用随机展开+梯度下降，高效估计随机微分方程参数。
Parameter Estimation in Stochastic Differential Equations via Wiener Chaos Expansion and Stochastic Gradient Descent
- 通过威纳混沌展开将随机方程转为确定性传播器系统。
- 从离散噪声观测中准确恢复参数，误差小且计算快。
- 适合生物生长等复杂随机系统建模，替代传统采样方法。
本研究针对随机微分方程（SDEs）的参数估计逆问题，通过最小化正则化偏差泛函并结合随机梯度下降（SGD）求解。为提升计算效率，采用威纳混沌展开（WCE）——一种将随机解投影到厄米多项式正交基上的谱分解技术。该方法将随机动力学转化为一组确定性函数构成的层级系统，称为“传播器”。由此，原本复杂的随机推断被降维为确定性优化问题，避免了传统基于模拟的方法（如MCMC或MLE）所需的高计算开销与采样过程。在多种非线性SDE模型（包括个体生物生长模型）上的数值实验表明，所提出的WCE-SGD框架即使在离散、含噪观测下仍能实现高精度参数恢复，显著提升了复杂随机系统建模的效率与可扩展性。
原文摘要 · Abstract (English)
This study addresses the inverse problem of parameter estimation for Stochastic Differential Equations (SDEs) by minimizing a regularized discrepancy functional via Stochastic Gradient Descent (SGD). To achieve computational efficiency, we leverage the Wiener Chaos Expansion (WCE), a spectral decomposition technique that projects the stochastic solution onto an orthogonal basis of Hermite polynomials. This transformation effectively maps the stochastic dynamics into a hierarchical system of deterministic functions, termed the \textit{propagator}. By reducing the stochastic inference task to a deterministic optimization problem, our framework circumvents the heavy computational burden and sampling requirements of traditional simulation-based methods like MCMC or MLE. The robustness and scalability of the proposed approach are demonstrated through numerical experiments on various non-linear SDEs, including models for individual biological growth. Results show that the WCE-SGD framework provides accurate parameter recovery even from discrete, noisy observations, offering a significant paradigm shift in the efficient modeling of complex stochastic systems.
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