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 homogeneous 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 equip 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.

动态模式分解 (DMD) 是一种强大的数据驱动技术，用于构建复杂动态系统的降阶模型。多项数值测试已经证明了 DMD 的准确性和效率，但主要用于由具有齐次边界条件的齐次偏微分方程 (PDE) 描述的系统。我们提出了一种扩展的动态模式分解 (xDMD) 方法来处理 PDE 中潜在的未知源/汇。受深度神经网络中类似想法的启发，我们为 xDMD 配备了两个新功能。首先，它有一个偏置项，它解释了 PDE 和/或边界条件的不均匀性。其次，xDMD 不是学习流程图，而是通过减去恒等运算符来学习残差增量。我们的理论误差分析表明 xDMD 相对于标准 DMD 的准确性有所提高。给出了几个数值例子来说明这个结果。