Stochastic Differential Equations

One hospital runs bloods and imaging at the same time. Another runs them one after the other, in either order, equally often. Knowing which actually happened, and how it is recorded in data, is critical for all operational managers. In process mining, the standard approach is to construct an event log, and attempt to discover concurrent and sequential processes in a data-driven way. We show this standard approach, built on the stochastic language of an event log, reports only the assumptions of its discovery algorithm, because every such log is explained equally well by a model with no concurrency at all. Further, before any data is acquired, we characterise when data can and cannot distinguish concurrent behaviour. Where it cannot, the distinction is recoverable from evidence the stochastic language discards, such as the times at which activities start and end, or object-centric records that fix an order within an execution. The remedy is therefore a choice of what is recorded, rather than a larger sample. This impacts decision making, as planning resource for truly concurrent services is very different from sequential services.

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.

In multi-agent environments, coordinating agents to prevent interference and ensure robust individual performance is a critical challenge. Previous research on social laws for multi-agent systems has primarily focused on deterministic, goal-based settings. This paper extends the concept of social laws to stochastic, reward-based environments, proposing a formalism for defining and verifying their robustness under various conditions. We introduce the notion of $α$-robustness, a measure of the guaranteed utility each agent retains while pursuing its optimal single agent policy, assuming all agents obey the social law. We then present an approach for robustness verification of social laws in stochastic settings, based on a reduction to solving a series of Markov decision processes. Empirical evaluations on toy environments illustrate the potential of our framework.

Diffusion samplers reverse a learned noising process using either stochastic (DDPM) or deterministic (DDIM) updates, which represent endpoints of a single family controlled by a scalar noise-injection variance that is applied identically at every spatial location. This uniform approach neglects the geometry of natural images: high-curvature regions such as edges and textures, where the denoiser is uncertain, benefit from stochastic correction, whereas smooth regions, where the score is precise, are degraded by injected noise. This work investigates whether each pixel requires stochastic correction at a given timestep and introduces Spatially Adaptive Noise Injection (SANI), a novel sampling framework that dynamically adjusts noise application on a per-pixel basis. SANI integrates a probabilistic gating mechanism with a derived spatially adaptive variance, ensuring that noise is injected precisely where needed to refine complex features while preserving well-formed structures. Experimental results and decoupling ablations demonstrate that SANI consistently improves Fréchet Inception Distance (FID) over the vanilla DDPM and DDIM endpoint samplers across diverse sampling timesteps, while remaining competitive with variance-learning baselines, highlighting the importance of spatial adaptivity in diffusion sampling.

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/}

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.

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%.

Multi-tenant tools commonly accept a tenant identifier and validate it against the caller's entitlement. For a large language model (LLM) agent, that pattern delegates resource selection to a process whose context may contain attacker controlled instructions. We formalize this stochastic deputy problem and present a structural defense: remove tenant identity from the Model Context Protocol (MCP) tool schema, bind scope to a verified credential, and enforce it below the agent. In a 373-trial ablation across eight model configurations and two transports, a correctly validated tenant parameter served every out-of-scope attempt: 26 of 26, or 26 of 41 plausible-pretext trials overall. With the parameter removed, no tool signature could express the read. Twelve of 56 trials instead escaped the interface by forging writable scope, showing that interface invariance requires cryptographically protected context. On a production dataset containing multiple GBs of data, set-valued scope caused a measured $57\times$ latency ratio under function-wrapped membership predicates; a JSON_TABLE lateral join recovered index access where the tenant key was indexed. The evaluation also exposes deployment limits, including an entitlement-size query-planner cliff and incomplete index coverage. The result is a tenant-isolation argument that depends on enforceable interfaces and credentials rather than model compliance.

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.

Generative Marginalization Models (MaMs) have been recently introduced as efficient neural sampling models for any-order autoregressive modelling of discrete distributions. By learning both the marginal and conditional probabilities of a persistent-block Gibbs sampler, MaMs enable fast posterior evaluation with a single neural network forward pass. While prior work has considered MaMs to be distinct from Generative Flow Networks (GFlowNets), a well-established paradigm for inference in discrete stochastic models, we show that they are equivalent. Then, we also extend MaMs' sampling strategy to non-autoregressive generative processes. In particular, we describe an automatic criterion for full-state rejuvenation of the Gibbs sampler, derived from the Gelman-Rubin statistic, which plays a key role in speeding up learning convergence. Our experiments show that our method, called Particle GFlowNets, markedly accelerates training in large combinatorial spaces.

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 oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq ε$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $σ^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(σ^2 ε^{-3} + ε^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with ``sparse'' noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

Leading benchmarks for formal theorem proving with large language models are small collections drawn from competition math, such as the IMO and Putnam, that poorly represent field-specific applications. We introduce StochBench, a Lean 4 benchmark of 450 graduate stochastic-processes problems at varying abstraction levels, each paired with its natural-language source. Addressing a field underrepresented in Mathlib, it covers finite and countable Markov chains, renewal processes, random walks, martingales, stopping times, queues, Brownian motion, stochastic calculus, weak convergence, and Poisson and continuous-time Markov processes. Our Opus 4.8-based agent achieves a 34.9% proof rate (157/450) under a 15-minute per-problem limit. StochBench better represents domain-specific applied mathematics while remaining challenging for advanced provers.

