Nonlinear differential equations are encountered as models of fluid flow, spiking neurons, and many other systems of interest in the real world. Common features of these systems are that their behaviors are difficult to describe exactly and invariably unmodeled dynamics present challenges in making precise predictions. In many cases the models exhibit extremely complicated behavior due to bifurcations and chaotic regimes. In this paper, we present a novel data-driven linear estimator that uses Koopman operator theory to extract finite-dimensional representations of complex nonlinear systems. The extracted model is used together with a deep reinforcement learning network that learns the optimal stepwise actions to predict future states of the original nonlinear system. Our estimator is also adaptive to a diffeomorphic transformation of the nonlinear system which enables transfer learning to compute state estimates of the transformed system without relearning from scratch.

翻译：非线性微分方程常被用于建模流体流动、尖峰神经元及现实世界中众多其他感兴趣系统。这些系统的共同特征在于其行为难以精确描述，且始终存在的未建模动力学对精确预测构成挑战。在许多情况下，由于分岔和混沌机制，模型会表现出极其复杂的行为。本文提出一种新型数据驱动线性估计器，利用Koopman算子理论提取复杂非线性系统的有限维表示。该提取模型与深度强化学习网络相结合，通过学习最优逐步动作来预测原始非线性系统的未来状态。此外，我们的估计器还具备对非线性系统微分同胚变换的自适应性，这使得无需重新学习即可通过迁移学习计算变换后系统的状态估计。