用神经网络学习随机哈密顿系统，保持其几何结构并提升长期预测精度。
Learning Stochastic Hamiltonian Systems via Stochastic Generating Function Neural Network
- 通过自编码器识别隐变量，解码器生成随机生成函数
- 在多类随机哈密顿系统上，长期预测误差显著低于基准模型
- 适合需要保持物理结构的复杂系统建模任务
本文提出一种新型神经网络模型——随机生成函数神经网络（SGFNN），用于从观测数据中学习随机哈密顿系统（SHSs）。SGFNN保持了底层系统固有的辛结构，并生成辛预测结果。模型采用自编码器框架，由编码器网络识别潜在系统的随机性，再由解码器网络基于编码器提取的随机变量，推断系统的随机生成函数。由此可生成符合辛结构的预测。我们在多种随机哈密顿系统上进行了数值实验，涵盖加性与乘性噪声、可分与不可分系统，以及单或多噪声情形。与基准模型（sFML）相比，SGFNN在各类预测指标上均表现出更高精度，尤其在长期预测中优势明显，同时有效维持了底层系统的辛结构。
原文摘要 · Abstract (English)
In this paper we propose a novel neural network model for learning stochastic Hamiltonian systems (SHSs) from observational data, termed the stochastic generating function neural network (SGFNN). SGFNN preserves symplectic structure of the underlying stochastic Hamiltonian system and produces symplectic predictions. Our model utilizes the autoencoder framework to identify the randomness of the latent system by the encoder network, and detects the stochastic generating function of the system through the decoder network based on the random variables extracted from the encoder. Symplectic predictions can then be generated by the stochastic generating function. Numerical experiments are performed on several stochastic Hamiltonian systems, varying from additive to multiplicative, and from separable to non-separable SHSs with single or multiple noises. Compared with the benchmark stochastic flow map learning (sFML) neural network, our SGFNN model exhibits higher accuracy across various prediction metrics, especially in long-term predictions, with the property of maintaining the symplectic structure of the underlying SHSs.
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