We present a framework for constructing physics and causally constrained neural models of turbulent dynamical systems from data. We first formulate a finite-time flow map with strict energy-preserving nonlinearities for stable modeling of temporally discrete trajectories. We then impose causal constraints to suppress spurious interactions across degrees of freedom. The resulting neural models accurately capture stationary statistics and responses to both small and large external forcings. We demonstrate the framework on the stochastic Charney-DeVore equations and on a symmetry-broken Lorenz-96 system. The framework is broadly applicable to reduced-order modeling of turbulent dynamical systems from observational data.

翻译：我们提出了一种基于数据构建湍流动力系统物理与因果约束神经模型的框架。首先，针对时间离散轨迹的稳定建模，我们构造了具有严格保能非线性的有限时间流映射。随后施加因果约束以抑制自由度间的虚假相互作用。所得神经模型能精确捕捉平稳统计特性以及对小尺度与大尺度外强迫的响应。我们通过随机沙尔尼-德沃尔方程和对称破缺洛伦兹-96系统验证了该框架。该框架广泛适用于基于观测数据的湍流动力系统降阶建模。