标题： 现代随机建模综述：SDE/SPDE数值方法、数据驱动识别以及生成方法在生物学和流行病学中的应用 显示英文标题

标题： A Review of Modern Stochastic Modeling: SDE/SPDE Numerics, Data-Driven Identification, and Generative Methods with Applications in Biology and Epidemiology

摘要： 这篇综述梳理了随机建模的发展，突出了非标准方法及其在生物学和流行病学中的应用。 它汇集了四个方向：(1) 描述随机演化的系统的核心模型；(2) 直接从数据中学习这些模型的关键部分；(3) 能够在连续时间中生成现实合成数据的方法；以及 (4) 在长时间运行中保持模拟稳定、准确和忠实的数值技术。 目标是实用的：帮助研究人员快速了解有哪些新内容，各部分如何结合，以及哪些重要空白仍然存在。 我们总结了在噪声和不完整观测下估计变化的感染或反应率的工具，建模空间传播，考虑突然跳跃和重尾，以及以对决策有用的方式报告不确定性。 我们还强调了值得近期关注的开放问题：当数据不规则时分离真实动态与噪声；在随机影响下学习具有稳定性保证的空间动态；将训练与实际应用中使用的数值方法对齐；在所有模拟中保持正性和守恒性；在大型研究中降低成本同时控制误差；估计罕见但重要的事件；以及采用清晰、可比较的报告标准。 通过围绕这些目标组织该领域，这篇综述提供了一个当前方法的简明指南，它们的实际应用，以及生物学和流行病学未来工作的最有希望的方向。 显示英文摘要

摘要： This review maps developments in stochastic modeling, highlighting non-standard approaches and their applications to biology and epidemiology. It brings together four strands: (1) core models for systems that evolve with randomness; (2) learning key parts of those models directly from data; (3) methods that can generate realistic synthetic data in continuous time; and (4) numerical techniques that keep simulations stable, accurate, and faithful over long runs. The objective is practical: help researchers quickly see what is new, how the pieces fit together, and where important gaps remain. We summarize tools for estimating changing infection or reaction rates under noisy and incomplete observations, modeling spatial spread, accounting for sudden jumps and heavy tails, and reporting uncertainty in a way that is useful for decisions. We also highlight open problems that deserve near-term attention: separating true dynamics from noise when data are irregular; learning spatial dynamics under random influences with guarantees of stability; aligning training with the numerical method used in applications; preserving positivity and conservation in all simulations; reducing cost while controlling error for large studies; estimating rare but important events; and adopting clear, comparable reporting standards. By organizing the field around these aims, the review offers a concise guide to current methods, their practical use, and the most promising directions for future work in biology and epidemiology.s.