Chain-of-thought reasoning provides a structured computation between a model's input and final answer. Yet it is often evaluated through endpoint accuracy, which ignores the path taken to reach that answer. An emerging line of work addresses this limitation using entropy profiles, which track how uncertainty evolves over the reasoning process but do not reveal which competing hypotheses account for that uncertainty. We introduce answer-distribution trajectories, a stochastic-dynamics-inspired representation that tracks the model's full predictive distribution over answers as reasoning unfolds. As a strictly finer representation than endpoint and entropy summaries, answer-distribution trajectories enable us to characterize a trace through a dynamical reasoning profile spanning exploration, revision, motion, and commitment, and to distinguish different dynamical mechanisms of reasoning success and failure. Across sixteen open-weight language models and four reasoning benchmarks, we show that traces with the same endpoint and similar entropy profiles can exhibit substantially different reasoning dynamics. We further find substantial variation in these dynamics both within and across models and tasks, with different objectives favoring different dynamical profiles. Additionally, we show that training and inference choices systematically reshape these profiles. Our results suggest that answer-distribution trajectories provide a rich framework for analysing and evaluating the dynamics of LLM reasoning.

In this paper we derive a Probably Approximately Correct (PAC)-Bayesian error bound for partially observed linear time-invariant (LTI) stochastic dynamical systems in state-space form with inputs and sub-Gaussian noise. Such bounds are widespread in machine learning, and they are useful for characterizing the predictive power of models learned from finitely many data points. The bound derived in this paper relates the expectation of prediction errors with the prediction error generated by the model on the data used for learning. In addition, we show that it can also be used to derive bounds for the parameter estimation error. In turn, this allows us to provide finite-sample error bounds for the prediction error and parameter estimation error for a wide class of system identification algorithms. Furthermore, as LTI systems are a sub-class of recurrent neural networks (RNNs), these error bounds could be a first step towards PAC-Bayesian bounds for RNNs.

Formulaic alpha discovery is a pool-dependent symbolic search problem in which informative feedback is observed primarily when a complete expression is evaluated. This delayed feedback creates two coupled difficulties: the retained alpha pool does not preserve the full history of realized evaluation feedback, and the value of an intermediate construction action is uncertain because its consequence depends on the formula eventually completed. We introduce AlphaRJM, which addresses these difficulties through Reward-Jump Memory, an event-driven latent state that remains fixed during token construction and updates only at terminal evaluation events using the realized pool reward and evaluation outcome, and an action-conditioned SDE return critic that represents future discounted discovery returns with stochastic particles. The particles guide action selection through their mean and uncertainty and are learned using a distributional Bellman objective combining energy-distance matching, mean calibration, and jump regularization. Empirically, AlphaRJM delivers strong and stable gains across multiple equity universes, forecasting horizons, and random seeds, while ablations confirm the complementary roles of persistent evaluation history, stochastic return modeling, and distributional supervision.

We study the time-uniform convergence of the raw iterate of standard stochastic gradient descent (SGD) for unconstrained smooth convex objectives. We prove that, under standard noise assumptions, the time-uniform convergence rate gets arbitrarily close to $\sqrt{\log n / n}$ but never reaches it. More specifically, we prove that for every positive, eventually nondecreasing sequence $h$ satisfying $h(n) = o(\sqrt{n})$, a bound of order $h(n)/\sqrt{n}$, holding simultaneously for all $n$ with probability at least $1-α$ and uniformly over the problem class, is achievable if and only if $$\sum_{j = 1}^{\infty} \frac{1}{h(2^j)^2} < \infty.$$ The constructive sufficiency result follows from a dyadic horizon-free schedule together with an additive conditional-restart inequality. The necessity counterpart applies to every deterministic nonnegative schedule and holds even for a one-dimensional analytic smooth convex objective with Gaussian noise.

We study the stochastic first-order oracle complexity for constrained or regularized convex-concave min-max optimization and stochastic monotone variational inequalities. We focus on the case when suboptimality is measured in terms of the gradient mapping, also known as, forward-backward or natural residual, an optimality notion that generalizes the gradient norm for unconstrained problems. In this setting, under standard unbiased oracle access with now-standard variance assumptions, the best-known complexity for making the norm of the gradient mapping less than $\varepsilon$ is $\widetilde{O}(\varepsilon^{-4})$, compared to the near-optimal $\widetilde{O}(\varepsilon^{-2})$ that is established in the unconstrained case. We bridge this gap to improve the gradient mapping complexity for constrained convex-concave min-max problems to $\widetilde{O}(\varepsilon^{-2})$. We then extend to prove the same complexity for problems without the bounded variance, by using the Blum-Gladyshev assumption.

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.

