Stochastic Representation of Time-Evolving Neural Network-based Wavefunctions

TL;DR

This work extends the stochastic representation framework to real-time quantum dynamics by coupling TDVP-based variational propagation with neural-network wavefunctions represented through stochastic samples. The method uses an adaptive, RBF-based network to learn time-evolving wavefunction values without grids, achieving strong agreement with grid benchmarks in 1D ionization scenarios and demonstrating potential in 3D with notable stability challenges. Key contributions include the real-time stochastic workflow, explicit RBF network architecture to handle spreading wavefunctions, and an adaptive sampling strategy that manages computational cost. The findings suggest a promising route for scalable quantum dynamics simulations, while also highlighting the need for improved stability and optimization techniques for higher dimensions.

Abstract

Solving the time-dependent Schrödinger equation (TDSE) is pivotal for modeling non-adiabatic electron dynamics, a key process in ultrafast spectroscopy and laser-matter interactions. However, exact solutions to the TDSE remain computationally prohibitive for most realistic systems, as the Hilbert space expands exponentially with dimensionality. In this work, we propose an approach integrating the stochastic representation framework with a neural network wavefunction ansatz, a flexible model capable of approximating time-evolving quantum wavefunctions. We first validate the method on one-dimensional single-electron systems, focusing on ionization dynamics under intense laser fields, a critical process in attosecond physics. Our results demonstrate that the approach accurately reproduces key features of quantum evolution, including the energy and dipole evolution during ionization. We further show the feasibility of extending this approach to three-dimensional systems. Due to the increased complexity of real-time simulations in higher dimensions, these results remain at an early stage and highlight the need for more advanced stabilization strategies.

Figures (7)

• Figure 1: Workflow and RBF neural network architecture. (a) Workflow of real-time propagation with stochastic representation. Each iteration consists of three key steps: first, the RBF neural network wavefunction is represented using stochastic samples; second, the wavefunction values at these sampled positions are propagated under the real-time evolution operator; and third, the RBF neural network is refined by fitting it to the propagated wavefunction values. The red line represents the real part of the wavefunction, and the blue line represents the imaginary part. (b) The RBF neural network architecture. The neural network take the sample coordinates as inputs and outputs the real part or the imaginary part of the wavefunction value $\psi(\mathbf{R})$. The first hidden layer is RBF layer, and the following hidden layers are fully connected layers with tanh activation functions.
• Figure 2: Computational details for one-dimensional calculations. (a) The two different laser fields(units in a.u.): laser A, $E_0=1.0, T =50,\tau=20.5,\omega=1.0$; laser B, $E_0=0.1, T =50,\tau=20.5,\omega=1.0/(2\pi)$. (b) The increasing sample sizes used in the four simulations, showing adaptive growth as required by the convergence criterion.
• Figure 3: Simulations of an electron in 1D space under four different conditions. (a) Gaussian potential with laser A; (b) Gaussian potential with laser B; (c) Soft Coulomb potential with laser A; (d) Soft Coulomb potential with laser B. The left column shows the time evolution of the energy, and the right column shows the time evolution of the dipole moment.
• Figure 4: Simulations of an electron under a soft Coulomb potential in three-dimensional space. (a) The external electric field. (b) The time evolution of energy. (c) The time evolution of dipole moment.
• Figure 5: Infidelity during simulations under different laser fields with soft Coulomb potential.