使用少于120行代码的Python实现线性动力系统的高效时空降阶模型
Efficient space-time reduced order model for linear dynamical systems in Python using less than 120 lines of code
AI总结：
本文首次推导了线性动力系统的时空Petrov-Galerkin投影及其块结构，并用少于120行Python代码构建了时空降阶模型，在实现百倍加速的同时保持高精度，误差界优于传统空间Galerkin方法。
摘要：
动力学问题的经典降阶模型（ROM）通常仅涉及对给定问题的空间降维。最近，一种针对线性动力学问题的新型时空ROM被开发出来，它通过在空间降维之外引入时间降维，进一步减小了问题规模，且精度损失不大。作者展示了对于大规模Boltzmann输运问题，实现了约一千倍的加速，相对误差小于0.00001。在本研究中，我们首次推导了线性动力系统的时空Petrov-Galerkin投影及其相应的块结构。利用这些块结构，我们用两个模型问题：带和不带线性源项的2D扩散和2D对流扩散，展示了时空ROM方法的构建便捷性。对于每个问题，我们演示了生成全阶模型（FOM）数据、构建时空ROM和预测降阶解的整个过程，所有这些均用少于120行Python代码完成。我们将Petrov-Galerkin方法与传统的Galerkin方法进行比较，表明时空ROM对于这些问题能实现O(100)的加速，相对误差为O(0.001)至O(0.0001)。最后，我们对时空Petrov-Galerkin投影进行了误差分析并推导了误差界，结果显示相比传统的空间Galerkin ROM方法有所改进。
英文摘要：
A classical reduced order model (ROM) for dynamical problems typically involves only the spatial reduction of a given problem. Recently, a novel space-time ROM for linear dynamical problems has been developed, which further reduces the problem size by introducing a temporal reduction in addition to a spatial reduction without much loss in accuracy. The authors show an order of a thousand speed-up with a relative error of less than 0.00001 for a large-scale Boltzmann transport problem. In this work, we present for the first time the derivation of the space-time Petrov-Galerkin projection for linear dynamical systems and its corresponding block structures. Utilizing these block structures, we demonstrate the ease of construction of the space-time ROM method with two model problems: 2D diffusion and 2D convection diffusion, with and without a linear source term. For each problem, we demonstrate the entire process of generating the full order model (FOM) data, constructing the space-time ROM, and predicting the reduced-order solutions, all in less than 120 lines of Python code. We compare our Petrov-Galerkin method with the traditional Galerkin method and show that the space-time ROMs can achieve O(100) speed-ups with O(0.001) to O(0.0001) relative errors for these problems. Finally, we present an error analysis for the space-time Petrov-Galerkin projection and derive an error bound, which shows an improvement compared to traditional spatial Galerkin ROM methods.
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