Stochastic Differential Equations

We study the problem of recovering the parameters of a multivariate Ornstein-Uhlenbeck (OU) process from steady-state observational and interventional data. In many applications, such as large-scale gene perturbation experiments, only stationary "snapshot" measurements are available, making standard stochastic differential equation estimation methods that rely on time-series trajectories inapplicable. We first establish an identifiability result: one intervention per strongly connected component (SCC) of the drift graph suffices to recover all OU process parameters generically up to a global scaling factor. This holds provided that the SCC condensation graph is connected with a single root and certain spectral nondegeneracy assumptions hold. We propose a recursive learning algorithm that orders SCCs topologically and, for each component, isolates its marginal dynamics and solves a linear system derived from the steady-state moment equations, leveraging parameters recovered for upstream components. Building on this theoretical foundation, we propose a regularized least-squares estimator that jointly minimizes residuals of the steady-state mean and covariance equations across observational and interventional data. Experimental results validate our theoretical findings in recovering parameters of the underlying OU process.

This letter presents a unified formulation and a controlled numerical comparison of generative-model approaches to the nonlinear filtering problem. Under this formulation the analysis step is realized by a transport of the forecast distribution to the posterior, the approaches differing only in how that transport is selected and learned. We derive three new filters, based on stochastic interpolants, their deterministic flow-matching limit, and Schrödinger bridges realized through forward--backward SDEs. We develop a two-stage tuning procedure that separates the training of the generative model from its online refinement. The resulting methods are compared against the optimal transport filter (OTF), the Knothe--Rosenblatt filter (KRF), the sequential importance resampling (SIR) particle filter and the ensemble Kalman filter (EnKF), in terms of accuracy, computational time, and sensitivity to ensemble size and state dimension. The results indicate that every generative filter resolves multimodal posteriors that the EnKF and SIR do not, that no single generative framework dominates, the preferred method being set by the available online budget and ensemble size, and that the filters differ in the regularity of the particle trajectories they produce.

Pretrained score-based diffusion models provide strong unconditional priors, yet enforcing measurement or physics consistency in inverse problems is often handled by heuristic guidance, intermittent projections, or task-specific conditional training, with limited guarantees of feasibility at the end of inference. We propose terminal-conditioned inversion for score-based SDE priors. Given a frozen Score-SDE prior and a task-defined terminal feasibility specification, we construct an associated backward stochastic differential equation whose adapted solution defines a principled inverse map from the terminal requirement to a prior state at a chosen noise level. Under standard regularity conditions, we establish existence and uniqueness of the adapted solution and obtain terminal consistency by construction. We further develop a practical neural BSDE solver that composes arbitrary pretrained diffusion priors with domain constraints without modifying the score-defined coefficients, producing an anchored prior state that enables neighborhood sampling for uncertainty characterization. Experiments on toy datasets validate stable terminal-conditioned inversion and distributionally consistent neighborhood sampling. As a real-world case study, we apply the framework to sparse-view CT reconstruction and achieve improved reconstruction quality over representative training-free baselines while satisfying strict measurement feasibility under the prescribed terminal specification. Project is available in: \href{https://laplacelab.github.io/BSDEDiffusion/}{https://laplace.center/icmlbsdeI/}

In many settings involving stochastic differential equations, including in diffusion based generative AI, our aim is to accurately generate samples from a terminal distribution. Typically, this is done by generating i.i.d. samples of diffusion paths. Given a fixed simulation budget, a reasonable way to gain efficiency may be to instead generate a tree of paths through appropriately split partial paths. This suggests improved performance, but one worries about the injected dependence. In this paper, we study this issue comprehensively. With Kolmogorov-Smirnov distance as a measure of accuracy, we identify the limiting errors of the associated empirical distributions as the simulation budget increases to infinity. We characterize a splitting strategy motivated by a corresponding asymptotic optimization problem. The theoretical results bring out the elegant underlying structure in the problem. Practical implementation involves two phases, an initial estimation phase and a final inference phase. Overall, we observe a 10-25% improvement in mean error over i.i.d. samples in many settings. In an exploratory CIFAR-10 study, our method reduces the maximum mean discrepancy by 8-13%.

Computed tomography (CT) and positron emission tomography (PET) provide complementary anatomical and functional information for cancer diagnosis and treatment planning. However, the widespread use of PET is limited by high radiation exposure, elevated costs, and restricted availability. To address these limitations, deep learning-based CT-to-PET translation has emerged as a promising approach for synthesizing PET-like information directly from CT images, although accurately modeling the large cross-modal gap remains challenging. In this work, we propose a 3D CT-to-PET translation framework based on latent Brownian Bridge Diffusion (BBDM). The method consists of two stages. First, a Variational Autoencoder (VAE) is trained on paired CT-PET patches, integrating contrastive learning to improve latent alignment between anatomical and metabolic representations. Second, a BBDM is trained in the latent space to translate CT latent representations into their corresponding PET counterparts. The translated PET latents are then decoded and stitched to reconstruct the final 3D PET volume. We evaluate the proposed approach on two publicly available datasets. Quantitative results based on image fidelity and lesion-level PET-specific metrics demonstrate improved performance compared with competing methods. In particular, the proposed approach improves PET signal fidelity, better preserves clinically relevant uptake patterns, and shows improved performance in preserving small-lesion metabolic activation, paving the way for virtual imaging applications.

Stochastic gradient descent (SGD) admits diffusion approximations that replace the complicated randomness of stochastic gradients by Gaussian noise, providing a powerful tool for understanding its dynamics and long-time behavior. We investigate whether an analogous approximation principle holds for optimization over probability measures, where the objective is a functional defined on the Wasserstein space P2. The nonlinear geometry and infinite-dimensional nature of P2 prevent a direct extension of the classical Euclidean theory. Using Lions differentiability, we lift the problem to a linear Hilbert space, where higher-order differential calculus becomes available. We then construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. By exploiting this moment matching through higher-order Taylor expansions, we show that the Gaussian approximation captures the SGD dynamics with second-order weak accuracy. Our result provides a rigorous foundation for replacing sample-driven randomness by analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

We introduce an amortized neural sampler that combines operator learning with flow methods for sampling. It maps SDE coefficient functions to pushforwards from a reference measure to the invariant measures, enabling efficient sampling across families of stochastic differential equations. Our framework shifts traditional sampling cost to an initial training phase, after which new SDE instances require only one encoder pass and a few ODE solver steps, independent of mixing time. To handle problems in high dimensions, we use Lagrangian trajectory sensors for the coefficient functions and cross attention in the architecture. We also theoretically establish the expressivity and resolution invariance of our framework. Experiments on 1D and 2D SDE families show competitive accuracy with substantial speedups over MCMC in regimes with slow mixing, transfer across sensor counts, and demonstration results on a 64D interacting particle SDE where traditional grid approaches are infeasible.

We study the large-depth behavior of residual networks whose weights are correlated across layers at initialization. Our results confirm and extend a conjecture of Marion et al. [2025], according to which correlated initializations should interpolate continuously between the Brownian stochastic differential equation arising from independent initialization and the ordinary differential equation arising from perfectly correlated initialization. When the initialization is obtained from the application of a feature function to a stationary Gaussian sequence with regularly varying correlation, we prove that there exists a unique critical scaling such that the infinite-depth limit is the solution of a Young differential equation driven by a Hermite process. Hermite processes reduce to the fractional Brownian motion if the feature function generating the initialization has Hermite rank one, which is the case for the identity function, for example. We show that the critical scaling and asymptotic limit are uniquely determined by the decay of correlations together with the Hermite rank of the feature function. Consequently, the correlation structure and Hermite rank of the initialization represent meaningful hyperparameters in the asymptotic regime. By contrast, under finite-variance iid initialization, the asymptotic driver is universally Brownian up to normalization regardless of the choice of distribution. Our proofs rely on a collection of novel results establishing a robust stability theory for Young differential equations in Banach spaces.

Speculative decoding losslessly accelerates autoregressive language models by verifying multiple draft tokens in parallel. Diffusion-based drafters further reduce proposal latency by predicting an entire token block in parallel, but their position-wise distributions are marginal rather than conditioned on tokens selected along each draft path. Existing recurrent correction incorporates causal information along a single draft chain, whereas diffusion-based tree construction broadens candidate coverage without carrying this correction along individual branches. We introduce DARTree, a training-free speculative decoding method that extends a pretrained AR correction head from chains to trees. DARTree first constructs a fixed-width candidate tree by expanding and scoring all nodes at each depth in a single batch, and then only applies best-first pruning to select the verification tree, decoupling AR-head inference from sequential heap operations. Across seven math, code, and chat benchmarks, DARTree achieves the highest average acceptance length and speedup in all four model--temperature configurations, accepting up to 12.97 tokens per verification round, 98.6% more than DFlash and 27.9% more than Domino in the same setting, and reaching up to 9.73$\times$ lossless speedup over locally measured autoregressive decoding.

This paper studies inverse sampling for Lévy-driven generative models from the perspective of Markov generators. Unlike conventional diffusion models, Lévy-driven dynamics involve infinite jump activities, which makes their reverse process nonlocal and difficult to characterize using score information alone. We address this challenge by analyzing the forward and reversed generators. It is derived that the reversed jump component generally becomes a state-dependent Markov jump process governed by a nonlocal density ratio. This observation motivates a structured reverse sampler that decomposes the dynamics into diffusion, small jump, and large jump components. Based on this characterization, we develop a computationally tractable sampler for a class of isotropic linear Lévy SDEs with symmetric $α$-stable jump components. For the jump component, the neural network is used only to amortize the rate of large jump activities, while jump amplitudes are generated from analytically derived conditional distributions, which improves interpretability and controllability. Efficient implementation techniques are further introduced under this setting to avoid expensive high-dimensional integration and sampling. The sampler is further adapted to approximate observation-guided sampling and applied to OFDM-SISO channel estimation under mixed Gaussian and impulsive noise. Simulations show robust estimation performance with a favorable tradeoff between complexity and performance.

In this paper we provide Monte Carlo and deep neural network approximations for stochastic representations of solutions to linear elliptic partial differential equations with constant diffusion, drift and killing. Building on the modified Walk-on-Spheres algorithm of Beznea et al. (arXiv:2209.01432), we introduce Monte Carlo estimators that explicitly incorporate sampled random times arising in the analyzed stochastic representations. We establish uniform error bounds for these estimators and show that, under suitable assumptions, a prescribed approximation accuracy is achieved with sample complexities growing at most polynomially in both the inverse accuracy and the problem dimension. Furthermore, we prove a deep neural network approximation result for the stochastic representations. Assuming suitable neural network representations of the boundary data and the distance function to the boundary, we use the constructed Monte Carlo to design deep neural networks that approximate the representation uniformly with a number of parameters growing at most polynomially in the inverse accuracy and the problem dimension. These results extend previous complexity analyses to a broader class of elliptic equations involving drift and killing.

Computational models of blood clotting improve understanding of thrombus formation, but their clinical application remains limited because many model inputs are difficult to measure and patient-specific data are often sparse. We present a computational framework based on latent neural differential equations that infers unknown model parameters from sparse measurements and forecasts thrombosis progression. We demonstrate the framework using data generated from a multiphysics blood-clotting model in which clot growth is governed by the coagulation cascade and diffusion. Four known biochemical inputs (fibrinogen and factors IX, VIII, and V), together with sparse early clot-size observations, are used to infer the tissue-factor parameter and predict subsequent clot growth. We compare seven probabilistic methods: stochastic neural ordinary differential equations (SNODE), stochastic neural functional differential equations (SNFDE), a latent neural-process baseline, a monotone probabilistic deep ensemble, empirical trajectory retrieval, PCA-ridge Gaussian posterior, and Gompertz-curve retrieval. SNODE achieved the best performance in inferring the unknown input and forecasting future clot-growth trajectories. SNFDE performed similarly and consistently outperformed the other non-differential models. Prediction accuracy improved as more observations became available, whereas longer forecasting horizons increased uncertainty and decreased accuracy. Latent neural differential equations thus effectively combine parameter inference and clot-growth forecasting from sparse measurements, providing a promising foundation for personalized thrombosis modeling.

Stochastic gradient descent for a loss function discontinuous across lower dimensional manifolds is analyzed by studying its differential equation limit.

Score-based diffusion models are typically formulated using continuous-time stochastic differential equations and measure-theoretic stochastic calculus. In this paper, we develop a hyperfinite formulation of score-based generative modeling within the framework of Nonstandard Analysis. Starting from an internal diffusion process on a hyperfinite grid, we derive the associated infinitesimal generator and establish its correspondence with the classical Fokker--Planck equation. We then obtain a hyperfinite backward-mean identity that yields the reverse-time drift and provides a constructive derivation of the reverse-time SDE. Building on these results, we show that minimization of an internal score-matching objective recovers the score function required by the reverse-time dynamics, thereby connecting score estimation with generative sampling directly at the hyperfinite level. Under suitable assumptions, we further derive a hyperfinite Girsanov formula and establish a relationship between likelihood optimization and Fisher-divergence objectives. Finally, we analyze the second-order consistency of the hyperfinite dynamics and show that the leading correction term depends explicitly on the fourth moment of the increment distribution, with the Gaussian value $κ=3$ eliminating the leading dispersion contribution. Taken together, these results provide a unified hyperfinite framework for diffusion-based generative modeling--while laying foundations for further extensions--that links discrete grid dynamics, reverse-time diffusion, score matching, and likelihood-based formulations within a common nonstandard setting.

Myocardial strain analysis of cardiac magnetic resonance (CMR) images provides an important tool for evaluating cardiac function. However, current techniques require either human-adjusted post-processing with suboptimal regional accuracy, or specialized acquisitions with limited availability. In this paper, we propose to leverage the power of generative models to synthesize high-quality motion-derived strain values from routinely acquired CMR sequences. Specifically, we develop a novel Brownian bridge diffusion model in motion space to learn the probabilistic mapping between standard CMR motion estimated from widely adopted registration methods and highly accurate motion provided by advanced strain imaging techniques. To promote the fidelity of anatomical structure in the generation process, our model is conditioned on the corresponding CMR images. We validate our method on large-scale multi-center CMR datasets including subjects of paired standard cine CMR and advanced strain imaging acquisitions. Experimental results demonstrate that our framework significantly improves the accuracy of motion prediction and strain analysis from standard CMRs compared to existing learning-based approaches. Our research represents a new paradigm for potentially developing cost-effective, clinically deployable AI tools for cardiac function assessment with enhanced strain accuracy in busy clinical workflows. Our code is publicly available at https://github.com/Rishov-MIA/Brownian-Bridge-strain-analysis.

