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Structure-Preserving Dynamic Mode Decomposition for Highly Oscillatory Dynamics of Semiclassical Schrödinger Equations

Yizhe Feng, Weiguo Gao, Jia Yin

Abstract

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.

Structure-Preserving Dynamic Mode Decomposition for Highly Oscillatory Dynamics of Semiclassical Schrödinger Equations

Abstract

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.

Paper Structure

This paper contains 28 sections, 5 theorems, 93 equations, 4 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Dynamic mode decomposition
  3. Notations
  4. The classical DMD
  5. The physics-informed DMD (piDMD)
  6. The classical DMD
  7. The piDMD
  8. Algorithm design
  9. Main idea
  10. The Crank--Nicolson DMD
  11. The method
  12. Error analysis
  13. Model order reduction
  14. The Semi-implicit DMD
  15. The method
  16. ...and 13 more sections

Key Result

Theorem 3.1

The CN-DMD preserves both the discretized mass and the discretized energy. In particular, Moreover, the discretized quadratic energy functional which is a finite-dimensional discretization associated with the Hermitian operator $\boldsymbol{A}$ of the continuous energy functional $\langle u^{\varepsilon}, \mathcal{A} u^{\varepsilon} \rangle_{L^2}$ in eq:energy_conservation, is conserved, i.e.,

Figures (4)

  • Figure 1: Forward wave propagation obtained by different DMD-type methods: (a) CN-DMD; (b) SI-DMD; (c) Classical DMD; (d) piDMD; compared with (e) True propagation. The red dotted line indicates the final time of the given data.
  • Figure 2: Training and prediction error under different levels of noise (a) $\sigma = 10^{-2}$; (b) $\sigma = 10^{-3}$; (c) $\sigma = 10^{-4}$; (d) $\sigma = 10^{-5}$.
  • Figure 3: (a) Mass variation. (b) Energy variation. The dotted line indicates the final time step of the train data.
  • Figure 4: Comparison of DMD-type methods for the nonlinear Schrödinger equation \ref{['eq:nonlinear_schrodinger_equation']}.

Theorems & Definitions (14)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Theorem 3.3
  • Proof 2
  • Theorem 3.4
  • Proof 3
  • Lemma 3.5
  • ...and 4 more