revisions converge to only 3--11 positions. With just five greedy-tail steps, generative perplexity more than halves under CTR-Reg for all three models while diversity is preserved, and these gains hold across sampling budgets. Our results identify correct-token retention as a key missing ingredient for self-correcting diffusion language models, and demonstrate an effective fix.

Trajectory crossing remains a critical bottleneck in Flow Matching (FM), and previous works typically view these crossings from a theoretical optimization perspective causing velocity averaging. They attempt to address it indirectly by post-hoc distillation or endpoint coupling, without explicitly regulating the intermediate trajectories. In this paper, we introduce a new network learning perspective: crossing points inherently induce large local Lipschitz constants in the target velocity field, leading to two drawbacks. First, high Lipschitz constants correspond to high-frequency signals in the velocity field that neural networks struggle to fit due to spectral bias. Second, they also imply drastic velocity variations, leading to severe numerical integration errors in few-step inference. To alleviate this, we propose CoFlow, a framework that introduces the contrastive learning paradigm into FM to explicitly repel trajectories during training, thereby lowering the local Lipschitz constants of the velocity field. Specifically, we formulate CoFlow from a Stochastic Differential Equation (SDE) perspective by injecting a repulsive drift term. This drift actively guides the forward process of positive samples away from negative trajectories, effectively reducing the local Lipschitz constant. Furthermore, we derive an equivalent stochastic interpolant formulation from this SDE, providing a simple and tractable design space to control the influence of negative samples. Extensive experiments on ImageNet 256x256 demonstrate that CoFlow significantly reduces FID compared to standard FM in few-step inference (e.g., 20 steps), with no added training overhead. The code can be accessed at: https://github.com/HKUST-LongGroup/CoFlow

Sampling from discrete, unnormalized distributions without access to data is a challenging problem. Neural samplers offer a promising approach by training generative models from density evaluations directly. Despite recent progress, existing discrete neural samplers are prone to mode collapse, come without convergence guarantees when trained via fixed-point iterations, and are often tied to a specific reference process such as masked or uniform diffusion. In this work, we introduce Discrete Gibbs Iterative Neural Sampler, a fixed-point neural sampler that addresses these limitations, enabling efficient, scalable learning, substantially reducing mode collapse in practice. Our framework builds on masked diffusion and also extends to transport between pairs of distributions. We demonstrate that the resulting method scales effectively to high-dimensional systems, supports amortized sampling across different conditions, and enables accurate estimation of alloy phase diagrams.

There has been a proliferation of sampling algorithms based on Wasserstein gradient flows (WGF) and forward-only diffusion processes (FODP), often accompanied by theoretical guarantees of exponentially fast convergence to the target distribution. These guarantees are frequently interpreted as evidence that such methods can efficiently sample complex multimodal distributions, often supported by empirical results. In this work, we argue that this interpretation is fundamentally misleading. By invoking the Jordan-Kinderlehrer-Otto (JKO) scheme and Otto calculus, we establish that the canonical WGF sampling dynamics and overdamped forward diffusion share the same density evolution and therefore inherit the same metastability and slow-mixing phenomena long understood in nonequilibrium statistical physics. We analyze this family of samplers using two complementary tools — spectral analysis and mean first-passage time (MFPT) analysis — and show that well-separated multimodality can induce exponentially long mixing times associated with small spectral gaps and rare inter-mode transitions. For the commonly adopted log-linear annealing schedule studied here, we find that introducing intermediate distributions does not remove the exponential scaling of the total transport time. The limitation is structural rather than implementation-specific: purely local, gradient-driven transport mechanisms can require exponentially long times to transport probability mass across well-separated modes. We argue that this represents a fundamental limitation of WGF- and FODP-based sampling in their standard forms, and motivates future development of fundamentally nonlocal mechanisms for efficient multimodal sampling.

