Modeling and controlling complex spatiotemporal dynamical systems driven by partial differential equations (PDEs) often necessitate dimensionality reduction techniques to construct lower-order models for computational efficiency. This paper explores a deep autoencoding learning method for reduced-order modeling and control of dynamical systems governed by spatiotemporal PDEs. We first analytically show that an optimization objective for learning a linear autoencoding reduced-order model can be formulated to yield a solution closely resembling the result obtained through the dynamic mode decomposition with control algorithm. We then extend this linear autoencoding architecture to a deep autoencoding framework, enabling the development of a nonlinear reduced-order model. Furthermore, we leverage the learned reduced-order model to design controllers using stability-constrained deep neural networks. Numerical experiments are presented to validate the efficacy of our approach in both modeling and control using the example of a reaction-diffusion system.

本研究解决了复杂时空动力系统建模与控制中的维数缩减问题，提出了一种深度自编码学习方法，能够有效构建非线性降阶模型。研究发现，此方法在模型学习和控制设计上均表现出色，特别是通过稳定性约束的深度神经网络设计控制器，展现了其潜在的广泛应用。 

桥接自编码器与动态模式分解用于偏微分方程的降阶建模与控制