论文标题
扩展动态模式分解不均匀问题
Extended Dynamic Mode Decomposition for Inhomogeneous Problems
论文作者
论文摘要
动态模式分解（DMD）是一种强大的数据驱动技术，用于构建复杂动力学系统的减少阶模型。多个数值测试证明了DMD的准确性和效率，但主要用于由具有均匀边界条件的部分微分方程（PDE）描述的系统。我们提出了一种扩展的动态模式分解（XDMD）方法，以应对PDE中潜在的未知来源/接收器。在深度神经网络中的类似思想中，我们为XDMD配备了两个新功能。首先，它具有一个偏差术语，该术语解释了PDE和/或边界条件的不均匀性。其次，XDMD没有学习流图，而是通过减去身份操作员来学习残差增量。我们的理论错误分析表明，相对于标准DMD，XDMD的准确性提高了。提出了几个数值示例，以说明这一结果。
Dynamic mode decomposition (DMD) is a powerful data-driven technique for construction of reduced-order models of complex dynamical systems. Multiple numerical tests have demonstrated the accuracy and efficiency of DMD, but mostly for systems described by partial differential equations (PDEs) with homogeneous boundary conditions. We propose an extended dynamic mode decomposition (xDMD) approach to cope with the potential unknown sources/sinks in PDEs. Motivated by similar ideas in deep neural networks, we equipped our xDMD with two new features. First, it has a bias term, which accounts for inhomogeneity of PDEs and/or boundary conditions. Second, instead of learning a flow map, xDMD learns the residual increment by subtracting the identity operator. Our theoretical error analysis demonstrates the improved accuracy of xDMD relative to standard DMD. Several numerical examples are presented to illustrate this result.
