Deep Stochastic Mechanics

TL;DR

This paper tackles the curse of dimensionality in time-dependent quantum dynamics by reframing the Schrödinger equation through stochastic mechanics and learning drift fields with neural networks. The Deep Stochastic Mechanics (DSM) framework replaces expensive Laplacian evaluations with a gradient-of-divergence formulation, yielding an $\mathcal{O}(d^2)$ loss-computation cost and practical $\mathcal{O}(d)$ forward passes, while ensuring strong convergence to the true quantum-density process. The authors provide a rigorous bound linking the learned-loss to process-trajectory proximity and demonstrate superior accuracy and scaling over PINNs and t-VMC across harmonic-oscillator and interacting-boson experiments, including high-dimensional, large-particle cases. Overall, DSM offers a scalable, diffusion-based means to simulate high-dimensional quantum systems, with potential impact on quantum chemistry, condensed matter, and quantum computing applications.

Abstract

This paper introduces a novel deep-learning-based approach for numerical simulation of a time-evolving Schrödinger equation inspired by stochastic mechanics and generative diffusion models. Unlike existing approaches, which exhibit computational complexity that scales exponentially in the problem dimension, our method allows us to adapt to the latent low-dimensional structure of the wave function by sampling from the Markovian diffusion. Depending on the latent dimension, our method may have far lower computational complexity in higher dimensions. Moreover, we propose novel equations for stochastic quantum mechanics, resulting in quadratic computational complexity with respect to the number of dimensions. Numerical simulations verify our theoretical findings and show a significant advantage of our method compared to other deep-learning-based approaches used for quantum mechanics.

Figures (11)

• Figure 1: An illustration of our approach. Blue regions in the plots correspond to higher-density regions. (a) DSM training scheme: at every epoch $\tau$, we generate $B$ full trajectories $\{ X_{ij}\}_{ij}$, $i=0, ..., N$, $j=1, ..., B$. Then, we update the weights of our NNs. (b) An illustration of sampled trajectories at the early epoch. (c) An illustration of sampled trajectories at the final epoch. (d) Collocation points for a grid-based solver where it should predict values of $\psi(x, t)$.
• Figure 2: Simulation results of PINN and our DSM method: (a) and (b) correspond to a particle in the harmonic oscillator with different initial phases; (c) corresponds to two interacting bosons in the harmonic oscillator. The left panel of these figures corresponds to the density $|\psi(x,t)|^2$ of the ground truth solution, our approach (DSM), PINN, and t-VMC. The right panel presents statistics, including the particle's mean position and variance.
• Figure 3: The one-particle density of a system of $100$ interacting bosons for varying interaction strength $g$. For a weaker interaction, the one-particle density is higher, indicating a more stable particle configuration. Conversely, for a stronger interaction, this value decreases, suggesting a more dispersed particle behavior.
• Figure 4: Empirical complexity evaluation of our method for the non-interacting system.
• Figure 5: Results for the singular initial condition case. DSM corresponds to our method.

Theorems & Definitions (30)

• Remark 4.1
• Theorem 4.2
• Proposition E.1
• proof
• Proposition E.2: \ref{['prop:alg']}
• proof
• Lemma F.1
• Lemma F.2
• proof
• Lemma F.3