A new knowledge-based and machine learning hybrid modeling approach, called conditional Gaussian neural stochastic differential equation (CGNSDE), is developed to facilitate modeling complex dynamical systems and implementing analytic formulae of the associated data assimilation (DA). In contrast to the standard neural network predictive models, the CGNSDE is designed to effectively tackle both forward prediction tasks and inverse state estimation problems. The CGNSDE starts by exploiting a systematic causal inference via information theory to build a simple knowledge-based nonlinear model that nevertheless captures as much explainable physics as possible. Then, neural networks are supplemented to the knowledge-based model in a specific way, which not only characterizes the remaining features that are challenging to model with simple forms but also advances the use of analytic formulae to efficiently compute the nonlinear DA solution. These analytic formulae are used as an additional computationally affordable loss to train the neural networks that directly improve the DA accuracy. This DA loss function promotes the CGNSDE to capture the interactions between state variables and thus advances its modeling skills. With the DA loss, the CGNSDE is more capable of estimating extreme events and quantifying the associated uncertainty. Furthermore, crucial physical properties in many complex systems, such as the translate-invariant local dependence of state variables, can significantly simplify the neural network structures and facilitate the CGNSDE to be applied to high-dimensional systems. Numerical experiments based on chaotic systems with intermittency and strong non-Gaussian features indicate that the CGNSDE outperforms knowledge-based regression models, and the DA loss further enhances the modeling skills of the CGNSDE.

翻译：本文提出一种融合知识驱动与机器学习的混合建模新方法——条件高斯神经随机微分方程（CGNSDE），旨在促进复杂动力系统建模并实现相关数据同化（DA）的解析计算。与标准神经网络预测模型不同，CGNSDE能有效兼顾前向预测任务与逆向状态估计问题。该方法首先利用基于信息论的系统性因果推断，构建一个尽可能捕获可解释物理机制的简单知识驱动非线性模型；随后以特定方式补充神经网络以增强该模型，不仅能表征难以用简单形式建模的剩余特征，还可利用解析公式高效计算非线性DA解。这些解析公式作为计算成本可控的额外损失函数训练神经网络，直接提升DA精度。该DA损失函数推动CGNSDE捕捉状态变量间的相互作用，从而增强建模能力。引入DA损失后，CGNSDE对极端事件的估计能力及不确定性量化性能均得到提升。此外，复杂系统中常见的状态变量平移不变局部依赖等关键物理性质，可显著简化神经网络结构并支持CGNSDE在高维系统中的实际应用。基于间歇性混沌系统及强非高斯特征的数值实验表明，CGNSDE优于传统知识驱动回归模型，且DA损失进一步强化了其建模技能。