diffusion approximation analysis

This work addresses the challenge of modeling a single nonstationary, non-ergodic stochastic differential equation (SDE) trajectory—a setting where conventional SDE identification methods fail due to their reliance on ergodicity or stationarity assumptions. Method: We propose Stochastic Sparse Identification of SDEs (SSISDE), the first data-driven algorithm capable of jointly estimating drift and diffusion functions while reconstructing Brownian increments from a single trajectory. SSISDE integrates stochastic Taylor expansions with the Girsanov transformation, constructing solvable estimators initialized from drift function approximations—bypassing the need for stationary or ergodic data. Contribution/Results: SSISDE establishes the first SDE modeling paradigm tailored to single-trajectory, nonstationary, non-ergodic regimes. It achieves high-fidelity model discovery on benchmark nonstationary linear and quadratic systems—including Black–Scholes dynamics—significantly improving estimation accuracy over existing approaches. This framework enables real-time, interpretable modeling of complex dynamical systems in domains such as finance and biophysics, where ergodic assumptions are fundamentally violated.

This work addresses the challenge of simulating sample paths for stochastic differential equations (SDEs) with gradient drift and unit diffusion coefficients under noisy observations, where existing methods often suffer from discretization bias or high sampling complexity. The authors propose an exact Gibbs sampling framework that enables unbiased path simulation without temporal discretization and naturally integrates Gaussian process tools to facilitate parameter inference. This approach achieves, for the first time, discretization-free MCMC sampling for a broad class of SDE models, handling both univariate and multivariate cases within a unified framework—without requiring rejection sampling or debiasing techniques. Empirical evaluations on synthetic and real-world data demonstrate clear advantages over particle MCMC methods, offering superior accuracy and computational efficiency.

A theoretical-practical gap persists in score-based diffusion models. Method: We propose a unified, reproducible SDE-based modeling framework that systematically integrates score matching, SDE/ODE solvers, denoising score estimation, and consistency modeling; notably, we introduce reinforcement learning into diffusion sampling for inference-path optimization. Contributions: (1) We establish theoretical consistency between sampling and score estimation under the SDE formulation; (2) we provide concise proofs of key theorems alongside practical algorithmic implementation guidelines; (3) we release modular, open-source code enabling rapid validation and extension to novel architectures. This work bridges the efficiency of score matching with scalable, RL-enhanced inference, delivering a foundational toolkit that balances theoretical rigor and engineering practicality for the design, analysis, and application of diffusion models.

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 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 study addresses the challenge of estimating diffusion parameters in stochastic differential equation (SDE) models when data and model are compatible only at specific scales. The authors propose an adaptive subsampling method based on the statistics of monotonic runs. By demonstrating that, for a broad class of additive-noise SDEs, the length of monotonic runs at infinitesimal scales approximately follows a geometric distribution with success probability 1/2, they establish a general criterion for selecting the subsampling rate without relying on multiscale diffusion asymptotics. The optimal sampling scale matching the SDE’s infinitesimal behavior is automatically determined solely from the statistical properties of monotonic increasing or decreasing segments in the observed time series. Validation on surrogate modeling of fiber lay-down trajectories in nonwoven fabric production demonstrates that the method yields highly accurate and model-consistent diffusion parameter estimates, proving effective in real-world industrial applications.

This study addresses the estimation of the time-homogeneous drift function in multivariate stochastic differential equations (SDEs) with known diffusion coefficients, based on high-frequency observations from multiple trajectories. To this end, the authors propose formulating drift estimation as a conditional denoising problem conditioned on historical observations and introduce a conditional diffusion model that dynamically generates new trajectories from which the drift estimator is extracted. This approach represents the first application of conditional denoising diffusion models to drift estimation in SDEs. It significantly outperforms classical methods in high-dimensional settings without relying on any specific neural network architecture, while achieving comparable performance to existing approaches in low-dimensional cases, thereby demonstrating both its effectiveness and scalability.

This work establishes a unified theoretical framework for diffusion models from the perspective of differential equations. Starting from a conditional Gaussian forward process, it derives the corresponding forward stochastic differential equation (SDE) and ordinary differential equation (ODE), and constructs a dynamical system that transports the data distribution to a standard Gaussian prior via marginalization. The framework then introduces a reverse SDE and a probability flow ODE, both driven by the marginal score function, thereby unifying score matching and noise prediction objectives. It rigorously demonstrates the equivalence of DDPM and DDIM in their training objectives while clarifying their fundamental distinction in sampling mechanisms—DDPM corresponds to a discretized reverse SDE, whereas DDIM implements a reverse ODE. Furthermore, the framework seamlessly incorporates mainstream sampling techniques such as DPM-Solver and classifier guidance, providing a coherent and rigorous continuous-time foundation for diffusion models.

This work addresses the computationally expensive inverse problem of parameter estimation for stochastic differential equations (SDEs) by proposing an efficient solution framework that, for the first time, integrates Wiener chaos expansion (WCE) with stochastic gradient descent (SGD). By projecting the stochastic solution onto a deterministic system of propagators via an orthogonal Hermite polynomial basis, the method constructs a regularized discrepancy functional amenable to SGD optimization. This transformation effectively converts the original stochastic inverse problem into a deterministic optimization task, substantially reducing computational complexity and data requirements. Numerical experiments on several nonlinear SDE models—including a biological individual growth model—demonstrate that the approach accurately and robustly recovers parameters even from sparse and noisy observational data, exhibiting strong scalability and practical promise.

This study addresses the limitations of existing Bayesian inference methods for switching stochastic differential equations (SSDEs), which typically rely on analytical transition densities and are constrained by noise assumptions and dimensionality. To overcome these bottlenecks, this work proposes an approximate Markov chain Monte Carlo (MCMC) sampling framework based on homogenization and factorized neural likelihood estimation (FNLE). By modeling state switching via continuous-time Markov chains and incorporating time-conditioned FNLE, the method achieves general and efficient inference without requiring analytical transition densities, effectively circumventing restrictions to linear drifts and low-dimensional settings. Experiments demonstrate that the proposed approach successfully recovers multi-model parameters on synthetic data and accurately detects state transitions in real-world datasets, thereby validating its broad applicability.