compute transient distributions

This work addresses the lack of finite-time theoretical analysis for discrete-time stochastic interpolation generative models by establishing, for the first time, an explicit error upper bound on their convergence rate. Methodologically, it models the sampling dynamics via discrete-time stochastic differential equations and systematically quantifies how the distance between source and target distributions and the gradient estimation error jointly govern the convergence rate. Based on this analysis, it proposes a theory-guided principle for designing accelerated sampling schedules. Key contributions include: (1) the first provably convergent discrete-time sampler for interpolation-based generative modeling; (2) explicit characterization of the convergence rate’s dependence on both distributional distance and gradient estimation accuracy; and (3) empirical validation on image generation tasks, where numerical experiments confirm the theoretical predictions and demonstrate significant improvements in sampling efficiency.

Traditional numerical methods struggle to efficiently solve the transient Fokker–Planck equation with arbitrary initial distributions and system parameters in a parallelizable manner, hindering comprehensive parameter-space exploration and transient analysis. This work proposes the Pseudo-Analytical Probabilistic Solution (PAPS) framework, which jointly models solutions across arbitrary multimodal initial conditions, system parameters, and time through a single training procedure. The key innovation lies in unifying initial, transient, and steady-state distributions into a Gaussian mixture representation, employing a constraint-preserving autoencoder to establish a bijective mapping between high-dimensional constrained parameters and a low-dimensional unconstrained latent space, and constructing a physics-informed evolution network to capture global dynamics across varying parameters and initial conditions. Experiments demonstrate that PAPS achieves high accuracy on benchmark systems, with inference speeds four orders of magnitude faster than GPU-accelerated Monte Carlo simulations, enabling real-time parameter sweeps and stochastic bifurcation analysis.

This work addresses key challenges in particle-based Bayesian inference for continuous-discrete state-space models (CD-SSMs): (i) absence of closed-form transition densities for Itô diffusions; (ii) non-resamplability of deterministic ancestral paths over continuous trajectories; and (iii) posterior collapse of diffusion parameters to Dirac distributions. To overcome these, we introduce a novel path-space Feynman–Kac formulation that unifies guided path proposals, differentiable reparameterization, and particle filtering/smoothing—enabling the first differentiable and resamplable particle inference framework for CD-SSMs without requiring closed-form transition densities. The method supports hypoelliptic diffusions and integrates seamlessly with probabilistic programming frameworks, enabling both online and offline parameter learning while scaling to high-dimensional settings. It has been implemented in mainstream probabilistic programming libraries and empirically validated across multiple numerical experiments, demonstrating superior robustness and accuracy compared to state-of-the-art alternatives.

This work addresses the problem of efficiently and accurately bridging arbitrary probability density functions within a bounded time horizon. Methodologically, it introduces a unified generative modeling paradigm based on stochastic interpolation processes, seamlessly integrating flow-based and diffusion-based models—supporting both deterministic ordinary differential equation (ODE) paths and stochastic differential equation (SDE) paths with tunable noise. A novel score-matching objective is derived for the first time; theoretical analysis proves that optimizing only a quadratic loss suffices for likelihood control, overcoming the traditional limitation of deterministic models requiring additional Fisher divergence regularization. By unifying Schrödinger bridge theory, the Fokker–Planck equation, and variational inference, the framework rigorously recovers the Schrödinger bridge solution under optimal interpolation and provides a unified estimator for both likelihood and cross-entropy.

This work addresses the modeling challenge of non-Markovian stochastic differential equations (SDEs), where temporal correlations induced by colored noise invalidate conventional Markovian SDE frameworks. We propose a generalized motion coordinate theory that unifies treatment of both Markovian and non-Markovian systems—driven by white or colored noise—via pathwise analysis and an extended state-space formulation. Innovatively, we construct the first generalized coordinate framework enabling exact short-time solutions and global long-time analytical characterization—including flow and perturbation analysis—for non-Markovian SDEs, circumventing the approximation limitations of traditional Markovian embedding approaches. Our methodology integrates rough path theory, analytical flow analysis, generalized Bayesian filtering derivation, and efficient numerical simulation. Key contributions include: (i) exact solutions for linear SDEs with analytically characterized perturbations; (ii) reconstruction of generalized Bayesian filtering; (iii) novel high-accuracy algorithms for simulation, filtering, and control; and (iv) smooth path approximations for rough SDEs.

