SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 480-509, January 2020.
The Koopman operator has emerged as a powerful tool for the analysis of nonlinear dynamical systems as it provides coordinate transformations to globally linearize the dynamics. While recent deep learning approaches have been useful in extracting the Koopman operator from a data-driven perspective, several challenges remain. In this work, we formalize the problem of learning the continuous-time Koopman operator with deep neural networks in a measure-theoretic framework. Our approach induces two types of models, differential and recurrent form, the choice of which depends on the availability of the governing equations and data. We then enforce a structural parameterization that renders the realization of the Koopman operator provably stable. A new autoencoder architecture is constructed, such that only the residual of the dynamic mode decomposition is learned. Finally, we employ mean-field variational inference on the aforementioned framework in a hierarchical Bayesian setting to quantify uncertainties in the characterization and prediction of the dynamics of observables. The framework is evaluated on a simple polynomial system, the Duffing oscillator, and an unstable cylinder wake flow with noisy measurements.

中文翻译：

---

具有保证稳定性的非线性动力学线性嵌入的物理信息概率学习

SIAM应用动力系统杂志，第19卷，第1期，第480-509页，2020年1月。
Koopman运算符已经成为分析非线性动力学系统的强大工具，因为它提供了坐标转换以使动力学全局线性化。尽管最近的深度学习方法对于从数据驱动的角度提取Koopman运算符很有用，但仍然存在一些挑战。在这项工作中，我们将在测度理论框架中用深度神经网络对学习连续时间Koopman算子的问题进行形式化。我们的方法归纳出两种类型的模型，即微分模型和递归模型，其选择取决于控制方程和数据的可用性。然后，我们强制执行结构化参数化，以使Koopman运算符的实现可证明是稳定的。构建了新的自动编码器架构，这样，仅学习动态模式分解的残差。最后，我们在分层贝叶斯环境中对上述框架采用均值场变分推断，以量化可观测物动力学表征和预测中的不确定性。该框架是在简单的多项式系统，Duffing振荡器和不稳定的圆柱尾流（带有噪声测量值）上进行评估的。