Major life events have proven difficult to predict. Does this reflect limits of theory, data, and algorithms, or the large role of chance? We examine one outcome - having a child within three years - through a near-ideal setting for prediction: a data challenge where 147 researchers predicted births for Dutch residents aged 18-45, using survey data and full-population registers. Methods ranged from logistic regression to a large language model and transformers. Predictions were moderately accurate (best F1: register 0.59, survey 0.76); advanced models did not outperform classical ones; and the larger registers did not beat the survey. Simulating the stochastic biology of conception and pregnancy, we estimated a predictive ceiling (survey F1 ~ 0.86-0.94, register 0.88-0.96). Observed performance falls short of this ceiling, implicating imperfect data, methods, and unmodelled chance, while the ceiling itself shows that chance in reproduction alone sets a non-trivial limit on predicting individual lives.

Large language models (LLMs) achieve strong reasoning performance, which depends critically on inference-time decisions. Yet these decisions are commonly handled by static, one-size-fits-all policies, limiting adaptation to diverse tasks and reasoning stages. Recent adaptive methods partially address this limitation, but they primarily adapt either decoding stochasticity (how the model explores) or reasoning compute (how long the model reasons) in isolation, leaving their interaction within a single reasoning trajectory unmodeled. To address this challenge, we shift toward a within-trajectory joint control view, and instantiate it in AutoCRAT, a decoder-side controller for frozen backbones. Using only signals available during decoding, AutoCRAT jointly adjusts sampling stochasticity and reasoning budget during generation. AutoCRAT operates over a discrete action space and updates control decisions only at semantic boundaries, improving stability while remaining responsive to the evolving reasoning process. Comprehensive evaluation across 6 benchmarks demonstrates that AutoCRAT (I) uses 13.8-52.7% fewer inference tokens on average than recommended static configurations, (II) surpasses recommended static and adaptive baselines by 1.5-4.5% in relative accuracy, and (III) enjoys strong cross-backbone transferability.

We consider the problem of sequential prediction of an $m$-ary sequence, where at each epoch, (i) the environment selects an outcome from an $m$-ary alphabet, (ii) the learner selects a probability distribution over the same alphabet (unaware of the outcome generated by the environment), and finally, (iii) the learner incurs a cost that depends on the probability assigned to the outcome. The cost function we consider captures the complexity of predicting the outcome generated by the environment, in a scenario where the aforementioned prediction is performed via comparative queries to a lying oracle. We consider both stochastic and adversarial environments, propose algorithms for both settings, and establish logarithmic upper bounds on their regret.

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.

Machine learning procedures are commonly evaluated in terms of predictive accuracy and computational efficiency. However, their achievable performance is fundamentally constrained by structural properties of the underlying data-generating process, which are formalized in terms of informational bounds. In this work we examine intrinsic limits of data-driven decision systems from an information-theoretic and interaction-based perspective. We analyze minimal achievable error in classification through Fano-type bounds and precision limits in parametric estimation via the Cramér-Rao inequality, emphasizing that such limits depend on the underlying model rather than on algorithmic sophistication alone. We further discuss how implicit assumptions, such as independence, ergodicity, and distributional stability, affect the validity of inferential procedures. Building on interaction-based modeling principles, we review typical frameworks such as Markov Random Fields and potential based representations for encoding dependence mechanisms. We also describe decision systems, including LLM-integrated agent architectures, as feedback-driven stochastic processes where state-dependent dynamics may induce emergent macroscopic behavior. This perspective highlights the importance of having adequate models for the data as a prerequi- site for expanding predictive capability, and situates algorithmic learning within the informational limits imposed by the models.

Streaming video generation holds strong potential for world modeling, where future frames must be inferred online sequentially to form a continuous video stream. However, streaming video diffusion models introduce a fundamental train-inference mismatch: inference follows a specialized denoising order, whereas advanced training strategies typically require diverse noise-level configurations. To address this trade-off between train-inference consistency and training coverage, we reformulate the video diffusion sampling as a frame-indexed stochastic process over noise levels. Within this stochastic process space, we construct a continuous training trajectory along which the sampling schedule progressively evolves from independent sampling to inference-consistent sampling. We further introduce a joint calibration algorithm and a temporal correlative sampling algorithm to ensure trajectory smoothness and cross-frame correlation. Building on these designs, we propose Stream Forcing, a unified training framework for streaming video generation that balances training sufficiency and inference efficiency. Extensive experiments demonstrate that Stream Forcing significantly improves generation quality with a 36.6% FVD improvement on the UCF-101 benchmark. Furthermore, our method facilitates robust zero-shot extrapolation to long-horizon video generation with a 27.9% FVD improvement on the UCF-101 benchmark.

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.

We present a stochastic coupled emulator of E3SM version 3, built on the SamudrACE framework, which couples an atmosphere emulator (ACE2) with a full-depth ocean emulator (Samudra). We replace the deterministic atmosphere emulator with its stochastic counterpart, ACE2S, and fine-tune the coupled system with a probabilistic objective, so that the atmosphere acts as a source of internal variability for the ocean. Trained on 105 years of a pre-industrial control simulation and evaluated on an independent 400 years, the emulator reproduces E3SMv3's mean climate state with biases much smaller than existing model-to-observation differences. Relative to a deterministic baseline, stochastic training maintains internal variability across timescales, most notably in the ENSO power spectrum, eddy-rich SST anomalies, and sea ice variability in the marginal ice zone. The emulator captures daily precipitation accurately up to the 99.99th percentile, but underestimates the rarest tropical extremes. These results show that stochastic coupled emulators can reproduce long-timescale variability with high fidelity, while extrapolation to unseen extremes remains a key challenge.