Counterfactual generation seeks to sample outcomes under a hypothetical intervention or decision using observational data collected under the factual assignment mechanism. We develop a flow-matching approach that combines a sample-split, doubly robust training objective with a learned coupling between observed source outcomes and target outcomes drawn from a fitted conditional outcome model. To enable finite-step generation, we leverage a score-corrected stochastic sampler based on a Gaussian-smoothed interpolation. Our main theoretical contribution is a coupling-sensitive KL bound for constant-step Euler discretization: the error is controlled by moments of the source--target displacement under the chosen coupling, rather than by global uniform regularity of the velocity field, and has near-linear dependence on the ambient dimension. We also establish finite-sample non-parametric guarantees for the learned velocity and score fields when both the conditional outcome model and the source-target coupling are estimated from data. These bounds separate approximation, coupling-replacement, nuisance-estimation, generalization, and Monte Carlo errors and, combined with the sampler analysis, yield an end-to-end guarantee for counterfactual generation. Experiments on synthetic and semi-synthetic image benchmarks support the coupling-dependent theory and show that, at finite discretization budgets, the stochastic sampler can outperform the corresponding deterministic ODE sampler.

Recent works on continuous diffusion for discrete data have demonstrated performance on par with comparable discrete diffusion models. However, these continuous counterparts lack key features that are essential to practical use as language models, namely variable-length generation and support for a key-value cache, and they still lag behind the frontier of autoregressive and discrete diffusion quality. In this work, we address these limitations. We do so by introducing a model parameterization that uses position-dependent noise schedules to define semi-autoregressive (SAR) continuous diffusion language models (DLMs). Together with efficient training and sampling algorithms, we call this framework Clock Diffusion, and we present two special cases of our method: block and sliding window generation. We then define ClockDLMs, a family of Gaussian DLMs based on sliding window Clock Diffusion that attain state-of-the-art diffusion likelihood bounds on OpenWebText, even beating the performant block SAR discrete diffusion models. ClockDLMs trained on TinyGSM also substantially outperform continuous baselines on the GSM8K benchmark and match and exceed comparable SAR discrete diffusion models. Finally, building on our parameterization, we propose more efficient samplers that we dub Cache Grab, which adapt techniques from accelerated inference in discrete diffusion, such as committing tokens whose probabilities exceed a confidence threshold and self-speculative decoding, further improving our models' quality and efficiency.

Post-training quantization for diffusion models increasingly exploits timestep, feature, and layer structure. While recent work has begun incorporating CFG structure into diffusion quantization, activation quantization still operates independently across conditional and unconditional coordinates, leaving cross-activation structure unexploited. We show that matched CFG activations form a strongly correlated two-dimensional source and that, under a fixed bit budget, the choice of branch coding basis materially affects quantization fidelity. Motivated by this observation, we introduce branch-space transform coding, which rotates matched CFG branches via an offline derived 2x2 orthogonal matrix, requiring minimal modifications to model parameters or the quantization pipeline. We further derive the Guidance-Correlation Branch Transform (GCBT), which jointly incorporates the CFG guidance direction and cross-branch second moments. Under an equal-rate quantization-noise surrogate, GCBT admits a closed-form per-layer solution without gradient optimization or angle search. Applied on top of existing diffusion PTQ methods, GCBT yields statistically significant fidelity gains in most evaluated comparisons with no statistically significant degradation, while leaving the underlying host quantization pipeline unchanged.

This work presents Diffusion Layer Integrated Gradients (DLIG), a token attribution method for diffusion language models (DLMs) that extends Integrated Gradients (IG~\cite{sundararajan2017axiomatic}) to arbitrary layers and denoising steps. DLIG attributes a DLM's progressive commitment to a self-generated or fixed completion for an input prompt. We establish direct correspondences between DLIG and the IG axioms of completeness, implementation invariance, linearity, and symmetry preservation. As a lightweight complement to interventional analysis, DLIG provides an inexpensive first check of mechanistic hypotheses across the denoising trajectory. We demonstrate this on word-sense disambiguation, multi-hop graph reasoning, and sentence infilling, revealing how DLMs draw on inputs across positions, layers, and denoising steps.