Symmetry is central to modern machine learning and physics: invariances and equivariances improve sample efficiency, robustness, and out-of-distribution generalization, while symmetry principles guide scientific modeling. Yet for stochastic dynamical systems the relevant continuous symmetries are rarely known, and symmetry discovery for SDEs has remained essentially unexplored. We introduce \textit{LieStoNet}, an end-to-end, \emph{template-free} framework for discovering Lie-point symmetries of SDEs directly from spatiotemporal trajectories, without prespecifying symmetry groups, templates, or canonical coordinates. Building on the seminal SDE Lie-symmetry theory of Gaeta and Quintero (1999), which formalizes Lie-point SDE symmetries and their relation to Fokker-Planck symmetries, LieStoNet learns neural surrogates for drift and diffusion from increments, then learns projectable generators by enforcing the SDE determining equations, separately regularizing for closure under Lie brackets, adherence to the Lie algebra axioms (bilinearity, antisymmetry, Jacobi), and a non-redundant independent basis. The surrogate also defines an associated Fokker-Planck equation, enabling optional discovery of its Lie-point symmetries in parallel. Across multiple canonical SDEs with known analytic symmetries, LieStoNet recovers generators consistent with the ground-truth symmetry algebra, providing interpretable symmetry discovery for noisy dynamics. Code is available at \href{https://github.com/sumit-sinha-seas/LieStoNet_Final.git}{this link}.

Local-stochastic volatility (LSV) combines vanilla marginals with richer smile dynamics, but calibration requires a slow, noisy and sequential McKean--Vlasov fixed point. We learn a projection-consistent operator for the calibration triple. Given finite quotes and a stochastic-volatility (SV) backbone, it jointly returns an implied-volatility surface subject to static-arbitrage constraints, its Dupire local volatility, LSV leverage and the conditional moment required by the projection identity. Starting from option-price marginals, we derive a division-free Dupire residual in log-implied-variance coordinates and a quotient Fokker--Planck equation after Gyöngy projection. Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) implementations enforce quote fit, static-arbitrage, Dupire and projection constraints. For the witness-augmented residual system, we prove conditional identification and empirical consistency under LSV existence and inverse residual stability. In controlled synthetic tests, forward-start and cliquet errors differ from a particle method by 0.1 and 0.2 percentage points, while calibration latency falls from 98.5 to 0.6 ms. Compared with the tested baselines, local-volatility root-mean-square error (RMSE) falls by 36% and leverage RMSE by 7-16%. These results support amortizing the LSV fixed point: the expensive solve moves offline, while online calibration reduces to a single projection-consistent operator evaluation.

This paper studies the policy gradient update for a multi-arm bandit problem in diffusion environment that is described by a stochastic differential equation (SDE) under the continuous-time reinforcement learning framework by Wang et al. (2020), Jia and Zhou (2022b). With the logit parameterization for the stochastic policy, we show that it converges almost surely to the optimal arm under an arbitrary constant learning rate. Furthermore, we derive the non-asymptotic regret upper bound when the constant learning rate is below a time-invariant threshold; and the regret bound has order $O(\log T)$. We improve the analysis in Lattimore (2026a) for the same SDE by constructing a novel Lyapunov function and demonstrate the transparency of analyzing policy gradient using the tools in SDEs. In addition, the same Lyapunov function is also helpful in analyzing the discrete-time policy gradient algorithm.

Neural stochastic differential equations (SDEs) have emerged as powerful tools for learning noisy or stochastic dynamics directly from data; however, existing approaches largely assume uncoupled and continuous noise, limiting their applicability to realistic stochastic drivers, and often scale poorly in time, requiring expensive autoregressive training. To address these limitations, we propose Neural Kolmogorov Equations (NKEs), a deterministic, infinite-dimensional reformulation of Neural SDEs based on the Kolmogorov Forward equation, transforming the learning problem from modelling individual stochastic trajectories to modelling the evolution of probability densities. NKEs learn general Lévy-type stochastic forcing directly through the operator structure of the KFE, and enable parallel-in-time training via a Lagrangian Galerkin projection and operator splitting. We evaluate NKEs on several stochastic benchmarks, including systems with coupled noise and jump processes, and verify that NKEs provide flexible models that accurately recover deterministic and stochastic dynamics with competitive predictive accuracy and improved training efficiency. Code and pretrained models will be released.

Over the past few years, diffusion-based Schrödinger bridge models have been proposed to approximate optimal transport dynamics between two prescribed boundary distributions, with successful applications to generative modeling. More precisely, these methods aim to estimate a path measure whose initial and terminal marginals match the two boundary distributions, while minimizing the Kullback-Leibler divergence with respect to a reference Markov process. In this work, we consider the generalized Schrödinger bridge problem, in which the reference process is a twisted Brownian motion, that is, a Feynman-Kac transform of a Brownian motion induced by a time-dependent differentiable potential. Building on the Iterative Markovian Fitting (IMF) paradigm, and in particular on its special case Diffusion Schrödinger Bridge Matching (DSBM), which corresponds to the zero potential case, we introduce Twisted Schrödinger Bridge Matching (TSBM), a diffusion-based method designed to handle both continuous- and discrete-time potentials. Unlike previous approaches, TSBM provides a rigorous extension of the IMF scheme to the generalized Schrödinger bridge problem. This derivation leads to a new bridge-matching loss that depends explicitly on the gradient of the potential and recovers the DSBM objective when the potential vanishes, yielding improved performance. We further introduce trajectory-based variance-reduction techniques that substantially stabilize optimization and may be useful beyond the present setting. Finally, we empirically demonstrate the benefits of TSBM for trajectory inference across increasingly high-dimensional settings, including crowd navigation and single-cell data. Code available at https://github.com/maxencenoble/twisted-sb-matching.

We address fundamental challenges in representing and computing $\mathbb{R}^{d}$-valued predictable square-integrable processes over $[0,T]$, collected in the space $\mathcal{H}^2_T(\mathbb{R}^{d})$. These processes are central to continuous-time stochastic control, reinforcement learning, and mathematical finance. Although Wiener-chaos expansions offer strong theoretical tools, traditional computational methods are hindered by the need for large chaos dictionaries and high-order iterated integrals. To overcome these obstacles, we introduce NeuralChaos -- a neural operator architecture that produces elements of $\mathcal{H}^2_T(\mathbb{R}^{d})$ using only finitely many evaluations of the driving Brownian motion, while preserving predictability and square-integrability. We prove that NeuralChaos is dense in $\mathcal{H}^2_T(\mathbb{R}^{d})$ and achieves the best $N$-term chaoslet approximation rates for compressible and Malliavin--Sobolev regular processes. Moreover, compressibility is shown to be typical for processes from $\mathcal{H}^2_T(\mathbb{R}^{d})$ under non-degenerate sub-Gaussian sampling. In contrast, we show that finite-dimensional Markovian neural SDE models constitute a meagre and Gaussian-null subset in $\mathcal{H}^2_T(\mathbb{R}^{d})$, regardless of discretization, whereas compressible processes are generic. Numerical experiments on a stochastic optimal control problem and dynamic hedging highlight the practical effectiveness of our approach. Our results enable more efficient and expressive modelling in stochastic analysis and mathematical finance.

Stochastic-process models are, as a rule, far easier to simulate than to condition. Non-linear observations, non-Gaussian likelihoods, black-box information, and global constraints all induce intractable conditional laws, requiring bespoke, model-specific constructions. We introduce LatentFlow, a single framework for conditioning stochastic processes, with no learned neural approximations and no training. Our starting point is to write the stochastic process as the deterministic image of a tractable latent innovation, $f_0 = T_{\vartheta}(ξ_0)$, with $ξ_0$ sampled from a simple reference distribution. This reduces process-level conditioning to latent-space inference: pull the likelihood back through $T_{\vartheta}$, sample the resulting latent law with a tractable guided probability flow, and push the samples forward. This construction is provably exact at the level of the target law; in practice, approximation enters only through finite terminal noising, Monte Carlo guidance, and time discretisation of the continuous-time dynamics, each of which is explicit and systematically reducible. As LatentFlow is training-free, conditioning reduces to solving a single reverse-time SDE. This enables conditional sampling in seconds on a single desktop CPU across model classes that have never shared a scalable method: classical spatial priors, nonlinear stochastic dynamics, mechanistic models from the physical and life sciences, stochastic PDEs, heavy-tails and extremes, point and discrete-state processes, and neural or simulator-defined processes.

