Nonlinear dynamics are ubiquitous in complex systems. Their applications range from robotics to computational neuroscience. In this work, the Koopman framework for globally linearizing nonlinear dynamics is introduced. Under this framework, the nonlinear observable states are lifted into a higher dimensional, linear regime. The challenge is to identify functions that facilitate the coordinate transformation to this raised linear space. This point is tackled using deep learning, where nonlinear dynamics are learned in a model-free manner, i.e., the underlying dynamics are uncovered using data rather than the nonlinear state-space equations. The main contributions include an implementation of the Linearly Recurrent Encoder Network (LREN) that is faster than the existing implementation and is significantly faster than the state-of-the-art deep learning-based approach. Also, a novel architecture termed Deep Encoder with Initial State Parameterization (DENIS) is proposed. By deriving an energy-budget control performance evaluation method, we demonstrate that DENIS also outperforms LREN in control performance. It is also on-par with and sometimes better than the iterative linear quadratic regulator (iLQR), which requires access to the state-space equations. Extensive experiments are done on DENIS to validate its performance. Also, another novel architecture termed Double Encoder for Input Nonaffine systems (DEINA) is described. Additionally, DEINA’s potential ability to outperform existing Koopman frameworks for controlling nonaffine input systems is shown. We attribute this to using an auxiliary network to nonlinearly transform the inputs, thereby lifting the strong linear constraints imposed by the traditional Koopman approximation approach. Koopman model predictive control (KMPC) is implemented to verify that our models can also be successfully controlled under this popular approach. Overall, we demonstrate the deep learning-based Koopman framework shows promise for optimally controlling nonlinear dynamics.

非线性动力学在复杂系统中普遍存在。它们的应用范围从机器人到计算神经科学。在这项工作中，引入了用于全局线性化非线性动力学的库普曼框架。在此框架下，非线性可观测状态被提升到更高维度的线性状态。挑战在于确定有助于坐标变换到该凸起线性空间的函数。这一点可以通过深度学习来解决，其中非线性动力学以无模型的方式学习，即使用数据而不是非线性状态空间方程来揭示潜在的动力学。主要贡献包括线性循环编码器网络（LREN）的实现，该网络比现有的实现速度更快，并且比最先进的基于深度学习的方法快得多。此外，还提出了一种称为具有初始状态参数化的深度编码器（DENIS）的新颖架构。通过推导能量预算控制性能评估方法，我们证明 DENIS 在控制性能方面也优于 LREN。它也与迭代线性二次调节器 (iLQR) 相当，有时甚至更好，后者需要访问状态空间方程。在 DENIS 上进行了大量实验以验证其性能。此外，还描述了另一种新颖的架构，称为输入非仿射系统的双编码器 ( DEINA )。此外，DEINA 在控制非仿射输入系统方面的潜在能力优于现有的 Koopman 框架。我们将此归因于使用辅助网络对输入进行非线性变换，从而解除了传统库普曼近似方法所施加的强线性约束。 库普曼模型预测控制（KMPC）的实施是为了验证我们的模型也可以在这种流行的方法下成功控制。总的来说，我们证明了基于深度学习的库普曼框架显示出优化控制非线性动力学的前景。