Developing reduced-order models for nonlinear parabolic partial differential equation (PDE) systems with time-varying spatial domains remains a key challenge as the dominant spatial patterns of the system change with time. To address this issue, there have been several studies where the time-varying spatial domain is transformed to the time-invariant spatial domain by using an analytical expression that describes how the spatial domain changes with time. However, this information is not available in many real-world applications, and therefore, the approach is not generally applicable. To overcome this challenge, we introduce sparse proper orthogonal decomposition (SPOD)-Galerkin methodology that exploits the key features of ridge and lasso regularization techniques for the model order reduction of such systems. This methodology is successfully applied to a hydraulic fracturing process, and a series of simulation results indicates that it is more accurate in approximating the original nonlinear system than the standard POD-Galerkin methodology.

由于系统的主要空间模式随时间变化，因此开发具有时变空间域的非线性抛物型偏微分方程（PDE）系统的降阶模型仍然是一个关键挑战。为了解决这个问题，已经进行了一些研究，其中使用描述空间域如何随时间变化的解析表达式将时空空间域转换为时不变空间域。但是，此信息在许多实际应用程序中不可用，因此，该方法通常不适用。为了克服这一挑战，我们引入了稀疏固有正交分解（SPOD）-Galerkin方法，该方法利用了脊和套索正则化技术的关键特征来简化此类系统的模型。