Discrete diffusion language models offer a compelling alternative to autoregressive generation for tasks demanding bidirectional reasoning and global constraint satisfaction. Yet they share a structural bottleneck: when decoding in parallel, each token is sampled independently from its marginal, severing the statistical dependencies among the tokens decoded together. Continuous diffusion language models avoid this by denoising a shared continuous state, but their denoiser sees only that state, so nothing ties it to a valid token configuration until it is finally decoded. To address this, we propose Hierarchical Continuous Diffusion Language Models (HC-DLM), which couple discrete token generation with a continuous latent trajectory in a single, principled denoising process, whose training objective is derived from a variational bound on the token likelihood. In contrast to recent methods that attach continuous context to a self-contained discrete chain, HC-DLM makes the latent the only persistent generative state: tokens are read out from it at every step and feed back as a scaffold for the next latent update. On structured reasoning (Sudoku), mathematical planning (Countdown) and language modeling (LM1B), HC-DLM improves over discrete and continuous diffusion baselines at matched model size, in puzzle accuracy on Sudoku and Countdown and in generative perplexity on LM1B. Project page: https://hc-dlm.github.io/.

Weather reanalysis products rely on computationally intensive numerical weather predictions followed by data assimilation that corrects the forecast toward observations. Recent advances in deep generative models offer a cheaper alternative that shifts much of this cost from inference to offline training. However, existing generative approaches have been evaluated on synthetic observations or under different datasets and evaluation schemes, making it unclear which design choices improve real-world data assimilation. We present the first controlled benchmark of generative data assimilation for single-time near-surface analysis from real weather station observations. Using 11,849 NOAA MADIS stations across the contiguous United States and four near-surface variables, we hold the dataset, observation operator, and deep learning architecture fixed, and measure spatial generalization at held-out stations. The benchmark compares the major design choices proposed for generative data assimilation, including diffusion versus flow matching, pixel versus latent-space formulations, and multiple inference-time conditioning strategies, against a classical 3D-Var baseline. The benchmark reveals three conclusions. First, the best generative methods outperform 3D-Var (35.7% vs. 33.3% RMSE improvement over ERA5), although 3D-Var receives the ERA5 field at the analysis time as its background and the generative methods receive none. Second, full-gradient guidance consistently outperforms stop-gradient and initial-noise optimization. Third, other choices provide little measurable benefit: diffusion and flow matching perform nearly identically under matched conditions, and latent-space variable mixing does not help. Both advantages widen when stations are sparse. Together, these results identify which components of generative data assimilation improve spatial generalization in near-surface analysis.

Amortized simulation-based inference (SBI), which is trained on radiative-transfer simulators, recovers exoplanet atmospheres accurately on synthetic James Webb Space Telescope (JWST) spectra but collapses when it comes to real reduced spectra. The flow posterior collapsed on real WASP-39b (importance-sampling effective sample size (ESS) = 1, best-fit \c{hi}2/N = 301), and such a failure is usually blamed on missing the forward-model physics, but ruling these levers out with nested sampling first (a temperature gradient, SO2 opacity, and a high-fidelity opacity set) leaves the fit unchanged, meaning the collapse is not from the simulator misspecification but instead from a missing latent, the planet radius. To fix this, we build MIRAGE, a radius-augmented flow-matching posterior calibrated against an independent nested-sampling reference with importance sampling and an optimal-transport map, which yields a physical and literature-consistent retrieval of real WASP-39b (with \c{hi}2/N from 301 to 0.06). This same method transfers unchanged across two instruments and three real JWST targets, including one spectrum self-reduced end-to-end from raw Mikulski Archive for Space Telescopes(MAST) data. The lesson is cross-domain, as a latent the simulator encodes but the inference omits can masquerade as misspecification.