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.

Accurate models of underwater vehicle motion are needed for autonomous execution of marine tasks like infrastructure inspection and scientific sampling. However, such motion is challenging to characterize using traditional physics-based methods. This paper presents a novel data-driven framework for learning stochastic underwater vehicle dynamics. Using Euler-Poincaré dynamics and the geometry of Lie groups, we develop a stochastic physics-informed neural network architecture that respects the physical and geometric constraints of underwater vehicles. Our approach leverages structure-preserving stochastic integration and builds upon moment matching and finite dimensional matching to ensure geometrically-consistent training. We evaluate our approach in simulation and on an underwater vehicle navigating dock pylons in a harbor environment. The results demonstrate that our method learns accurate and robust dynamics models, enabling safe model-based control in challenging marine environments.

As firms increasingly deploy machine learning for strategic decision-making, understanding algorithmic interactions has become central to operations research and economics. This paper studies learning in infinite-horizon, nonzero-sum linear-quadratic stochastic games under a radically uncoupled information structure, where players are either unaware of opponents or strategically oblivious, observing only a common state and their own action history. Under this minimal information, we analyze an asynchronous decentralized learning process in which each player independently runs a single-agent $ε$-greedy iterated least-squares algorithm. We prove that, despite being unable to identify the system parameters, players' learning dynamics converge almost surely to the complete-information Nash equilibrium and characterize the convergence rate. We then apply the framework to a dynamic Cournot competition with sticky prices. Numerical experiments validate the theoretical results and show that learning under limited information reduces firm profits under both low and high price stickiness, while total surplus declines and market concentration increases when price stickiness is high. Publicly revealing aggregate market output substantially accelerates convergence and mitigates these welfare losses.

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 cortical self-reconstruction (SCSR) enables personalized mapping of gray matter atrophy, a hallmark of neurodegenerative disorders such as Alzheimer's disease (AD), onto high-resolution cortical surfaces. Unlike conventional normative modeling approaches, which typically operate at a coarse regional level and remain inherently constrained by the covariates included during training, SCSR estimates an individualized healthy reference directly from the observed cortical thickness at the vertex level. This allows the detection of subtle, subject-specific deviations from healthy cortical shape. In this work, we investigate the generalization and transferability of SCSR, originally trained on UK Biobank (UKB) data, to an independent Chinese population dataset. Specifically, we evaluate the ability of SCSR-derived Z-scores to discriminate between healthy scans, individuals with mild cognitive impairment (MCI), and patients with AD, while also assessing model robustness across the lifespan. We compare four training strategies: direct application of the UKB-trained model, fine-tuning on Chinese data, training from scratch, and joint training on UKB and Chinese cohorts. As reconstruction backbones, we consider both a multilayer perceptron (MLP) and a Spherical UNet (SUNet). Our results demonstrate that SCSR provides robust detection of cortical atrophy in the Chinese population across all evaluated models. The highest discriminative performance was achieved by the fine-tuned SUNet model (average pairwise AUC = 0.848), followed closely by the UKB-trained SUNet. Moreover, reconstruction errors remained low across the lifespan, even when the training population exhibited a substantially narrower age distribution, indicating strong cross-population transferability.

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

Persistence diagrams (PDs) provide stable and interpretable summaries of multiscale topological structure. While substantial progress has been made in the statistical analysis of PDs, existing literature often treats diagrams as static objects and provide limited frameworks for probabilistic modeling and stochastic evolution on PD space. We introduce a reinforcement learning framework for stochastic dynamics on PD space, where diagrams evolve through topology aware local edit operations. The dynamics define controlled Markov processes on spaces of finite PDs with variable cardinality. We establish conditions under which the induced Markov chains are irreducible, aperiodic, and geometrically ergodic, implying the existence of unique stationary probability laws on PD space. To guide the dynamics toward scientifically relevant topological targets, we formulate objectives that encompass distribution matching, task specific topological statistics, and structure-preserving compression. The resulting rewards balance task specific distributional targets, diagram fidelity, and complexity reduction, and yield a framework for adaptive topological simplification and probabilistic modeling. Experiments on synthetic and neuroimaging PDs demonstrate that the proposed framework can preserve dominant topological structure while reducing diagram complexity.

When a student must learn concepts connected by prerequisite dependencies, when does the order of instruction matter, and what does it cost to find the best one? We study instructional sequencing as a stochastic shortest-path problem in which attempting a concept succeeds with a state-dependent probability and failure leaves the learner state unchanged. We first prove that this stochasticity can be eliminated exactly: the problem collapses to a deterministic shortest-path problem on the lattice of prerequisite order ideals, preserving optimal values and actions. The collapse removes stochastic complexity but not combinatorial complexity: optimal sequencing remains NP-hard -- via reduction from feedback arc set in tournaments -- even with no prerequisite edges, unit costs, uniform binary nonnegative transfer, and success probabilities at least $1/2$. Hardness is not uniform: when realizable transfer preferences remain jointly acyclic with the prerequisites, any topological order of the residual joint graph is optimal, and fixed prerequisite width yields polynomial-time exact dynamic programming. A computable diagnostic, $mΔ$, bounds the value of sequencing before optimization. On 70,893 interactions from an introductory CS course, the diagnostic certifies a doubly easy regime -- little value to optimize and little space to search -- while constructed transfer instances realize the challenging regime, where myopic sequencing suffers large regret yet exact A* with a consistent heuristic expands only linearly many states on that family.

