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：本文提出一种基于深度学习的新型数值模拟方法，用于求解含时薛定谔方程。该方法受随机力学与生成扩散模型启发，不同于现有方法在问题维度上呈现指数级计算复杂度的特点，我们通过从马尔可夫扩散过程中采样，自适应地利用波函数的潜在低维结构。根据潜在维度的高低，我们的方法在高维场景中可能具有显著更低的计算复杂度。此外，我们提出了随机量子力学的新方程，使得计算复杂度随维度呈线性增长。数值实验验证了理论发现，并表明我们的方法相比其他基于深度学习的量子力学方法具有显著优势。