Flow matching models excel at generative modeling, and many downstream applications require their samples to satisfy prescribed constraints, such as observed measurements and physical laws. However, existing constrained samplers often face a trade-off: enforcing constraints can substantially displace samples from the pretrained data distribution. To address this trade-off, we introduce MintFlow, a training-free constrained sampling framework that formulates constraint enforcement as a minimal intervention on the pretrained flow trajectory. MintFlow seeks the minimal perturbation of an intermediate flow state such that its subsequent evolution under the pretrained flow field satisfies the target constraint. By minimally perturbing the flow state while keeping the pretrained flow field unchanged, MintFlow enforces the constraint while minimizing unnecessary deviation from the pretrained distribution. An adjoint formulation yields a closed-form expression for this perturbation, eliminating expensive iterative optimization. Furthermore, MintFlow adaptively selects the intervention time to balance the required perturbation magnitude with its amplification by the remaining flow. Across a range of tasks in generative vision and physical system modeling, MintFlow achieves competitive constraint satisfaction while preserving the pretrained generative distribution substantially better than state-of-the-art constrained methods.

Single-image HDR reconstruction requires inferring missing detail while preserving the visible content of an LDR image. Differences in sensor dynamic range and exposure cause LDR images to lose varying amounts of information in shadows and highlights. We present ExpandDiff, a conditional diffusion pipeline that jointly reconstructs clipped shadows and highlights. To account for this variation, we introduce Dynamic Clipping Synthesis (DCS), which randomly samples shadow and highlight clipping percentiles when constructing training inputs from HDR targets. A pixel-space diffusion model guided by spatially-adaptive normalization then predicts perceptually encoded HDR through a bounded output head, reconstructing both clipping directions in one sampling trajectory. On the SI-HDR benchmark, ExpandDiff variants improve HDR reconstruction accuracy by 3.43 dB in PU21-PSNR over the strongest evaluated competing method, and by 7.34 dB under two-sided clipping. The code and supplementary material are available at https://memreandiran.github.io/expanddiff/.

Generative models are transforming Bayesian computational imaging, yet the field still lacks physics-aware foundation models. Current practice falls into two camps. Large foundation image models are deployed as plug-and-play priors with zero-shot approximate likelihood guidance, which introduces significant bias and computational cost. Physics-aware generative models avoid this bias, but each is tied to a specific dataset, task and instrument. We introduce BAM (Bayesian Anything Model), a lightweight foundation model for few-step, physics-aware posterior sampling that generalises robustly to unseen data and tasks, zero-shot or with minimal finetuning. BAM upgrades the operator-conditioned Reconstruct Anything Model (RAM) backbone (Terris et al.) into a conditional flow map, so instrument physics is specified at inference time rather than fixed during training. BAM has just 36M parameters and is pre-trained jointly on large image corpora and libraries of forward operators. A single network then draws posterior samples in a few steps, with no likelihood approximation and no guidance weights to tune. Across linear inverse problems on FFHQ, AFHQ, LSUN, DIV2K and the Kohler camera-shake benchmark, BAM outperforms in just 3 steps both specialised models and leading zero-shot methods in sample quality, at a fraction of their computational cost. BAM gives the community an accessible entry point to generative computational imaging, lowers the economic and environmental cost of training imaging models, and opens a new path for research on physics-aware Bayesian computational imaging. Official page: https://bayesian-anything-model.github.io/

Density estimation is a fundamental problem in statistics and machine learning. In this work, we introduce BayesNDE, a neural density estimator based on Bayesian generative modeling. BayesNDE learns a Bayesian generative model and evaluates its density without requiring invertible networks or Jacobian-determinant computation. For each observation, it infers a sample-specific latent posterior to construct an adaptive proposal that focuses computation on regions contributing most to its density. Bridge sampling then combines samples from this proposal with separate posterior samples to estimate the density. Experiments on nonlinear and multimodal synthetic datasets show improved estimation of density values and better recovery of the density structure compared to the state-of-the-art neural density estimators. Applications to real-world datasets further demonstrate improved anomaly detection. Together, these results highlight BayesNDE as a flexible and effective neural density estimator, demonstrating how posterior inference can turn generative models into tools for density estimation. The code and tutorials are available at https://github.com/liuq-lab/BayesNDE.