Control policies optimized in simulation can perform poorly in the real system when the parameters $x$ of the simulator are estimated from limited data but the resulting parameter uncertainty is not represented inside the simulation. A common way to incorporate such ambiguity is to simulate each trajectory of the system under a randomly drawn value for $x$. Since the policy cannot observe the drawn value, it must initially choose controls that perform well across many possible parameter values. However, if the policy progressively observes the system, it can often gradually infer the value of $x$, so that ambiguity vanishes. Over time, the policy then specializes to its estimate of $x$ and loses its robustness. This is undesirable in many real systems, where latent factors are expected to shift. In financial markets, for example, a policy hedging a derivative payoff should remain robust to changes in the volatility regime. To induce such continual robustness, we propose training policies in simulators where ambiguity varies with the system's state but does not systematically decay over time. We formalize this requirement as stationary ambiguity: the simulator should induce a stationary filter process over the latent state. We show how to construct such simulators and demonstrate, on hedging problems, that policies trained under stationary ambiguity preserve robustness to latent factors over time, leading to strong performance on real market data. As a modeling principle, stationary ambiguity informs many simulator design decisions: which models make realistic simulators, how their parameters should be randomized, and how simulator and policy should be initialized. While our experiments focus on hedging, stationary ambiguity may also be useful for other sequential control problems driven by exogenous stochastic processes with shifting latent structure.

Unit excitation (UE) is a common assumption in stochastic saddle avoidance: the stochastic error must have a uniformly positive component along every direction, in expectation. This condition gives a direct way to rule out convergence to strict saddles, but it also oversimplifies the actual noise structure, and does not match many stochastic optimization regimes. In overparameterized or interpolation models, the noise may vanish near stationarity. In finite-sum problems, the stochastic gradient noise may lie in a low-dimensional, data-dependent subspace. In these (common) scenarios, UE is naturally not satisfied. In this paper, we prove an abstract almost sure avoidance theorem for stochastic recursions without UE. The theorem replaces UE-type requirements by verifiable pathwise conditions. In applications, these conditions follow, e.g., from local smoothness and finite-moment assumptions under standard i.i.d. sampling, or from the finite-sum structure under without-replacement sampling. Since the stochastically sampled maps generally do not share a fixed point, the celebrated center-stable manifold argument used in deterministic analyses is not directly applicable. Instead, we use a path-dependent change of variables together with a pathwise Lyapunov--Perron-based proof strategy. As applications, we obtain strict saddle avoidance for stochastic mirror descent (including SGD) and for random reshuffling. For nonsmooth composite objectives, we prove avoidance results for a proximal-type stochastic gradient method. Combining these insights with suitable iterate convergence guarantees, this allows establishing convergence to local minimizers of the original objective function.

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.

Classical neural networks frequently produce overconfident predictions on ambiguous or out-of-distribution (OOD) data, a liability that grows with each AI system deployed in safety-critical real-world scenarios. Bayesian neural networks (BNNs) provide a principled framework for uncertainty-aware prediction by replacing deterministic parameters with probability distributions, but repeated sampling increases latency, memory traffic, and energy consumption. Photonic probabilistic computing offers a promising alternative by exploiting intrinsic optical stochasticity for fast and parallel sampling. However, photonic BNNs are not ideal samplers: analog constraints on quantization, programming error, dynamic range, and representable mean and variance restrict the variational families that can be implemented in hardware. In this work, we study which hardware-imposed constraints limit scalable photonic BNN inference, how these constraints can be represented, and which ranges can be tolerated by photonic BNNs beyond small proof-of-concept networks. We formulate photonic BNN inference as constrained stochastic variational inference and perform a systematic ablation study over stochasticity location, stochasticity modality, quantization, programming error, and mean/variance bounds. From these results, we derive concrete co-design guidelines that distinguish hardware constraints that can be compensated by training from those requiring hardware or architecture intervention. We validate these guidelines under coupled, hardware-realistic constraints on Dirty-MNIST, CIFAR-10, and CINIC-10, using Fashion-MNIST and SVHN as OOD benchmarks, showing that hardware-aware training recovers predictive performance and uncertainty quality whenever the required variational family remains representable, whereas violations of representational limits require targeted hardware modifications.

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.