This work addresses the challenge of uniformly modeling multiple action delays—including stochastic, deterministic, and general delays—in concurrent systems under discrete time. The paper proposes the dtphPBC framework, which introduces discrete phase-type distributions into Petri Box Calculus for the first time. By leveraging the transition probability matrix of discrete-time Markov chains with an absorbing state, the approach precisely characterizes diverse delay behaviors. A structured operational semantics seamlessly integrates instantaneous and timed actions, yielding a step semantics grounded in labeled probabilistic transition systems. This model coherently accommodates both zero-delay and positive-delay actions, enabling accurate representation of complex timing characteristics. The framework’s semantic consistency and constructiveness are demonstrated through illustrative examples.

Existing probabilistic programming languages lack native support for dynamic systems—particularly state-space models—hindering the broader adoption of Bayesian methods in this domain. This work introduces dynestyx, a library that provides first-class, unified, and user-friendly support for state-space models within a probabilistic programming framework. dynestyx enables flexible specification of priors, accommodates both discrete- and continuous-time dynamics, handles mixed-effects data, and facilitates joint Bayesian inference over latent states and model parameters with full uncertainty quantification. By doing so, this contribution substantially enhances the accessibility, flexibility, and practical utility of dynamic system modeling across statistics, signal processing, and machine learning.

This study addresses the failure of conventional delay coordinate reconstruction and convergent cross mapping in stochastic systems due to inherent uncertainty. To overcome this limitation, we propose a conditional moment-based stochastic delay coordinate reconstruction method. The core innovation lies in extending Takens' embedding theorem to stochastic settings for the first time, proving that Mori-Zwanzig orthogonal noise terms vanish in coefficient space and thereby establishing a theoretical foundation for moment closure approximation. By integrating the Fokker-Planck equation with Koopman operator theory, the proposed approach achieves causal diagnosis through finite-order moment approximations of future distributions. Experimental evaluations on coupled logistic systems successfully identify causal relationships under both additive and multiplicative coupling schemes, validating the effectiveness of the method.

This study addresses the absence of a unified closed-form solution and systematic numerical validation for the M/PH₂/1 queueing system by deriving, for the first time, explicit analytical expressions for both the queue-length and sojourn-time distributions. It fully specifies the coefficient formulas for the PH₂-type service-time distribution and establishes PH₂ as the boundary of algebraic solvability. The approach integrates algebraic derivation, matrix-analytic methods, and symbolic computation, complemented by large-scale numerical experiments implemented via BuTools. Results demonstrate that queue-length computations achieve machine precision even under extreme conditions—such as high traffic intensity (ρ ≈ 0.999) and very high service-time variability (Cₛ² > 10⁴)—while sojourn-time calculations retain at least 10 significant digits of accuracy, thereby enabling reliable sensitivity analysis, threshold optimization, and tail-probability guarantees.

This work addresses the challenge of efficiently sampling from Gibbs distributions in complex energy landscapes characterized by barriers or metastable states. The authors propose a hybrid stochastic dynamics framework that employs two distinct sampling dynamics in different regions of the state space, coupled at their interface through a natural transmission condition that preserves the target distribution. By introducing a regularization mechanism, they establish—for the first time—the exponential convergence rate of this hybrid dynamics. In radially symmetric potentials, the method significantly reduces the mean escape time compared to conventional approaches. Both theoretical analysis and numerical experiments demonstrate that the proposed scheme offers marked improvements over traditional sampling strategies in terms of convergence speed and the ability to overcome metastability.