# Deep Stochastic Mechanics

arXiv: https://arxiv.org/abs/2305.19685 · HTML: https://mlsanity.com/paper/2305.19685/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 23 of its 86 references that are in the map.
- First posted on arXiv 31 May 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 23 of 23)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Diffusion Models: A Comprehensive Survey of Methods and Applications](https://mlsanity.com/paper/2209.00796/) (2022)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Non-convex learning via Stochastic Gradient Langevin Dynamics: a nonasymptotic analysis](https://mlsanity.com/paper/1702.03849/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Why and When Can Deep -- but Not Shallow -- Networks Avoid the Curse of Dimensionality: a Review](https://mlsanity.com/paper/1611.00740/) (2016)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)
- [Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space](https://mlsanity.com/paper/2005.05409/) (2020)
- [Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learning](https://mlsanity.com/paper/2008.13333/) (2020)
- [Convergence rates for the stochastic gradient descent method for non-convex objective functions](https://mlsanity.com/paper/1904.01517/) (2019)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Probability flow solution of the Fokker-Planck equation](https://mlsanity.com/paper/2206.04642/) (2022)
- [NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems](https://mlsanity.com/paper/2112.10526/) (2021)
- [Deep Neural Networks with Trainable Activations and Controlled Lipschitz Constant](https://mlsanity.com/paper/2001.06263/) (2020)
- [Interpolating between BSDEs and PINNs: deep learning for elliptic and parabolic boundary value problems](https://mlsanity.com/paper/2112.03749/) (2021)
- [Dimension-free convergence rates for gradient Langevin dynamics in RKHS](https://mlsanity.com/paper/2003.00306/) (2020)
- [Embed and Emulate: Learning to estimate parameters of dynamical systems with uncertainty quantification](https://mlsanity.com/paper/2211.01554/) (2022)
- [CNNs Avoid Curse of Dimensionality by Learning on Patches](https://mlsanity.com/paper/2205.10760/) (2022)

## Built on by (most cited first, 3 of 3)

- [Hamilton-Zero: A Neural Tensor-Network Foundation Model for Ground States of Arbitrary Quadratic Qubit Hamiltonians](https://mlsanity.com/paper/2608.11911/) (2026)
- [Intrinsic-Metric Physics-Informed Neural Networks (IM-PINN) for Reaction-Diffusion Dynamics on Complex Riemannian Manifolds](https://mlsanity.com/paper/2601.00834/) (2025)
- [Quantum Dynamics via Score Matching on Bohmian Trajectories](https://mlsanity.com/paper/2604.25137/) (2026)

## Abstract (arXiv)

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 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.