We present a data-driven learning approach for unknown nonautonomous dynamical systems with time-dependent inputs based on dynamic mode decomposition (DMD). To circumvent the difficulty of approximating the time-dependent Koopman operators for nonautonomous systems, a modified system derived from local parameterization of the external time-dependent inputs is employed as an approximation to the original nonautonomous system. The modified system comprises a sequence of local parametric systems, which can be well approximated by a parametric surrogate model using our previously proposed framework for dimension reduction and interpolation in parameter space (DRIPS). The offline step of DRIPS relies on DMD to build a linear surrogate model, endowed with reduced-order bases (ROBs), for the observables mapped from training data. Then the offline step constructs a sequence of iterative parametric surrogate models from interpolations on suitable manifolds, where the target/test parameter points are specified by the local parameterization of the test external time-dependent inputs. We present a number of numerical examples to demonstrate the robustness of our method and compare its performance with deep neural networks in the same settings.

翻译：我们提出了一种基于动态模态分解（DMD）的数据驱动学习方法，用于未知且具有时变输入的非自治动力系统。为规避逼近非自治系统时间相关Koopman算子的困难，我们采用由外部时变输入局部参数化导出的修正系统作为原始非自治系统的近似。该修正系统由一系列局部参数化系统构成，可通过我们先前提出的参数空间降维与插值（DRIPS）框架构建的参数化替代模型进行有效逼近。DRIPS离线阶段依赖DMD构建线性替代模型，该模型配备由训练数据映射观测量所得的降阶基（ROBs）。随后离线阶段通过在适当流形上的插值构造一系列迭代参数化替代模型，其中目标/测试参数点由测试外部时变输入的局部参数化指定。我们通过多个数值算例验证了该方法鲁棒性，并在相同条件下将其性能与深度神经网络进行了对比。