Score-based diffusion models have proven effective in image generation and
have gained widespread usage; however, the underlying factors contributing to
the performance disparity between stochastic and deterministic (i.e., the
probability flow ODEs) sampling schemes remain unclear. We introduce a novel
formulation of diffusion models using Feynman's path integral, which is a
formulation originally developed for quantum physics. We find this formulation
providing comprehensive descriptions of score-based generative models, and
demonstrate the derivation of backward stochastic differential equations and
loss functions.The formulation accommodates an interpolating parameter
connecting stochastic and deterministic sampling schemes, and we identify this
parameter as a counterpart of Planck's constant in quantum physics. This
analogy enables us to apply the Wentzel-Kramers-Brillouin (WKB) expansion, a
well-established technique in quantum physics, for evaluating the negative
log-likelihood to assess the performance disparity between stochastic and
deterministic sampling schemes.

利用费曼路径积分的新方法来描述分数评分的扩散模型，推导了反向随机微分方程和损失函数，通过应用量子物理中的 Wentzel-Kramers-Brillouin (WKB) 展开技术来评估随机和确定性采样方案之间的性能差异。

费曼路径积分理解扩散模型

Understanding Diffusion Models by Feynman's Path Integral

This paper proposes a novel deep generative model, called BSDE-Gen, which
combines the flexibility of backward stochastic differential equations (BSDEs)
with the power of deep neural networks for generating high-dimensional complex
target data, particularly in the field of image generation. The incorporation
of stochasticity and uncertainty in the generative modeling process makes
BSDE-Gen an effective and natural approach for generating high-dimensional
data. The paper provides a theoretical framework for BSDE-Gen, describes its
model architecture, presents the maximum mean discrepancy (MMD) loss function
used for training, and reports experimental results.

该论文提出了一种新颖的深度生成模型 BSDE-Gen，将 BSDE 的灵活性与深度神经网络的高能力相结合，特别适用于高维复杂目标数据的生成，特别是在图像生成领域。该模型将随机性和不确定性纳入生成建模过程，使 BSDE-Gen 成为生成高维数据的有效和自然方法。论文提供了 BSDE-Gen 的理论框架，描述了其模型架构，介绍了用于训练的最大均值差异（MMD）损失函数，并报告了实验结果。

反向随机微分方程结合的深度生成建模

Deep Generative Modeling with Backward Stochastic Differential Equations

High-dimensional partial differential equations (PDEs) are ubiquitous in
economics, science and engineering. However, their numerical treatment poses
formidable challenges since traditional grid-based methods tend to be
frustrated by the curse of dimensionality. In this paper, we argue that tensor
trains provide an appealing approximation framework for parabolic PDEs: the
combination of reformulations in terms of backward stochastic differential
equations and regression-type methods in the tensor format holds the promise of
leveraging latent low-rank structures enabling both compression and efficient
computation. Following this paradigm, we develop novel iterative schemes,
involving either explicit and fast or implicit and accurate updates. We
demonstrate in a number of examples that our methods achieve a favorable
trade-off between accuracy and computational efficiency in comparison with
state-of-the-art neural network based approaches.

本文研究高维偏微分方程，提出了一种基于张量列的近似方法和迭代方案，能够在精确性和计算效率之间达到有利的折衷。

使用张量列车格式解决高维抛物偏微分方程

Solving high-dimensional parabolic PDEs using the tensor train format

We propose a new algorithm for solving parabolic partial differential
equations (PDEs) and backward stochastic differential equations (BSDEs) in high
dimension, by making an analogy between the BSDE and reinforcement learning
with the gradient of the solution playing the role of the policy function, and
the loss function given by the error between the prescribed terminal condition
and the solution of the BSDE. The policy function is then approximated by a
neural network, as is done in deep reinforcement learning. Numerical results
using TensorFlow illustrate the efficiency and accuracy of the proposed
algorithms for several 100-dimensional nonlinear PDEs from physics and finance
such as the Allen-Cahn equation, the Hamilton-Jacobi-Bellman equation, and a
nonlinear pricing model for financial derivatives.

该论文提出了一种基于强化学习和神经网络的算法用于解决高维情况下的偏微分方程和反向随机微分方程等数学问题，并在物理和金融学领域的各种非线性情况下进行了测试和优化。