We introduce Sticky Jump Diffusions (SJDs), continuous-time Markov processes on $\mathbb R^d$ whose discrete anchors are token embeddings. In forward time, anchors release their mass at a hazard rate and the released mass diffuses in the continuous ambient space; time reversal couples a score-driven SDE with a sticky jump kernel whose rate and destination are fixed by flux balance with the forward law. We estimate the score and the per-anchor reverse hazards from a single denoising classifier via Denoising Hazard Matching, the hazard analogue of denoising score matching, with simulation-free cross-entropy training. SJD recovers masked diffusion, continuous diffusion, and hybrid diffusion as limits. Its reversal explains features that each family treats as given: the mask of masked diffusion carries no evidence about the source token because the unsticking kernel of every anchor collapses to the same absorbing point; the terminal projection of continuous diffusion is required due to the absence of atoms in its forward marginal, without which flux balance yields no reverse jumps; and the update rules of hybrid diffusion (commit rate, destination, and drift) all follow from flux balance rather than from separate design. Beyond these limits, the unsticking kernel becomes a design space: a cross-position blending corrupts each position toward a blend of its neighbors' clean values or embeddings, turning dependency structure such as spatial locality or a constraint graph into an inductive bias of the corruption itself, and improves over the identity-kernel hybrid on CIFAR-10, Text8, and Sudoku.

Deep generative models are increasingly used as simulators for downstream decision-making under data scarcity, but in risk-sensitive applications their usefulness depends on rare adverse scenarios rather than typical samples. Standard generative objectives prioritize bulk distributional fidelity, leaving low-probability tails vulnerable to localized optimization noise and making tail-dependent functionals unstable under finite simulation budgets. We introduce Diachronic Sample Integration (DSI), a test-time inference framework that ensembles generated samples across checkpoints from a stochastic training trajectory. DSI targets a checkpoint-mixture distribution that averages checkpoint-specific tail fluctuations rather than relying on a single brittle endpoint. We formalize this mechanism through a finite-budget bias-variance theory. Empirically, across multivariate synthetic processes and high-frequency trading data, DSI substantially reduces tail-estimation error compared to single-checkpoint baselines under fixed simulation budgets, outperforming standard diffusion and state-of-the-art tail-aware baselines without modifying the generative objective.

The stationary distribution of reflected Brownian motion (RBM) plays an important role in the analysis of high-dimensional stochastic systems, yet closed-form solutions are known only for a few special cases. Computing important performance metrics, such as tail probabilities, is even more intractable, despite their practical relevance. In this paper, we develop a deep learning approach that accurately and efficiently learns the Laplace transform of high-dimensional RBMs based on the basic adjoint relationship (BAR). Our framework combines a careful design of the loss function, training data sampling procedure, and neural network architecture. We evaluate the proposed method on RBM instances with known ground-truth tail probabilities and demonstrate near-perfect prediction in high-dimensional settings, highlighting its potential as a general tool for analyzing stochastic systems beyond analytically tractable regimes. Our code can be found at https://github.com/zhangz73/NN4MGF.

Diffusion models offer a powerful framework for sampling from complex probability densities by learning to reverse a noising process. A common approach involves solving for the time-reversed stochastic differential equation (SDE), which requires the score function of the evolving sample distribution. The logarithm of this distribution's density is governed by a Hamilton-Jacobi-Bellman (HJB) type partial differential equation (PDE). However, current methods for solving this PDE, such as PINNs or trajectory-based techniques, often suffer from long training times and significant sensitivity to hyperparameter tuning. In this work, we introduce a novel and efficient solver for the underlying HJB equation based on the functional tensor train (FTT) format. The FTT representation leverages latent low-rank structures to efficiently approximate high-dimensional functions, enabling both model compression and rapid computation. By integrating this efficient representation with a backward-in-time iterative scheme derived from backward stochastic differential equations (BSDEs), we develop a fast, robust and accurate sampling method. Our approach overcomes primary bottlenecks of existing techniques, enabling high-fidelity sampling from challenging target distributions with improved efficiency.

Brownian Bridge Diffusion Models (BBDM) offer an appealing framework for image restoration and inverse problems by constructing a stochastic bridge from the clean signal directly to the degraded observation, rather than to pure noise. Despite their promise, the choice of bridge schedule is typically inherited from heuristics, and a principled analytical framework for schedule design has been lacking. In this work, we develop such a framework by offering a novel analysis of BBDM reverse dynamics under a Mixture-of-Gaussians (MoG) prior. This setting yields a closed-form ideal posterior and a corresponding MMSE denoiser, while the BBDM-induced reconstruction law is captured analytically through a tractable surrogate. Building on these expressions, we formulate two complementary schedule-design objectives: a Wasserstein criterion targeting perceptual quality and an MSE criterion targeting reconstruction fidelity. Our work exposes an inherent tradeoff between the two and proves the existence of universal schedules for both that are independent of the degradation and prior. Extensive experiments on controlled MoG settings confirm full alignment between theory and practice, and experiments on the FFHQ dataset across inpainting, deblurring, and super-resolution tasks validate the practical value of our schedule-design criteria.

The use of ordinary and stochastic differential equations has led to substantial progress in generative machine learning with applications to, for example, image, video and biomolecule generation. This paper provides a self-contained and informal introduction to the differential equations, the probabilistic framework for using them in generative modeling and the Fokker--Planck equation that governs the temporal evolution of the marginal distribution of the stochastic variables of the differential equations. The variational lower bound on the log-likelihood (the evidence lower bound, ELBO) is derived and used as a general starting point for a discussion of diffusion models, score matching, and flow matching. All of these approaches may be viewed as specific parameterizations of the most general variational approach. A one-dimensional density modeling problem is used as a simple example to compare different parameterizations.

We consider sparse multivariate stochastic systems that evolve in continuous time according to a causal mechanism and present methodology to recover the system's time-infinitesimal transition mechanism from mere cross-sectional data. This observational paradigm is motivated by applications such as gene expression analysis, where destructive experimental techniques may only allow recording data once over a cell's lifetime. Precisely, we assume the system follows a time-homogeneous diffusion process that has reached an equilibrium distribution at observation time. Further, we assume the causal mechanism is fully described by the diffusion drift, is acyclic, and its causal structure graph is known. In this setting, we prove that the full causal mechanism, i.e., the drift function, can be non-parametrically identified under a weak non-explosion criterion. We derive a non-parametric kernel estimator for this challenging inverse problem and prove its consistency. Moreover, we propose a cross-validation scheme for hyperparameter tuning, illustrate the behavior of our estimator in simulations, and we discuss connections with irreversible generative diffusion models and low-frequency sampled data.

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

This paper develops an a posteriori error analysis framework for decoupled neural approximations of fully coupled forward--backward stochastic differential equations (FBSDEs). It provides an a posteriori error-analysis for the idealized discrete adapted trajectory. The main feature of the proposed formulation is the use of an auxiliary control process in the forward coefficients, which may differ from the backward component approximated by the neural network. This decoupling is useful in practical deep learning implementations, but it creates a control mismatch that must be included in the error analysis. We first establish a continuous-time stability estimate for fully coupled FBSDEs under perturbations of the drift, diffusion, generator, terminal condition, and auxiliary control input. We then transfer this estimate to the discrete-time setting and derive computable a posteriori error bounds depending only on the terminal defect, the pathwise residual, and the control mismatch. When the auxiliary control is identified with the backward approximation, the mismatch term vanishes and the bound reduces to the standard two-term form. Numerical experiments on a linear--quadratic FBSDE with an explicit reference solution and a multidimensional Burgers-type FBSDE without a reference solution illustrate the diagnostic role of the proposed indicators and the contribution of the mismatch penalty to the consistency and reproducibility of the numerical approximations.

