A Global Spacetime Optimization Approach to the Real-Space Time-Dependent Schr\"odinger Equation

TL;DR

This paper introduces a neural network framework that globally optimizes the real-space time-dependent Schrödinger equation for fermionic systems, enabling accurate, flexible simulations of complex quantum dynamics without stepwise propagation.

Contribution

It presents a novel spatiotemporal neural network approach that treats time as an explicit input, formulating the TDSE as a global optimization problem for the first time.

Findings

Achieves excellent agreement with reference solutions on five benchmark problems.

Demonstrates stable simulation of multi-electron dynamics over extended times.

Supports highly parallelizable training, improving computational efficiency.

Abstract

The time-dependent Schr\"odinger equation (TDSE) in real space is fundamental to understanding the dynamics of many-electron quantum systems, with applications ranging from quantum chemistry to condensed matter physics and materials science. However, solving the TDSE for complex fermionic systems remains a significant challenge, particularly due to the need to capture the time-evolving many-body correlations, while the antisymmetric nature of fermionic wavefunctions complicates the function space in which these solutions must be represented. We propose a general-purpose neural network framework for solving the real-space TDSE, Fermionic Antisymmetric Spatio-Temporal Network, which treats time as an explicit input alongside spatial coordinates, enabling a unified spatiotemporal representation of complex, antisymmetric wavefunctions for fermionic systems. This approach formulates the TDSE as a global optimization problem, avoiding step-by-step propagation and supporting highly parallelizable training. The method is demonstrated on five benchmark problems, achieving excellent agreement with reference solutions across all cases. These results demonstrate the method's accuracy and flexibility within the bound-state manifold across various dimensions and interaction regimes. While the current localized Ansatz inherently restricts the description of extensive ionization and continuum states, the method demonstrates the capability to stably simulate coherent multi-electron dynamics over extended time windows. Our framework offers a highly expressive alternative to traditional basis-dependent or mean-field methods, opening new possibilities for ab initio simulations of time-dependent quantum systems, with applications in quantum dynamics, molecular control, and ultrafast spectroscopy.