This paper introduces a novel deep-learning-based approach for numerical simulation of a time-evolving Schr\"odinger 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 linear 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.

翻译：本文提出了一种新型的基于深度学习的方法，用于数值模拟随时间演化的薛定谔方程，该方法受随机力学和生成扩散模型的启发。与现有方法相比，现有方法的计算复杂度随问题维度呈指数增长，而我们的方法通过从马尔可夫扩散中采样，能够适应波函数的潜在低维结构。根据潜在维度的不同，我们的方法在高维情况下可能具有远低于现有方法的计算复杂度。此外，我们提出了随机量子力学的新方程，使得计算复杂度与维度数量呈线性关系。数值模拟验证了我们的理论发现，并表明我们的方法相比其他用于量子力学的深度学习方法具有显著优势。