We present a set of tools for mapping general stochastic programs to thermodynamic hardware designed for energy-efficient stochastic sampling. Given a target stochastic program expressed as a Directed Factor Graph (DFG) of stochastic channels, or equivalently as a Parametrized Stochastic Circuit (PSC), we first introduce a method to approximately compile each factor in the DFG to an Energy-Based Model (EBM) that is native to the hardware. We then analyze how the error of the compiled DFG accumulates from the per-factor errors, and introduce two training refinements, context matching and trajectory-level REINFORCE post-training, which can reduce the residual error left by training each factor in isolation. The \texttt{thermalizers} framework takes a stochastic program expressed in the \texttt{torx} library and replaces its factors with thermodynamic kernels implemented and sampled using the \texttt{thrml} library. We demonstrate it on several example applications, including a market simulator that learns the joint day-to-day dynamics of a panel of financial time series from recorded market history alone, a probabilistic model from mathematical ecology, Gibbs sampling of an EBM the hardware cannot natively express, and a sequential Bayesian design loop over a Gaussian stochastic circuit.

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.

Safe navigation under uncertain time-dependent blockage requires anticipating observations before committing to motion. We present StochSIPP, an exact contingent planner for temporal roadmaps with uncertain edge and vertex statuses revealed locally during execution. StochSIPP uses SIPP to generate certified-safe macro-actions that terminate at the next observation or the goal, and bounded AND/OR search over a cached action--observation graph to select actions for every reachable observation outcome. Optimistic and robust SIPP relaxations provide admissible lower and upper bounds for bounded AND/OR search. When every interval declared deterministically safe is truly safe, sensing is exact, and execution follows the planned timing, the resulting policy is provably collision-free. With correct independent probabilities and complete action and outcome generation, it minimizes expected arrival time within the roadmap and horizon. Experiments on controlled roadmap instances show that StochSIPP preserves the observed success of safe fixed-path baselines while reducing arrival time, and solves gated scenarios in which conservative fixed-path planners return no plan. A scalability study further reveals rapid growth as the number of simultaneously observed uncertain statuses increases.

Thermodynamic computing is an emerging hardware paradigm, in which stochastic physical dynamics serve as the direct computational primitive. The recent explosion of generative AI has only sharpened the search for alternative approaches to compute, and, as we show in this work, thermodynamic computing turns out to be well suited to this space. An important class of methods realises a function as the stationary expectation of an ergodic stochastic process: the answer is encoded in the time-averaged statistics of an equilibrating trajectory. To date, this equilibration-style class has been formulated exclusively through Langevin dynamics, restricting its implementations to analogue substrates and the engineering challenges those bring. In this work, we propose a substrate-independent formalisation of the equilibration-style formulation, in which the only object of design is the dynamical generator L* of an arbitrary ergodic process. The formalisation makes three hardware-level properties of the formulation explicit: the precision of a result is a knob set by how long the dynamics are run, sample averages decompose across independent trajectories, and dependent stages of a computation operate concurrently rather than serially, a property we call sequential parallelism. We instantiate the formalisation by fabricating a prototype digital thermodynamic computing chip, named CN101, that implements the formulation through discrete accumulator dynamics on standard CMOS using stochastic computing principles. We characterise CN101's success across conventional generative AI workloads in the form of VAEs and flow matching, applied to both image generation and scientific problems. Together, the formalisation and its digital instantiation show that the equilibration-style formulation is substrate-independent, and that its computational properties can be exploited on standard digital hardware.

A calibrated stochastic world model can reveal how uncertain a future is without revealing why it branches. The same conditional future law can arise because an observation aliases physical states or because dynamics remain random after the declared full state is fixed. We prove that ordinary transitions cannot identify these two sources, even for a perfect probabilistic predictor. ClosurePairs makes them identifiable by crossing compatible microstates with repeated exogenous disturbances and estimating state, noise, and state-noise interaction variance. The central consequence is operational: under finite hierarchical sampling, forecast difficulty governs the useful compute scale, while the alias/process composition provides complementary information about its direction-resolving the current state or sampling future randomness. ClosurePairs recovers source attribution at unchanged likelihood, reduces equal-budget decomposition error in a nonlinear interaction benchmark, and supports observation-only routing. On exact-marginal MetaWorld twins, an output-only allocator is at chance while a Closure-supervised probe on frozen JEPA-WM features routes 89.8-100%. In an independent ManiSkill PushCube confirmation, a stochastic RSSM's outputs and latents remain at chance, whereas an RGB-only Closure probe routes 100% under both ID and geometry/camera OOD over five seeds, matching direct allocation rather than exceeding it. Across five unseen allocation menus, the same Closure probe routes 92.5%/90.4% ID/OOD with no new oracle labels, versus 37.9%/32.9% for a frozen direct allocator. ClosurePairs is therefore an identifiable, reusable mechanism target that cannot be recovered from forecast quality alone.

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.