In this paper, we develop a fractional stochastic neural network with residual dynamics driven by fractional Brownian motion. By introducing a discrete stochastic maximum principle for the network, we construct the corresponding adjoint recursion. For deterministic network parameters, we prove mean square convergence of projected samplewise stochastic gradient descent. Numerical experiments include a closed form convergence test, noisy regression with uncertainty quantification, long memory time series generation and image classification under structured perturbations. The results identify settings in which fractional drivers improve long memory recovery or robustness relative to Brownian and deterministic baselines.

In next-generation wireless networks, the growing density of devices and limited spectrum resources pose severe jamming challenges to fragile legitimate communication links in the wireless electromagnetic environment. Crucially, when jamming overlaps with pilot and data symbols in both time and frequency domains, it inflicts a severe bottleneck on receiver-side joint estimation and detection. Existing schemes often lack an effective framework to combat such jamming contamination, thereby failing to guarantee reliable transmission. To address this issue, we propose a Brownian bridge diffusion-based joint channel estimation and data detection framework (BBD-JCED) for jamming-resilient receivers. Specifically, the proposed framework comprises two core modules: the first extracts jamming features in the short-time Fourier transform (STFT) domain and suppresses jamming samples, thereby improving the signal-to-jamming-plus-noise ratio (SJNR) of the received signal; the second introduces a Brownian bridge diffusion (BBD) process to model the evolution of the suppressed signal and the encoded bits in the presence of channel estimation errors, thereby enabling enhanced joint channel estimation and data detection. To alleviate the computational burden of the BBD process in the second module, we further derive a fast ordinary differential equation (ODE) solver that enables its low-complexity iterative evolution. Finally, we design a multi-module training algorithm to improve the data recovery capability of the proposed framework. Simulation results demonstrate that the proposed framework achieves superior bit recovery performance compared with baseline schemes while maintaining a lower number of model parameters and competitive computational complexity.

Causal representation learning for time series has developed strong identifiability results in discrete-time latent causal models, but identifiability in continuous-time latent stochastic differential equation (SDE) models remains largely open. We address this gap using environment-induced shifts in diffusion covariance. We study additive-noise latent SDEs observed through an unknown nonlinear diffeomorphism, with shared drift but environment-specific diffusion covariance. We show that two diagonal diffusion regimes with pairwise distinct coordinate-wise variance ratios identify the latent coordinates up to permutation and scaling, without any sparsity assumption on the drift. We first prove this result for linear Ornstein--Uhlenbeck systems and then extend it to general additive-noise latent SDEs. Under mild smoothness, the instantaneous drift-Jacobian causal graph is identifiable up to the same permutation. We propose a two-stage estimator for latent disentanglement and optional graph recovery; experiments on synthetic systems confirm the predicted identifiability boundary, and an application to Hardanger Bridge monitoring data illustrates the approach on real sensor trajectories.

AI-driven image-to-image synthesis is rapidly advancing, with growing applications in medical imaging. Multi-modal image analysis plays a crucial role in optimizing examination quality, yet acquiring multiple imaging modalities in clinical settings remains resource-intensive and time-consuming, especially for 3D imaging. To address this challenge, we propose a novel image-to-image translation model based on Brownian Bridge Diffusion Models (BBDM), which synthesizes magnetic resonance imaging (MRI) sequences from 2D axial slices. Our approach integrates a variational encoder-guided diffusion mechanism, leveraging probabilistic image distributions to enhance synthesis quality. Evaluated on the BraTS 2021 dataset, our Probabilistic-BBDM (Prob-BBDM) achieves superior performance across multiple translation tasks, reaching up to 88.46% SSIM and 26.09 dB PSNR, with consistent improvements over baselines. Notably, our diffusion process requires only 4 steps, making it computationally efficient while maintaining high-quality synthesis. To further validate generalizability, we test Prob-BBDM on an external third-party dataset, demonstrating consistent performance across domains. Additionally, we assess the clinical utility of the synthesized slices by using them as input to a pre-trained segmentation model. Tumor segmentation yields a Dice score of 88.71% and an HD95 of 3.49 mm, confirming that the synthesized slices preserve critical diagnostic information. These results highlight the potential of Prob-BBDM for high-quality, efficient, and generalizable MRI synthesis, offering a promising step toward improved medical image translation.

In this paper, we introduce two neural-network-based numerical schemes for solving systems of coupled ergodic Backward Stochastic Differential Equations (eBSDEs), motivated by the approximation of optimal strategies within the framework of forward utilities in a regime-switching stochastic factor model. Our approach builds on the representation of such models through systems of eBSDEs introduced in [HLT20]. We first establish a link between the solution of the system of ergodic BSDEs and that of an associated multidimensional BSDE with random terminal time, given by the hitting time of the positive recurrent stochastic factor. Building on this representation, we introduce a locally additive deep learning scheme obtained by minimizing aggregated local error terms. We then present a new Deep Galerkin Method (DGM) inspired algorithm that minimizes the residual of the associated ergodic PDE system, relying on a representation of the ergodic cost. Finally, we apply this framework to regime-switching forward utilities in a stochastic factor model. We first derive a general consistency SPDE that characterizes regime-switching forward utilities and retrieve their representation with systems of ergodic BSDEs in the homothetic case. Numerical experiments demonstrate the performance of the proposed methods, with a particular focus on the impact on forward preferences of taking into account regime switches.

While longitudinal brain PET imaging is the gold standard for quantifying the spatiotemporal accumulation of Beta-amyloid, its widespread clinical utility is constrained by high operational costs and cumulative radiation risks. Recent deep generative models show promise in longitudinal image synthesis; however, they often fail to capture subtle pathological progression due to identity drift and a persistent bias toward trivially replicating baseline signal intensities rather than modeling temporal transition. To this end, we propose Delta-Diffusion, a novel progression-aware framework that redefines longitudinal PET synthesis as a conditional Poisson Diffusion Bridge (PDB) process. Unlike standard diffusion models that start from Gaussian noise, our PDB formulation is mathematically anchored to the subject's baseline PET, effectively transforming the generative task into a conditional distribution transition of the amyloid trajectory. To handle heteroscedastic nature of PET imaging, we introduce a physically-grounded Poisson perturbation within a Diffusion Transformer (DiT). This architecture uses adaptive scale-shift modulation to precisely calibrate the synthesis with the elapsed clinical interval and structural MRI context. A volume-of-interest balanced objective is designed to emphasize sparse, high-risk regions of amyloid accumulation. Validated on two cohorts with 542 subjects, Delta-Diffusion demonstrates superior performance in capturing longitudinal variations in amyloid deposition compared to state-of-the-art methods, offering a robust computational framework for tracking disease progression.

The remarkable success of score-based diffusion models has spurred significant efforts to establish their theoretical foundations. However, existing complexity bounds for score approximation rely heavily on restrictive assumptions like Lipschitz continuous densities or smooth manifold supports, which are routinely violated by the singularities, sharp boundaries, and disjoint clusters inherent to real-world perceptual data. This work establishes a universal score approximation theorem that works for any distribution supported on any compact set of upper Minkowski dimension $d$. Using a novel discrete-mixture formulation, we prove that the score function can be approximated with a ReLU network whose complexity grows exponentially only with $d$, thus breaking the exponential curse of ambient dimensionality. Combined with existing theories on accurately solving the backward diffusion SDE for arbitrary compact distributions, our work shows that diffusion models readily adapt to irregular, non-smooth data structures, explaining their competence in real-world generative tasks.

