QPKO: Differentiable QP-Embedded Deep Koopman Framework for Modeling Nonlinear Systems

Deep learning has been widely regarded as a powerful tool for Koopman operator theory-based modeling, as it provides a promising architecture for data-driven learning of observable functions. To fully leverage this advantage, a well-designed training paradigm is required. However, the existing training paradigms typically either incur high optimization complexity or hinder effective end-to-end training, limiting modeling accuracy and training efficiency. To address this issue, we propose a differentiable quadratic programming (QP)-embedded deep Koopman framework (QPKO). In QPKO, a QP problem, which comprises a one-step accuracy-oriented objective function and a set of multi-step accuracy-oriented constraints, is formulated to introduce a mapping from observable functions to the global linear model. By doing so, the global linear model no longer needs to be treated as an independent trainable component, thereby effectively reducing optimization complexity. This QP-based mapping is implemented as a differentiable and computationally efficient module by leveraging OptNet (a differentiable QP layer), enabling effective end-to-end training. Experiments on four nonlinear dynamical systems show that QPKO achieves satisfactory improvements in modeling accuracy, training efficiency, and control performance.

From climate system to power system, many real-world dynamical systems behave in highly nonlinear ways. This makes controlling and analyzing them highly challenging. Koopman operator theory provides a useful workaround: It suggests that complex nonlinear behavior can be represented by a simpler linear model in the transformed space. This allows us to use the well-established techniques designed for linear systems to handle these complex nonlinear ones. Therefore, many deep learning-based frameworks have been developed to learn this transformation and the corresponding linear model in a data-driven manner. However, they fail to leverage the connection between the transformation and the model, which slows the learning process and limits the model’s accuracy. To address this, we propose a new framework, called QPKO. It develops this connection using a tractable mathematical optimization, whose objective is to enhance both the one-step and multi-step prediction accuracy of the model, and leverages this connection efficiently during the learning process. We test QPKO on four nonlinear systems, and find it exhibits faster learning speed, higher modeling accuracy, and better control performance. This work provides a new perspective for Koopman operator theory-based modeling, and offers an effective tool to support the control and analysis of complex nonlinear systems.