We present PrefPI (Preference-Guided Policy Iteration), an iterative framework for steering pretrained generative robot policies using only relative preferences over self-generated trajectories. Unlike prior preference-learning methods that primarily sharpen modes already represented by the policy, we study steering beyond the initial effective support, where desired behaviors are rarely or never observed under the initial policy. Our key idea is to formulate preference learning as preference-conditioned generative modeling: preferred trajectories define a conditional distribution, whose density ratio with the broader behavior prior provides an implicit preference signal amplified by classifier-free guidance (CFG). Repeating this preference-conditioned modeling and guidance step yields a form of preference-guided policy iteration, turning incremental improvements toward previously inaccessible behaviors. Across diffusion policies and the PI0.5 flow- matching VLA in simulation and the real world, PrefPI produces substantial behavioral shifts with limited feedback. In particular, PrefPI increases object transport height from 10.7 cm to 19.8 cm on real hardware with only 150 preference-labeled trajectories.

We establish near-linear accuracy bounds for the classical Moreau--Yosida unadjusted Langevin algorithm (MYULA). The target is $π\propto e^{-f-g}$, where $f\in C^2(\mathbb{R}^d)$ is $m$-strongly convex with Lipschitz gradient and $g$ is convex and globally Lipschitz. Under an explicit parameter-dependent step-size condition, we bound the invariant-measure bias relative to the Moreau-smoothed target by $\widetilde O(h)$, with only logarithmic dependence on the inverse smoothing parameter in the error coefficient. Combining this estimate with the Moreau approximation bias and Wasserstein contraction gives $\widetilde O(\varepsilon^{-1})$ iterations to make the $N$th-iterate law $μ_N$ satisfy $\sqrt m\,W_2(μ_N,π)\le\varepsilon$, for fixed model parameters and initialization. We bound the stationary error directly, without assuming third derivatives or a Lipschitz Hessian. Each iteration uses one gradient evaluation and one exact proximal evaluation. The key idea in our analysis is to convert a second-order stationary residual into a Wasserstein bound using a Poisson-based estimate.

Exact stochastic equations for non-equilibrium dynamics are rarely accessible. We show that the long-time evolution of stochastic dynamics can be predicted from configuration pairs at a fixed short time lag, without knowledge of the equation of motion. Generative diffusion models learn the finite-time transition kernel from these pairs, and iterating it propagates the dynamics far beyond the training lag. For two-dimensional Model B, the diffusive dynamics of a conserved order parameter, the learned kernels reproduce dynamic critical scaling and self-similar $t^{1/3}$ coarsening. Agreement with direct simulations persists on lattices twice the largest training size and for initial ensembles absent from training. For driven colloids in a periodic optical potential, ten minutes of measured trajectories suffice to predict the particle current and mean passage time over the next twenty minutes within experimental uncertainty. Short-time observations thus contain the information needed to predict emergent non-equilibrium dynamics at much longer times.

Diffusion large language models (dLLMs) generate text by denoising a sequence or successive blocks, allowing several tokens to be revealed in parallel. Reinforcement learning with verifiable rewards (RLVR) reuses terminal feedback across these decisions, even as their conditioning context changes. We propose stepwise risk-sensitive GRPO (StepRS-GRPO), which varies the risk coefficient of the group-advantage transformation across denoising states while retaining the underlying trainer. For binary rewards, we show that this transformation is exactly a prompt- and state-dependent rescaling of centered outcome advantages. A capability-based calibration suggests a coefficient scale, while endpoint and interpolation ablations guide schedule selection. Across multiple dLLM backbones and mathematical reasoning benchmarks, StepRS-GRPO improves both pass@1 accuracy and pass@k coverage over centered GRPO, while increasing answer diversity. In our ablation studies, mass-matched controls support the contributions of state allocation and schedule direction, and the gains persist after matching the root mean square (RMS) of the advantages to that of centered GRPO. Reasoning-trace diagnostics further show that the diversity gains from StepRS-GRPO extend beyond final-answer strings.