Multivariate time series anomaly detection (MTSAD) is critical for a wide range of application areas, such as industrial monitoring, cybersecurity, or healthcare. Real-world data is often sparse, irregularly sampled or partially observed, yet existing methods assume uniformly sampled time series. We propose a generative approach based on Latent SDEs that projects the observed time series on a continuous-time stochastic dynamical system, directly being able to handle missing observations and irregular sampling, while also naturally capturing possible cyclic behavior that many real-world use cases inherently possess. Experiments on six anomaly benchmark datasets show that our proposed method ranks first among state-of-the-art baselines. We further demonstrate that our method remains robust under severe data sparsity, while performance significantly degrades for the tested baseline methods. These results highlight latent SDEs as a natural inductive bias for anomaly detection in multivariate time series, especially in presence of real-world irregularities.

Temporal difference (TD) learning with linear function approximation is a core method for policy evaluation. Its classical continuous-time description is an ordinary differential equation (ODE), which captures the asymptotic mean dynamics but neglects stochastic fluctuations determining the error floor. We introduce a stochastic differential equation (SDE) approximation for linear TD(0) under Markovian noise. The resulting model distinguishes the contraction dynamics governed by the projected Bellman operator from the influence of Markovian sampling. As a consequence, the model explains the constant-stepsize error floor through the interaction between Markovian long-run covariance and the contraction geometry of the projected Bellman operator.

Diffusion Flow Matching (DFM) has recently emerged as a versatile framework for generative modeling, yet its theoretical convergence properties remain only partially understood. In this work, we provide refined and novel convergence guarantees for Brownian motion based DFMs, focusing on the discretization error. Our analysis is conducted under the Kullback-Leibler (KL) divergence and the 2-Wasserstein distance. Under finite-moment conditions and a mild score integrability assumption, we derive KL convergence bounds with improved dimensional dependence compared to prior work, achieving, up to our knowledge, state-of-the-art scaling under minimal conditions. We further extend the analysis to the 2-Wasserstein distance: under an additional first-order score integrability assumption and a weak log-concavity condition, we obtain convergence guarantees with dimensional dependence consistent with the KL case.

Recovering dynamical systems from noisy observations is a recurring challenge across scientific domains, including neuroscience and physics. Latent stochastic differential equations (SDEs) address this by modeling the system as an unobserved state that evolves according to a learnable SDE and generates the observations. Variational inference (VI) provides a tractable objective for fitting latent SDEs. Traditional VI algorithms evaluate this objective by numerical simulation over a time discretization, trading fidelity for computational cost. A recent class of algorithms, simulation-free VI, sidesteps this tradeoff by parameterizing the posterior through its instantaneous marginals rather than its drift. In this work, we show that the efficiency of existing simulation-free VI algorithms comes at a price: their parameterizations restrict the approximate posterior to a subset of the SDEs available to simulation-based methods, degrading posterior inference and parameter learning. We propose Helmholtz-SDE, a simulation-free VI algorithm that closes this gap by optimizing over path laws compatible with a prescribed collection of marginals. Helmholtz-SDE recovers dynamics more faithfully than prior simulation-free methods, with the largest gains under high posterior uncertainty. It further matches the performance of simulation-based VI at a fraction of the runtime.

Constructing mathematically tractable function spaces that capture hierarchical compositional representations remains a central challenge in statistical learning theory. We introduce Brownian kernel ladders (BKLs), a recursively defined hierarchy of integral reproducing kernel Hilbert spaces generated through Brownian-kernel integral constructions. Starting from linear functionals, each layer is obtained by integrating Brownian kernels over probability measures supported on subsets of the previous layer, yielding a recursive function-space model in which depth is encoded directly through the hierarchy. Based on this framework, we define canonical BKL spaces together with an associated complexity functional. We establish several analytical and statistical properties of these spaces. In particular, we show that BKL spaces form quasi-Banach spaces, satisfy depth-dependent Hölder regularity estimates, and exhibit strict monotonicity with respect to depth. We further prove existence results for regularized empirical risk minimization and derive Gaussian complexity bounds that remain uniformly controlled with respect to both the ambient dimension and the hierarchy depth. A key ingredient of the analysis is a combinatorial proof technique based on recursive subset decompositions and Brownian-kernel threshold representations. These estimates yield excess-risk guarantees of near-parametric order for regularized empirical risk minimization over BKL spaces. Our results provide a mathematically tractable hierarchical function-space framework for studying compositional representations in deep learning.

Diffusion models are typically trained with objectives that focus on local denoising targets at individual time steps (or adjacent pairs), which do not enforce consistency between predictions along the denoising trajectory. This lack of cross-time consistency can degrade performance, especially for few-step samplers. We introduce a temporal difference (TD) objective that penalizes inconsistency of the model's multi-step progress along the denoising path. By reformulating the diffusion process as a Markov reward process and casting denoising as a policy evaluation problem in reinforcement learning, we derive a unified TD approach that applies to both discrete- and continuous-time diffusion formulations. We further propose a principled sample-based reweighting method that stabilizes training. Empirically, we show that using our TD training can significantly improve sample quality measured by FID, with stronger advantages when the number of sampling steps is small, highlighting its practical utility under low-computation-budget scenarios. We provide ablation studies to justify our design choices, including pairwise loss reweighting, regularization weight, and one-step stride. Overall, our TD approach can be a general drop-in that enforces cross-time consistency and improves generation quality across different diffusion generative models.

Speculative decoding speeds up LLM inference by using a draft model to generate tokens, with an acceptance-rejection scheme that ensures that the output matches the target distribution. Adapting this to continuous diffusions is difficult because speculative sampling requires drawing from a residual distribution. While straightforward in discrete spaces, efficiently sampling this residual in continuous space is non-trivial. Consequently, existing diffusion adaptations either use computationally inefficient sampling techniques or rely on an alternative scheme. In this work, we introduce a novel scheme that efficiently implements the original speculative sampling mechanism for diffusion models. Our approach offers a critical advantage over current methods: it enables us to adapt block verification from LLMs to diffusions -- which provably improves the acceptance rate of drafts. Furthermore, we formalize and analyze the Free Drafter, a heuristic self-speculative drafter for diffusions that requires no training. By enabling block verification, our Free Drafter yields up to a 6.3% speedup over existing speculative methods with no additional training and negligible overhead beyond the existing parallel verification pass.

Sound source distance estimation (SDE) is a critical capability in human-robot interaction. An inappropriate interaction distance not only reduces the reliability of speech acquisition and understanding, but also compromises the naturalness and comfort of the interaction. Most existing SDE methods rely on microphone arrays, however, multi-microphone systems typically require careful hardware synchronization, geometric calibration, and additional space and computational resources, which limits applicability to size-constrained and computability-limited embodied platforms. To alleviate these issues, we propose Fast-SDE, a lightweight single-microphone SDE framework that is suited for deployment on robot platforms with limited computational resources and strict size constraints. Specifically, Fast-SDE employs a subband-based backbone that decomposes the frequency axis into multiple subbands, rather than processing the entire spectrum with a wide full-band backbone. A shared subband encoder then maps each subband to a compact latent representation and learns the relationship between acoustic structure and time-frequency patterns. Finally, a lightweight regression head converts the fused subband representations into the estimated distance. Extensive simulation and real-world experiments demonstrate the merits of the proposed method. To benefit the broader research community, we have open-sourced our code at https://github.com/JiangWAV/FAST-SDE.

