We propose a novel learning framework for Koopman operator of nonlinear dynamical systems that is informed by the governing equation and guarantees long-time stability and robustness to noise. In contrast to existing frameworks where either ad-hoc observables or blackbox neural networks are used to construct observables in the extended dynamic mode decomposition (EDMD), our observables are informed by governing equations via Polyflow. To improve the noise robustness and guarantee long-term stability, we designed a stable parameterization of the Koopman operator together with a progressive learning strategy for roll-out recurrent loss. To further improve model performance in the phase space, a simple iterative strategy of data augmentation was developed. Numerical experiments of prediction and control of classic nonlinear systems with ablation study showed the effectiveness of the proposed techniques over several state-of-the-art practices.

翻译：我们提出了一种新颖的非线性动力系统Koopman算子学习框架，该框架受控制方程信息引导，并保证长期稳定性与噪声鲁棒性。与现有框架中采用临时可观测函数或黑盒神经网络构建扩展动态模态分解（EDMD）可观测量的方法不同，我们的可观测函数通过Polyflow从控制方程中获取信息。为提升噪声鲁棒性并保证长期稳定性，我们设计了Koopman算子的稳定参数化方法，并结合滚动循环损失的渐进式学习策略。为进一步提升模型在相空间中的性能，开发了一种简单的数据增强迭代策略。通过对经典非线性系统进行预测与控制的数值实验及消融研究，证明了所提技术相对于多种先进实践方法的有效性。