Oversmoothing is a fundamental limitation of deep graph neural networks (GNNs), where repeated message passing causes node representations to become increasingly similar, eventually collapsing toward a low-dimensional subspace. This phenomenon limits the effective depth of message-passing architectures and motivates the search for mechanisms that preserve representation diversity. In this paper, we study a recurrent graph neural network in which independent Gaussian noise is injected after every propagation step and analyze the resulting architecture as a stochastic dynamical system. Under a standard global contraction assumption on the deterministic update, we prove that the hidden representations form a geometrically ergodic Markov chain admitting a unique invariant probability measure. Our main theoretical result establishes an explicit positive lower bound on the expected stationary Dirichlet energy, proportional to both the noise variance and the spectral gap of the underlying graph. Consequently, the stationary representations cannot collapse onto the constant manifold, providing a rigorous guarantee that asymptotic oversmoothing is prevented in the sense of non-vanishing Dirichlet energy. Our analysis reveals persistent stochastic perturbations as a fundamentally different mechanism for combating oversmoothing, complementing existing deterministic approaches based on residual connections, normalization, and graph rewiring. Finally, numerical experiments on both linear and nonlinear recurrent graph neural networks closely match the theoretical predictions, illustrating the emergence of a stationary distribution and the predicted dependence of the limiting Dirichlet energy on the noise intensity.

Engression (Shen and Meinshausen, 2024) learns a conditional distribution by fitting a generative model $Y = f(X,\varepsilon)$ under the energy score, a strictly proper scoring rule. We provide a theoretical error analysis of engression implemented with deep neural networks. We decompose the excess risk into three components: the approximation error, the stochastic error, and the Monte Carlo error. Based on this decomposition, we establish convergence rates under the assumption that the target conditional generator admits a compositional smoothness structure.

A capability appears in a language model when the last parts of its circuit align in one stochastic attempt, and getting all but one right is worth nothing. We show this no-partial-credit joint alignment is the rate-limiting step of capability formation. Two fingerprints: in a shortcut-free apparatus a five-part circuit missing three waits as long as a three-part circuit missing three (1.19-1.37), so the wait counts missing parts, not size; and on Pythia across seven capabilities and three scales, ablating one part leaves a median 17% of the capability in 32 of 32 discriminating cells, where partial credit predicts 50-83% (p = 2e-10), while a random non-part head leaves 100%. One rare event whose barrier grows with missing parts yields a rate equation -- sites x attempts x drive x exp(-beta*K), minus destruction -- read three ways, each preregistered with frozen constants. Forward: a capability flat at baseline ignites at a step of our choosing once the mix passes a concentration floor (10/10 above, 0/12 below), and while still flat its arrival is datable from its precursor to 5% median error on six held-out models. Backward: the delay to learn a withheld capability grows with waiting until, past a critical step, it never ignites -- yet validation loss falls smoothly throughout, so standard monitors are blind to it. We locate the damage (heads commit to the base data) and isolate the cure: re-initializing only the query-key slices restores learnability (6/6) while the value slices do nothing (0/6). We prove the mechanism in a controlled gated-attention model: occupation forces a deadline whose consequences need no mixing assumption. Completed: SGD's noise fails the fluctuation-dissipation test, so we install one and anneal, melt and pin circuits on schedule. Scope: conjunction circuits in transformers to 1.4B.

Image inpainting aims to recover missing regions while preserving structural consistency. We propose a non-parametric method without network training based on data-guided stochastic dynamics. Starting from a masked image, the missing pixels are evolved through a reverse-time stochastic differential equation with a kernel-weighted correction estimated directly from a reference dataset. This empirical correction guides the reconstruction toward high-density regions of the data distribution without training a neural network or fitting a parametric density model. Experiments on MNIST, Fashion-MNIST, and MVTec show that the proposed method outperforms Mean Fill, Telea, and Navier-Stokes inpainting in PSNR, SSIM, and visual quality. On CelebA, it remains competitive and produces plausible completions for structure-sensitive occlusions. These results demonstrate the effectiveness of empirical reference statistics as a non-parametric prior for image inpainting.

Volatility forecasting is dominated by persistence and measurement noise, leaving limited residual structure for nonlinear models to exploit. We introduce Susceptible Architectures (SUSA), a reservoir-design principle for volatility forecasting, and its two concrete implementations, based on complex-valued open-chain and periodic reservoirs and regime-conditioned experts to interpret reservoir features across calm, onset, recovery, and persistent-stress states. We also implement open-system $q$-qubit counterparts in Qiskit while retaining a common AR-Ridge anchor and a bounded residual correction trained under QLIKE. We evaluate models on 16 U.S. equity and exchange-traded-fund series using three disjoint chronological training, validation, and test folds, a 12-observation input window, and a five-observation forecast horizon. The proposed models perform competitively with GARCH, achieving statistically significant QLIKE improvements for specific assets (IWM, XLP). Also models' forecasts complement HARQ-style predictions: a stacked ensemble improves mean QLIKE by 0.0116 over its strongest constituent and wins in 75% of test scenarios.

We consider the problem of learning from a single finite trajectory of an ergodic stochastic dynamical system. More precisely, we study discrete-time autonomous stochastic systems defining time-homogeneous Markov processes. We first focus on estimating the optimal one-step prediction function by nonlinear least squares, and derive high-probability guarantees measured with respect to the invariant measure of the process. These results make explicit how the non-independent and non-identically distributed nature of trajectory data modifies the classical statistical learning analysis. We then extend the framework to higher-order systems and finite-state spaces. Finally, we show that the same least squares and concentration arguments naturally extend to learning Koopman operators. Our approach combines tools from statistical learning theory and quantitative ergodic theory for Markov chains. It relies, in particular, on a concentration inequality for Hilbert-space-valued additive functionals of uniformly geometrically ergodic Markov chains.