Recent one-step generative models accelerate sampling by learning deterministic flow maps of the underlying dynamics. These methods rely on learning from ordinary differential equations, leaving open how to define an exact distillation procedure for stochastic dynamics. We introduce the Itô map, an any-step stochastic flow map that takes an intermediate state and Brownian path and predicts future states in a single pass. The Itô map formulation yields novel estimators for inference-time control by providing cheap, differentiable access to posterior samples. Empirically, Itô maps produce diverse, conditionally valid endpoint samples from fixed intermediate states and support strong steering performance on synthetic and image-generation benchmarks. These results establish any-step SDE integration as a useful primitive for posterior sampling and stochastic control.

Recovering 3D human poses for multiple individuals from different camera views is a fundamental bottleneck for analyzing interacting behaviors. Existing self-supervised approaches leverage synthetic catalogues of 3D poses; however, this leads to poor generalization in real-world scenarios due to distribution shifts. We therefore introduce DisPOSE, a self-supervised framework that approximates the inherently discrete multi-view person-assignment problem as a generative diffusion process over the space of polystochastic tensors. By employing differentiable Sinkhorn projections during denoising, our model learns to guide solutions toward valid and feasible assignments based on 2D image priors. The complete 3D skeletons of localized individuals are then regressed using a Hypergraph-Convolutional Decoder that explicitly models relational structures and articulated joints across multiple views. The proposed approach outperforms current state-of-the-art self-supervised methods on standard datasets and demonstrates strong performance on a newly proposed benchmark featuring highly occluded scenes from surgical operating rooms. Our diffusion-based localization demonstrates high label efficiency, retaining 99% of its performance with only 10% of the pseudo-labels. Notably, disentangling the assignment and root regression components while maintaining differentiability makes DisPOSE nearly agnostic to different camera arrangements.

Hydrodynamic models of stochastic particle systems represented by coarse-grained stochastic partial differential equations (SPDE), such as the regularized Dean-Kawasaki (DK) equation, do not accurately capture the short-time system dynamics that is dominated by non-Markovian effects, and low particle density regimes where the distributions are highly non-Gaussian. We develop a generative flow matching method that directly models the probability distribution of fluxes from particle simulations that explicitly incorporates non-Markovian and non-Gaussian effects. As a demonstration, we use this method to simulate the Kramers first passage time problem for a system of non-interacting Brownian particles. We show the model accurately captures the short-time behavior and provides better predictions of the statistical moments of the number density when compared against the solution of the Markovian baseline, regularized DK equation.

One of the primary challenges in Bayesian inference on the parameters of a diffusion model from discrete observations is the unavailability of an analytical expression for the transition density function between consecutive observation times, which is needed to derive the likelihood function. Extending previous studies that solve Fokker-Planck (FP) type partial differential equations with Normalizing Flows, we propose a new Normalizing Flow architecture to learn the transition density function of the diffusion process between two observation times. We do so by solving in a Neural Galerkin framework the associated FP equation with a Dirac mass as initial condition, over a specified training distribution of the initial datum and the coefficients of the diffusion. We specifically focus on processes whose diffusion matrix vanishes in certain inaccessible boundary regions, such as Stochastic Volatility models that satisfy a Feller condition. The product of the obtained transition densities evaluated along the observed trajectory approximates the likelihood function, thereby enabling cheap posterior sampling via Markov chain Monte Carlo (MCMC). After the offline training phase, inference becomes significantly more efficient, as it avoids the need to solve the FP equation in real time for each parameter proposed by the MCMC sampler or to rely on other likelihood-free methods for Bayesian inference that involve repeated simulation of diffusion bridges.

We study the robustness of stochastic mirror descent (SMD) under heavy-tailed noise, focusing on whether the method retains its convergence guarantees when run with infinite-variance stochastic gradient input. To address this question in a principled manner, we begin by introducing a continuous-time model of SMD as a stochastic differential equation (SDE) driven by a centered Lévy noise process with finite $p$-th order moments, $1 < p \leq 2$. This scheme -- which we call the Lévy mirror flow (LMF) -- arises naturally as the scaling limit of SMD in the presence of heavy-tailed noise. In particular, when $p < 2$ -- the heavy noise regime -- the trajectories of LMF generically exhibit jump discontinuities of arbitrary magnitude which, if frequent enough, lead to infinite variance. Nonetheless, despite this highly singular behavior, we show that LMF attains $ε$-optimality within $\mathcal{O}(ε^{-p/(p-1)})$ time in the convex case, and within $\mathcal{\tilde O}(ε^{-1/(p-1)})$ time for (relatively) strongly convex objectives. These guarantees provide a transparent characterization of the impact of frequent long jumps on the convergence of the process, and percolate to a series of matching discrete-time guarantees for several variants of SMD under heavy-tailed noise.

Parameter estimation in stochastic differential equations is a classical statistical problem of much importance in many scientific fields. Recent work of Tapia Costa et al. (2026) introduced a novel technique for estimating the drift when the diffusion parameter is known, using discrete samples from multiple trajectories. Their method treats drift estimation as a denoising problem, and leverages tools from (conditional) score-matching diffusion models. Although their experiments showed promising results across different drift classes, the question of theoretical guarantees for their estimator was left unanswered. In this note, we address this gap by exploiting techniques from diffusion model theory. More concretely, we derive an explicit risk bound for the time-averaged mean-squared error of said drift estimator. Our bound decomposes the risk into the (i) Euler-Maruyama discretization, (ii) score/denoiser approximation, (iii) noise initialization, and (iv) sampling variance, revealing the trade-offs between the different hyperparameters and sources of error in the estimator.

Accurate Remaining Useful Life prediction is critical for industrial predictive maintenance. However, real-world deployment is challenging due to the irregular nature of sensor observations, characterized by asynchronous sampling, burst missingness, and temporal jitter. Compounding this issue, purely data-driven models often generate physically implausible degradation trajectories that violate the irreversible nature of damage accumulation. To address this, we propose PC-MambaSDE, a unified continuous-time framework for robust RUL prediction under irregular observations. Specifically, we design a Mask-Aware Continuous Mamba Encoder that explicitly leverages observation masks to extract context-rich control signals. Furthermore, we introduce a Physics-Guided Latent SDE with parametrically rectified hybrid drift, superimposing a global physical bias to enforce monotonic degradation even amid severe observation gaps. Additionally, we formulate RUL prediction as a boundary value problem via a Terminal Degradation Penalty, which decouples a Health Index dimension and applies a penalty loss to guide trajectories toward the failure state. Theoretically, we prove that our variational objective is mathematically equivalent to minimizing the KL divergence via Girsanov's theorem, and we guarantee the global asymptotic stability of the learned dynamics through Lyapunov analysis. To enable rigorous evaluation, we develop a Hybrid Irregularity Generation Scheme that simulates realistic industrial imperfections. Extensive experiments on public benchmarks demonstrate that PC-MambaSDE significantly outperforms state-of-the-art methods, particularly under extreme observation scarcity, validating the efficacy of embedding physical priors into continuous-time latent dynamics.

Flow and diffusion models generate high-quality samples in many modalities; however, many network evaluations are required during inference due to numerical integration of an underlying differential equation. Flow maps alleviate this problem by learning the solution map of the differential equation directly, enabling few-step sampling. Yet, current methods are restricted to approximating the solution map of ODEs. These methods can be used to learn the transition kernel of an SDE, thereby obtaining a solution map that recovers the marginal distributions of the process (weak convergence) rather than the solution path (strong convergence). We propose Strong Stochastic Flow Maps (SSFMs) as a novel framework for learning the strong solution map of additive-noise SDEs, directly generalizing deterministic flow maps to the stochastic setting. Further, a polynomial approximation to Brownian motion is introduced and shown to converge pathwise. These results enable a simulation-free training objective for the solution map of diffusion models. We demonstrate that SSFMs outperform previous stochastic flow map methods on image generation and enable few-step sampling of molecular systems.

