The engineering design process often relies on mathematical modeling that can describe the underlying dynamic behavior. In this work, we present a data-driven methodology for modeling the dynamics of nonlinear systems. To simplify this task, we aim to identify a coordinate transformation that allows us to represent the dynamics of nonlinear systems using a common, simple model structure. The advantage of a common simple model is that customized design tools developed for it can be applied to study a large variety of nonlinear systems. The simplest common model -- one can think of -- is linear, but linear systems often fall short in accurately capturing the complex dynamics of nonlinear systems. In this work, we propose using quadratic systems as the common structure, inspired by the lifting principle. According to this principle, smooth nonlinear systems can be expressed as quadratic systems in suitable coordinates without approximation errors. However, finding these coordinates solely from data is challenging. Here, we leverage deep learning to identify such lifted coordinates using only data, enabling a quadratic dynamical system to describe the system's dynamics. Additionally, we discuss the asymptotic stability of these quadratic dynamical systems. We illustrate the approach using data collected from various numerical examples, demonstrating its superior performance with the existing well-known techniques.

翻译：工程设计过程通常依赖于能够描述潜在动态行为的数学建模。本文提出了一种数据驱动的非线性系统动力学建模方法。为简化这一任务，我们旨在识别一种坐标变换，使得能够通过一种通用、简单的模型结构来表示非线性系统的动力学。通用简单模型的优势在于，为其开发的定制化设计工具可广泛应用于各类非线性系统的研究。最简单的通用模型可视为线性模型，但线性系统往往难以准确捕捉非线性系统的复杂动力学行为。受提升原理启发，本文提出以二次系统作为通用结构。根据该原理，光滑非线性系统可在适当坐标下表示为无近似误差的二次系统。然而，仅从数据中寻找这些坐标极具挑战性。本研究利用深度学习，仅通过数据即可识别此类提升坐标，使得二次动力系统能够描述系统动力学行为。此外，我们讨论了此类二次动力系统的渐近稳定性。通过多个数值算例采集的数据验证了该方法，结果表明其性能显著优于现有已知技术。