Low-rank adaptation (LoRA) optimizes $J(B,A)=\mathcal L(W_\mathrm{base}+sBA)$ over two adapters $B \in \mathbb{R}^{m \times r}$ and $A \in \mathbb{R}^{r \times n}$ that form a low-rank update to a frozen pretrained weight matrix $W_\mathrm{base} \in \mathbb{R}^{m \times n}$. The prior analysis shows LoRA-GD takes $\exp\{\mathcal{O}(ε^{-2})\}$ oracle calls to find an $ε$-stationary point such that $\|\nabla J(B,A)\|\leq ε$ in the deterministic setting. We sharpen the analysis and show that $\mathcal{O}(ε^{-4})$ full-gradient evaluations suffice for the same first-order criterion. We further study stochastic LoRA under unbiased gradient estimates and finite variance. We propose LoRA-NSGDM, which finds an $ε$-stationary point with $\mathcal{O}(ε^{-8})$ stochastic oracle complexity. Under the additional mean-square smoothness condition, we use variance reduction strategy and propose LoRA-STORM, which improves the stochastic oracle complexity to $\mathcal{O}(ε^{-6})$.

This paper addresses multi-objective kinodynamic planning in environments with stochastic hybrid adversaries that probabilistically transition to adversarial modes based on the ego state. The goal is to construct the Pareto-front of paths that trade off execution cost and the probability of safety constraint violation (risk). Existing chance-constrained planners evaluate risk over open-loop trajectories, yielding overly conservative solutions that fail to account for ego-agent reactivity. To address this limitation, we shift the planning space to sequences of closed-loop policies, and integrate sample-based risk evaluation directly into tree construction via Monte-Carlo particle rollouts. We first introduce Stochastic Multi-Objective RRT (SMO-RRT), for which we prove probabilistic completeness, followed by Stochastic Multi-Objective Stable Sparse RRT (SMO-SST), which leverages selective pruning to improve numerical performance at the cost of completeness. For both algorithms, we derive a finite-sample bound on the probability of chance constraint violation for systems with non-Gaussian, state-dependent uncertainty, enabling probabilistically safe planning in a broad class of environments applicable to multi-agent systems, social navigation, and autonomous driving.

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.

Gaussian graphical model selection is usually studied under independent sampling, but in many applications the data arise as a single trajectory of a dependent stochastic process. We study exact recovery of the graph from one trajectory of random-scan Gaussian Glauber dynamics. Existing techniques for this problem either inherit the mixing time of the chain, which can be super-polynomial in the dimension $p$ without strong assumptions, or are suboptimal in the minimum normalized edge strength $κ$. We propose two algorithms that are mixing-free and attain the $κ^{-2}$ dependence of the information-theoretic lower bounds. Both instantiate a shared dueling-neighborhood search meta-algorithm with a local statistic built directly from the update sequence. For every fixed precision matrix and deterministic initialization, the first algorithm fits a least-squares regression at the updates of each node and has pointwise recovery horizon $\widetilde O(pd^{2}/κ^{2})$, where $d$ is the maximum degree. Its horizon depends logarithmically on a local conditioning quantity and on the initialization potential. The second algorithm is based on counting occurences of a specific update pattern and requires $\widetilde O(pd^{4}/κ^{2})$ updates, with no dependence on any condition number. The central technical challenge is that both statistics are built from dependent, non-stationary observations. Our analysis tackles this by demonstrating how to extract fresh Gaussian innovations from the update sequence, which yields mixing-free control of appropriate quantities. Neither the algorithms nor their analyses invoke stationarity, a spectral gap, or mixing conditions.

Quantum machine learning models define probabilistic input--output maps through coherent quantum evolution and measurement. While such models can exhibit computational advantages, their internal functioning and decision making generally resists interpretation in terms of stochastic trajectories through intermediate configurations. In contrast to classical (Markovian) stochastic processes, quantum dynamics generically violates the Chapman--Kolmogorov divisibility condition, preventing a decomposition into probabilistically meaningful intermediate transitions. We develop a probabilistic framework for representing quantum learning models as stochastic processes over configuration spaces where the dynamics are modeled as linear maps on probability distributions. Starting from a fixed POVM, arbitrary quantum channels induce transition kernels on the associated probability representation. For informationally complete POVMs, and in particular SIC-POVMs, these kernels are Markovian but generally quasi-stochastic, with non-classicality appearing as negativity. By contrast, projective spaces admit positive stochastic kernels but generally require non-Markovian dynamics due to the failure of Chapman--Kolmogorov divisibility. This yields a trade-off between negativity and dependence on past configurations, i.e. quantum dynamics can be represented either by Markovian quasi-stochastic maps or by positive stochastic processes with higher Markov order. We discuss how such representations of quantum dynamics can be interpreted as stochastic walks through a memory space in the spirit of Projective Simulation, a model of learning and agency in which decisions arise from random walks over an episodic memory network. We further outline how finite-order stochastic kernels can approximate such quantum deliberation processes and show in what regimes the classical machine learning model is recovered.

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.