A Global Spacetime Optimization Approach to the Real-Space Time-Dependent Schrödinger Equation

TL;DR

FASTNet introduces a real-space, time-aware neural ansatz for solving the time-dependent Schrödinger equation in many-electron systems by enforcing fermionic antisymmetry and treating time as an explicit input. Framing TDSE as a global spacetime residual minimization, the method enables highly parallel training without stepwise propagation and uses a pretraining scheme over overlapping time subintervals to tackle long-time evolution. Through benchmarks in 1D and 3D settings, including interacting fermions, hydrogen dynamics, and laser-driven molecules, FASTNet achieves high accuracy, often surpassing traditional basis sets and capturing complex correlated dynamics. The work offers a flexible, ab initio-capable alternative to mean-field and basis-dependent methods, with potential impact on quantum dynamics, molecular control, and ultrafast spectroscopy.

Abstract

The time-dependent Schrödinger 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 four benchmark problems: a 1D harmonic oscillator, interacting fermions in a time-dependent harmonic trap, 3D hydrogen orbital dynamics, and a laser-driven H$_2$ molecule, achieving excellent agreement with reference solutions across all cases. These results confirm our method's scalability, accuracy, and flexibility across various dimensions and interaction regimes, while demonstrating its ability to accurately simulate long-time dynamics in complex systems. 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.

Figures (5)

• Figure 1: Architecture of FASTNet for time-dependent electronic wavefunctions. (a) Overview of the network structure. (b-e) Detailed views of the embedding, permutation-equivariant block, phase, and envelope modules, respectively.
• Figure 2: Illustration of the pre-training strategy. The time domain is divided into overlapping intervals, with each interval trained using the previous interval's parameters as initialization. The overlap ensures smooth transition between intervals, allowing for efficient and accurate training of long-time dynamics.
• Figure 3: Real and imaginary parts, as well as absolute errors, of the 1D harmonic oscillator wavefunctions at $t=\pi$ for various initial states. Blue solid lines denote analytical solutions, red dashed lines show FASTNet predictions, and orange lines indicate the absolute error $|\Psi_{\mathrm{ref}} - \Psi_{\mathrm{net}}|$. FASTNet accurately reproduces both amplitude and phase across all cases.
• Figure 4: Results for $N = 2$ and $N = 3$ fermions. a) Monopole breathing mode; b) Absolute error between FASTNet and analytical solutions.
• Figure 5: Time evolution of the induced $x$-component of the electronic dipole $\mu_x(t)$ for stretched H$_2$ driven by the laser field. Results are shown for TD-HF and FCI with STO-3G and cc-pVDZ bases, together with tVMC and FASTNet.