A central aim of deep learning theory is to characterize how neural networks make predictions in the regime of simultaneously large model and training set size. Since the limits of diverging number of model parameters and dataset size do not commute it is not clear a priori what limits exist. In this work, we shed new light on these questions by studying Bayesian inference in deep non-linear MLPs in the regime where the number of training samples ($P$), the input dimension ($N_0$), the hidden layer width ($N$), and the number of hidden layers ($L$) can all be large. We build on the Neural Covariance SDE (Li et al., 2022) to analyze predictive posteriors in the regime where $LP/N\inΘ(1)$, playing the role of an effective network depth. Our framework covers both smooth and ReLU activation functions and applies to arbitrary temperature. We find to first order in $LP/N$ a simple criterion for which data generating processes benefit from depth in the sense that larger $LP/N$ increases the Bayesian model evidence. We also give a novel derivation of a prior result from the physics literature that at least to first order in $LP/N$, the Bayesian predictive posterior is remarkably simple and is simply equivalent to that of a data-dependent kernel method.

Diffusion models achieve state-of-the-art image synthesis, with their generative trajectories fundamentally exhibiting a spectral bias, resolving low-frequency global structures early and high-frequency fine details later. Conventional stochastic differential equation (SDE) solvers fail to account for this dynamic, naively injecting uniform white noise throughout the entire process and misusing the finite energy budget. In this work, we establish a mathematical framework that reconsiders SDE inference as a targeted, frequency-decoupled energy transfer. Leveraging this framework, we introduce Colored Noise Sampling (CNS), a novel, training-free stochastic solver. Rather than injecting uniform white noise, CNS utilizes a dynamic, timestep- and frequency-dependent schedule that more efficiently allocates injected energy toward structurally unresolved frequency bands. By actively exploiting the model's inherent spectral bias, CNS systematically steers the generated distribution toward the true data manifold. Extensive experiments demonstrate that CNS significantly outperforms standard ODE and SDE baselines as a strictly plug-and-play, inference-time sampler substitution across diverse architectures (SiT, JiT, FLUX). Compared to standard sampling on ImageNet-256, CNS achieves substantial unguided FID reductions, improving from 8.26 to 6.27 on SiT-XL/2, 32.39 to 26.69 on JiT-B/16, and 11.88 to 8.31 on JiT-H/16, while yielding consistent relative FID improvements with Classifier-Free Guidance. Project page is available at https://hadardavidson.github.io/CNS/.

Extending discrete-time causal Prior-data Fitted Networks for time series to continuous time invites writing the mechanism as a stochastic differential equation (SDE) -- but if the SDE is integrated \emph{once per observation gap}, the trajectory law depends on when it is observed, and the prior remains a discrete-time Markov model in SDE clothing. We propose a precise continuity criterion -- trajectory-law invariance to the observation schedule -- together with a three-tier taxonomy (discrete; naive observation-grid integration; fine-grid integration with decoupled observation) and a construction realising the top tier on a random DAG with OU or small-MLP nonlinear drifts, irregular observation schedules, and hard / soft / time-varying interventions. A $2 \times 2$ encoder $\times$ integrator ablation, run independently on a linear and a nonlinear prior, finds fine-grid integration beats naive on 8/8 cells (sign-consistency $p < 1/256$) with the gap growing as the eval grid refines; the encoder axis is null with fine integration but time-aware-leading with naive. We release the prior and a preliminary zero-shot protocol on pharmacokinetic and physical-system data.

Reliable uncertainty quantification in continuous-time (CT) representation learning remains nascent, particularly within CT attention literature. We introduce the Neuronal Stochastic Attention Circuit (NSAC), a novel biologically-inspired CT attention architecture that reformulates attention logit computation as the solution of an Ornstein-Uhlenbeck stochastic differential equation modulated by input-dependent, nonlinear interlinked gates derived from repurposed C. elegans Neuronal Circuit Policies (NCPs) wiring mechanism. It induces a Gaussian distribution over logits that propagates principled stochasticity through a logistic-normal distribution over attention weights to yield probabilistic output. A two-term objective function combining Gaussian negative log-likelihood with an epistemic-separation regularizer enforces higher predictive variance under distributional shifts and enables joint quantification of aleatoric and epistemic uncertainty. Theoretically, we provide: (i) state stability bounds; (ii) closed-form guarantees; and (iii) frozen-coefficient error approximation. Empirically, we implement NSAC in a diverse set of learning tasks including: (i) irregular CT function approximation; (ii) multivariate regression; (iii) long-range forecasting; (iv) Industry 4.0; and (v) lane-keeping of autonomous vehicles. We observe that NSAC remains competitive against several baselines in terms of accuracy and produces informative uncertainty estimates while being interpretable at the neuronal cell level.

Time series driven by unobserved latent states frequently exhibit abrupt jump discontinuities whose timing and magnitude cannot be predicted from observed history alone. Classical jump-diffusion models offer a principled mathematical framework but assume rigid parametric forms, while recent neural jump models operate on fully observed trajectories without inferring the hidden states that govern the dynamics. We propose \textit{Deep ZakaiJ}, a latent-state model for partially observed jump-diffusion systems that embeds the Zakai nonlinear filtering equation into a neural encoder--decoder architecture. The encoder recursively updates a belief over the latent state via Strang splitting into three interpretable substeps: prior propagation, diffusion innovation, and jump innovation, yielding a differentiable, first-order-accurate approximation of the exact filtering evolution. The decoder is a structured jump-diffusion model explicitly conditioned on the filtered belief, preserving the separation between continuous dynamics and discontinuous shocks. On synthetic, financial, and oceanographic datasets, \textit{Deep ZakaiJ} improves distributional forecasts while remaining competitive in point accuracy, achieving calibrated predictive intervals and recovering interpretable latent structure in synthetic and qualitative case studies.

Reinforcement learning (RL) has become an effective way to improve prompt alignment and perceptual quality in diffusion and flow-matching generators. A critical step for applying online RL to flow matching is turning the deterministic sampling trajectory into a stochastic policy, typically by replacing the reverse-time Ordinary Differential Equation (ODE) with a Stochastic Differential Equation (SDE). The stochastic sampler, controlling the exploration behavior and denoising dynamics, is thus part of the policy, and its design can significantly affect the reward optimization performance. We break down the sampler design into two interdependent components: choosing the right amount of stochastic exploration, and discretizing the resulting SDE faithfully at the small step counts used in RL. To address the first component, we analyze the inherent tension between exploration and stability in denoising and derive an SDE schedule that balances the two. Turning to the discretization challenge, we use a toy example to show that existing samplers can deviate from the flow-matching process, either by introducing excessive discretization noise or by relying on heuristic rules that do not guarantee convergence to the data distribution. To address these issues, we propose Precise, a new stochastic sampler that balances effective exploration with stability. Crucially, Precise keeps the denoising trajectory SDE-consistent through a novel approximation that freezes the clean-latent posterior mean, resolving the excess noise issue in standard samplers. Extensive experiments demonstrate that this formulation leads to significantly faster and more stable reward optimization via reinforcement learning, achieving state-of-the-art alignment scores (e.g., PickScore, HPSv2.1) while requiring 13.1-53.2% less wall-clock training time to match the best in-domain performance of prior samplers.