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：我们提出了一种框架，用于从数据中构建受物理学和因果约束的湍流动力系统神经模型。首先，我们构造了一个具有严格保能非线性的有限时间流映射，以实现时间离散轨迹的稳定建模。随后，我们施加因果约束，以抑制自由度之间的虚假相互作用。由此得到的神经模型能够准确捕捉平稳统计特性以及对小尺度与大尺度外部强迫的响应。我们在随机Charney-DeVore方程和对称破缺Lorenz-96系统上验证了该框架。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。