One-step generative models construct a static generator through iterative training-time transport. Existing transport objectives primarily assess distributional motion, although a neural generator needs to realize the requested sample displacements jointly through shared parameter updates. The training-time construction raises the question: once training becomes the iterative process that constructs the final one-step map, what to optimize: the next distributional move, or the route by which the finite generator learns the final map? To address the question, we introduce Training Dynamics Action (TDAction), which selects transport targets according to local shared-parameter realization cost while retaining a prescribed level of distributional progress. We formulate the cost as a soft-terminal control problem and derive a closed-form Batch Tangent Action-to-Go value that accounts for parameter effort and terminal mismatch. The criterion captures cross-sample interactions omitted by independent pairwise costs; under isotropic mobility, the criterion agrees with quadratic Euclidean assignment for deterministic balanced couplings. Randomized tangent probes provide a low-rank implementation that constructs shared detached targets without adding an inference-time trajectory. Controlled studies examine the relationship between generator geometry, transport selection, and realized local action. On ImageNet $256\times256$, TDAction attains an FID below $1.1$ without distillation.

We introduce Grand Canonical Generators (GCG), a generative framework that extends Boltzmann generators to the grand canonical ensemble. We present two designs. The first conditions a variable-size generative model on the chemical potential, sampling particle number and configuration jointly. The second factorizes the grand canonical distribution into a particle-number distribution and the corresponding canonical Boltzmann density. This factorized formulation can use any existing Boltzmann generator for the canonical component, encodes the known linear chemical-potential dependence analytically, and yields a tractable likelihood that supports self-normalized importance sampling (SNIS). Empirically, GCG accurately reproduces grand canonical observables on a Lennard--Jones fluid and methane adsorption in a zeolite, demonstrating generalization across chemical potentials and correction via SNIS and grand canonical Monte Carlo.

Recent joint audio-video generative models can synthesize realistic videos with synchronized sound, but typically generate audio as a single mixed track. This limits source-level control and differs from practical audiovisual workflows, where speech, music, sound effects, and ambient sounds are represented as separate editable tracks. We introduce Soundwich, a training-free framework that transforms a frozen joint audio-video flow-matching model into a generator of multiple synchronized, independently editable audio stems coupled to a shared video. Soundwich generates separate audio stems with explicit control over their temporal activity. To keep separately generated sounds coherent, we introduce a shared scene representation that communicates global audiovisual context across stems while preserving their source-level separation. We further route cross-modal interactions between each audio stem and its corresponding visual source, improving audiovisual consistency. The resulting stems remain synchronized with the video and can be independently retimed, muted, replaced, or remixed. Experiments and human evaluations show improved temporal control, source separation, and naturalness, while enabling flexible source-level editing within coherent audiovisual generation. Code is available at https://github.com/CodyNing/Soundwich.

Identifying which speaker a listener is attending to in a noisy room — the cocktail-party problem — is the missing ingredient for next-generation hearing aids and brain-computer interfaces: it tells the device whose voice to amplify. Auditory attention decoding (AAD) reads this answer from EEG, but the literature splits into disconnected pieces: directional-AAD classifies side but does not map side to stream; regression-based source-AAD ranks candidate streams by a single Pearson correlation that is intrinsically noisy at the 1-5 s windows real devices need; and envelope reconstruction has no native AAD rule. We argue the right object is not any single statistic but the conditional likelihood of the attended envelope given EEG, and we make this practical with NEUROTOKEN: a single network whose three heads share one EEG front-end, with a conditional flow-matching head (ATTUNEFLOW) that scores candidates by an integrated velocity-residual likelihood ratio. Two inference-time ensembles — QUADTRACK (four complementary statistics) and ENV-FLOW (z-normalised QUADTRACK+ATTUNEFLOW) — absorb per-statistic failure modes for free. On KU Leuven, DTU, and NJU at 5 s, ATTUNEFLOW lifts per-segment source-AAD by 9%-16% over the strongest non-generative baseline and shrinks across-subject variance by ~3x; trial-level fusion exceeds 93% on two of three datasets. In parallel reproductions we show that canonical 95-97% direction-AAD numbers collapse by 17%-45% under a strict trial-disjoint protocol, clarifying both the true ceiling and why a likelihood-based formulation is needed.