stochastic trajectory reconstruction

This study addresses the challenge of reconstructing nonstationary time series when only sparse extremal observations—such as glacial moraine positions—are available. The authors propose a novel data-driven framework that integrates record theory with Brownian motion-based stochastic simulation to reconstruct full temporal trajectories. A neural-inspired Bayesian inference (NBI) method is introduced to automatically optimize the hyperparameters of the trajectory generator. Applied to the historical fluctuations of the Bossons Glacier in France, the approach successfully reconstructs a high-confidence, continuous time series of glacier terminus positions without requiring additional prior assumptions. This work establishes a new paradigm for studying glacier dynamics and climate response and is readily generalizable to other scientific domains where only extreme-value records are accessible.

To address the challenge of modeling irregularly sampled time series, this paper proposes a continuous-time generative framework based on trajectory flow matching, unifying stochastic differential equations (SDEs) and jump processes. Methodologically, it pioneers the adaptation of the flow matching paradigm—originally developed for image generation—to irregular time series, explicitly modeling discrete jump events. A learnable scaled Gaussian jump kernel is introduced, and a closed-form solution for the KL divergence under its joint dynamics with the SDE is derived, enabling efficient end-to-end optimization. Experiments on multiple benchmark datasets demonstrate that the method significantly outperforms existing continuous-time generative models, achieving both high-fidelity sample generation and theoretical rigor.

For stochastic processes exhibiting complex features—such as skewed marginal distributions, non-Gaussian heavy tails, and long-range dependence—parameter estimation lacks closed-form solutions and thus heavily relies on simulation-based inference (SBI). However, existing SBI methods suffer from excessive simulation requirements, architectural complexity, and severely undercalibrated credible intervals. This paper proposes a sample-efficient and robust SBI framework: it innovatively combines stepwise parameter dimensionality decomposition with Chebyshev polynomial approximation to achieve high-accuracy, low-dimensional posterior density modeling. Complementary diagnostic tools and a post-hoc calibration mechanism enable model reuse across time series of varying lengths, substantially reducing training overhead. Experiments on trawl processes demonstrate that our method maintains high inferential accuracy even under poor MCMC mixing, while yielding credible intervals that strictly attain nominal coverage.

This work addresses the challenge of generating conditional stochastic processes in continuous spatiotemporal settings from arbitrary observation subsets—such as irregularly sampled data or future frames—by proposing an autoregressive generative framework based on non-Markovian diffusion bridges. The method unifies physical time and state evolution within a single continuous stochastic differential equation (SDE), innovatively initializing from neighboring states, injecting noise proportionally to temporal intervals, and explicitly embedding time into the SDE dynamics. The SDE is derived via path-space measure transformation, and training is performed using a path- and time-dependent denoising score matching algorithm. Empirical evaluations on video generation and weather forecasting demonstrate significant improvements over existing approaches, particularly under low-step sampling and irregular conditioning scenarios.

This work unifies diffusion sampling and stochastic localization under a single theoretical framework, addressing the lack of rigorous theoretical connections between them and the limited applicability of existing algorithms. Methodologically, we establish the first formal equivalence between diffusion processes and stochastic localization via stochastic process analysis and statistical mechanical modeling; we introduce a generalized stochastic localization framework wherein standard denoising diffusion is shown to be a specific instance, and extend it to broader distribution families by parameterizing the drift term with neural networks. Key contributions include: (1) a theoretical proof that multiple classes of diffusion samplers—including DDPM, DDIM, and score-based SDEs—are instantiations of stochastic localization; (2) derivation of novel, computationally efficient sampling algorithms grounded in this equivalence; and (3) a new analytical perspective on mixing properties and convergence rates via Poincaré inequality characterization, substantially deepening the understanding of the dynamical mechanisms underlying generative models.

Industrial research agents often generate experimental trajectories containing invalid or incomplete information, rendering them unreliable for direct decision-making. This work proposes an evidence-oriented framework that automatically transforms such trajectories into structured evidence through a context-isolated generate–verify–repair pipeline. The approach introduces intervention-level claim categorization—distinguishing actionable repairs, diagnostic safeguards, and retained discoveries—and incorporates end-to-end provenance tracking to enable claim scoping and auditability. Experimental results demonstrate that the resulting candidate solutions outperform existing baselines. Audits further reveal that trajectory evolution is non-monotonic, and that applicability assessment constitutes a key performance bottleneck for the controller.

研究了在固定区间[0,a]内的布朗运动和布朗桥过程，通过精确插值和外推法解决了条件分布问题，并提出了统一的精确采样方法。

研究通过无训练方法将冻结的时间序列基础模型边际结合成多变量预测样本路径，以改善依赖关系诊断。

This work investigates the theoretical underpinnings of memorization and overfitting in stochastic interpolation generative models. Focusing on continuous-time stochastic differential equations and their Euler discretization, it provides the first rigorous theoretical definitions of overfitting and underfitting in generative modeling and derives closed-form expressions for the optimal velocity field and score function. The analysis reveals that generated samples can be expressed as training samples perturbed by three controllable error terms, whose bias is jointly determined by the discretization step size and estimation error. Synthetic experiments corroborate the theoretical prediction that generated samples cluster around the training data distribution, highlighting the critical roles of error accumulation and noise modeling in the model’s reconstruction capability.

This work proposes a neural-network-free feature-augmented forecasting approach for time series generated by Itô-type stochastic differential equations. The core idea lies in statistically reconstructing the unknown drift and diffusion coefficients directly from observed data: uniform reconstruction is employed alongside a non-uniform variant—equivalent to a stochastic Taylor expansion—to capture the state-dependent structure of these coefficients. A Gaussian mixture model is further integrated to achieve statistical separation of the reconstructed components. The resulting reconstruction parameters are incorporated as additional features into an autoregressive model, substantially enhancing predictive accuracy. Empirical results validate both the informational richness of these statistically derived features and the practical utility of the proposed methodology.