用自适应随机傅里叶特征提升随机微分方程学习效率
An Adaptive Random Fourier Features approach Applied to Learning Stochastic Differential Equations
- 引入自适应随机傅里叶特征与马尔可夫采样优化参数更新
- 在多项式、朗之万等4类模型上优于传统Adam方法
- 适合需要快速高精度建模随机动力系统的研究者
本文提出一种基于自适应随机傅里叶特征（ARFF）的训练算法，结合马尔可夫采样与重采样，从快照数据中学习随机微分方程的漂移与扩散项。研究针对伊藤扩散过程，采用基于欧拉-马鲁亚姆积分的似然损失函数，评估了多项式模型、阻尼朗之万动力学、随机易感-感染-康复模型及随机波动方程等基准问题。在所有案例中，基于ARFF的方法在损失下降和收敛速度上均达到或超过传统Adam优化器的表现，凸显其在数据驱动随机动力系统建模中的潜力。
原文摘要 · Abstract (English)
This work proposes a training algorithm based on adaptive random Fourier features (ARFF) with Metropolis sampling and resampling \cite{kammonen2024adaptiverandomfourierfeatures} for learning drift and diffusion components of stochastic differential equations from snapshot data. Specifically, this study considers Itô diffusion processes and a likelihood-based loss function derived from the Euler-Maruyama integration introduced in \cite{Dietrich2023} and \cite{dridi2021learningstochasticdynamicalsystems}. This work evaluates the proposed method against benchmark problems presented in \cite{Dietrich2023}, including polynomial examples, underdamped Langevin dynamics, a stochastic susceptible-infected-recovered model, and a stochastic wave equation. Across all cases, the ARFF-based approach matches or surpasses the performance of conventional Adam-based optimization in both loss minimization and convergence speed. These results highlight the potential of ARFF as a compelling alternative for data-driven modeling of stochastic dynamics.
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