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Deep learning-based moment closure for multi-phase computation of semiclassical limit of the Schrödinger equation

Jin Woo Jang, Jae Yong Lee, Liu Liu, Zhenyi Zhu

TL;DR

This work tackles the semiclassical limit of the Schrödinger equation by recasting it as a Liouville/Vlasov problem through the Wigner transform and addressing the computationally challenging moment closure for multi-phase solutions. It introduces a two-stage neural network framework: Stage 1 learns a surrogate for the highest moment derivative $\partial_x m_2$ from lower moments and their spatial derivatives, and Stage 2 employs physics-informed neural networks (PINNs) to enforce the moment equations and recover $m_0$ and $m_1$ using the learned Stage 1 flux. The authors provide convergence guarantees for both the loss functions and NN approximations and validate the approach with 1D and 2D numerical experiments across varying phase counts, demonstrating accuracy and computational efficiency relative to traditional closures. This data-driven closure of the Liouville moment system enables robust, scalable simulations of multi-phase semiclassical dynamics with potential impact on quantum transport and high-frequency wave propagation problems.

Abstract

We present a deep learning approach for computing multi-phase solutions to the semiclassical limit of the Schrödinger equation. Traditional methods require deriving a multi-phase ansatz to close the moment system of the Liouville equation, a process that is often computationally intensive and impractical. Our method offers an efficient alternative by introducing a novel two-stage neural network framework to close the $2N\times 2N$ moment system, where $N$ represents the number of phases in the solution ansatz. In the first stage, we train neural networks to learn the mapping between higher-order moments and lower-order moments (along with their derivatives). The second stage incorporates physics-informed neural networks (PINNs), where we substitute the learned higher-order moments to systematically close the system. We provide theoretical guarantees for the convergence of both the loss functions and the neural network approximations. Numerical experiments demonstrate the effectiveness of our method for one- and two-dimensional problems with various phase numbers $N$ in the multi-phase solutions. The results confirm the accuracy and computational efficiency of the proposed approach compared to conventional techniques.

Deep learning-based moment closure for multi-phase computation of semiclassical limit of the Schrödinger equation

TL;DR

This work tackles the semiclassical limit of the Schrödinger equation by recasting it as a Liouville/Vlasov problem through the Wigner transform and addressing the computationally challenging moment closure for multi-phase solutions. It introduces a two-stage neural network framework: Stage 1 learns a surrogate for the highest moment derivative from lower moments and their spatial derivatives, and Stage 2 employs physics-informed neural networks (PINNs) to enforce the moment equations and recover and using the learned Stage 1 flux. The authors provide convergence guarantees for both the loss functions and NN approximations and validate the approach with 1D and 2D numerical experiments across varying phase counts, demonstrating accuracy and computational efficiency relative to traditional closures. This data-driven closure of the Liouville moment system enables robust, scalable simulations of multi-phase semiclassical dynamics with potential impact on quantum transport and high-frequency wave propagation problems.

Abstract

We present a deep learning approach for computing multi-phase solutions to the semiclassical limit of the Schrödinger equation. Traditional methods require deriving a multi-phase ansatz to close the moment system of the Liouville equation, a process that is often computationally intensive and impractical. Our method offers an efficient alternative by introducing a novel two-stage neural network framework to close the moment system, where represents the number of phases in the solution ansatz. In the first stage, we train neural networks to learn the mapping between higher-order moments and lower-order moments (along with their derivatives). The second stage incorporates physics-informed neural networks (PINNs), where we substitute the learned higher-order moments to systematically close the system. We provide theoretical guarantees for the convergence of both the loss functions and the neural network approximations. Numerical experiments demonstrate the effectiveness of our method for one- and two-dimensional problems with various phase numbers in the multi-phase solutions. The results confirm the accuracy and computational efficiency of the proposed approach compared to conventional techniques.

Paper Structure

This paper contains 19 sections, 3 theorems, 77 equations, 11 figures, 7 tables.

Table of Contents

  1. Introduction
  2. The Wigner Transform and Semiclassical Limit
  3. One-Dimensional Moment System for Multi-Phase Solutions
  4. Motivation and Goal of This Work
  5. Particle Method for the Vlasov Equation
  6. Neural Network Approaches
  7. A Two-Stage Moment Closure Model
  8. Physics Informed Neural Networks (PINNs)
  9. Model Design
  10. Loss Functions
  11. Convergence of the Loss and Solutions for Stage 2
  12. Numerical Experiments
  13. Test I: Moment closure for 1D Problem
  14. Test II: Moment Closure for 1D Problem with Multi-Valued Solutions
  15. Test III: Moment Closure for 2D Problem
  16. ...and 4 more sections

Key Result

Lemma 1

Suppose a solution $(m_0,m_1)$ of the moment system 2-2moment satisfies $m_0,\ m_1 \in C^1([0,T]\times \Omega)$. Let the activation function $\sigma$ be any non-polynomial function in $C^1(\mathbb R)$. Then for any $\delta>0$ there exists a two-layer neural network for $k=0$ and $1$, such that where the domain $K$ denotes $[0,T]\times\Omega$.

Figures (11)

  • Figure 1: Framework of the two-stage moment closure model of solving moment system.
  • Figure 2: Problems I Stage 1 with different $t$. Moment $\partial_x m_2$ for different schemes and reference solutions at different time steps.
  • Figure 3: Problems I Stage 2 with different $t$. Moments $m_0,m_1$ for our proposed method and reference solutions at different time steps.
  • Figure 4: Problems II Stage 1 with different $t$. Moment $\partial_x m_2$ for different schemes and reference solutions at time steps $T=0.05, 0.10, 0.15, \text{and}, 0.20$.
  • Figure 5: Problems II with different $t$. $m_0$ for our proposed method and reference solutions at different time steps.
  • ...and 6 more figures

Theorems & Definitions (6)

  • Lemma 1: Li, Theorem 2.1, li1996simultaneous
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark