保持结构的动态模式分解用于半经典薛定谔方程的高频动力学
Structure-Preserving Dynamic Mode Decomposition for Highly Oscillatory Dynamics of Semiclassical Schrödinger Equations
AI总结：
本文提出两种新的数据驱动DMD方法，用于高效准确预测半经典薛定谔方程的高频动力学，通过学习薛定谔算子并显式纳入质量与能量守恒定律，确保物理真实性并内置模型降阶能力。
摘要：
我们提出两种新的数据驱动动态模式分解（DMD）-型方法，即Crank-Nicolson DMD和半隐式DMD，以高效准确地预测半经典薛定谔方程的高频动力学。不同于许多现有DMD型方法直接建模波函数的动力学，我们的方法基于学习薛定谔算子并显式纳入质量与能量守恒定律。这种做法确保了物理真实性，并赋予所得到的方法内置的模型降阶能力，而无需额外的降维预处理。对训练和预测误差进行了分析以提供理论保证。广泛的数值实验展示了所提出方法的噪声鲁棒性、计算效率以及对其他方程的可转移性。
英文摘要：
We propose two novel data-driven dynamic mode decomposition (DMD)-type methods, the Crank--Nicolson DMD and the semi-implicit DMD, to predict the highly oscillatory dynamics of the semiclassical Schrödinger equations efficiently and accurately. Unlike many existing DMD-type methods which directly models the dynamics of the wave function, our approach is based on learning the Schrödinger operator while explicitly incorporating mass and energy conservation laws. This approach ensures physical fidelity and endows the resulting methods with built-in model order reduction capabilities, without the necessity for additional dimensionality-reduction preprocessing. An analysis of training and prediction errors are given for theoretical guarantees. Extensive numerical experiments demonstrate the noise robustness, computational efficiency, and transferability to other equations of the proposed methods.
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