Score-based generative models (SGMs) are powerful tools to sample from
complex data distributions. Their underlying idea is to (i) run a forward
process for time $T_1$ by adding noise to the data, (ii) estimate its score
function, and (iii) use such estimate to run a reverse process. As the reverse
process is initialized with the stationary distribution of the forward one, the
existing analysis paradigm requires $T_1\to\infty$. This is however
problematic: from a theoretical viewpoint, for a given precision of the score
approximation, the convergence guarantee fails as $T_1$ diverges; from a
practical viewpoint, a large $T_1$ increases computational costs and leads to
error propagation. This paper addresses the issue by considering a version of
the popular predictor-corrector scheme: after running the forward process, we
first estimate the final distribution via an inexact Langevin dynamics and then
revert the process. Our key technical contribution is to provide convergence
guarantees in Wasserstein distance which require to run the forward process
only for a finite time $T_1$. Our bounds exhibit a mild logarithmic dependence
on the input dimension and the subgaussian norm of the target distribution,
have minimal assumptions on the data, and require only to control the $L^2$
loss on the score approximation, which is the quantity minimized in practice.

本文研究基于分数的生成模型（SGMs）中遇到的向后过程收敛性问题，提出了基于预测校正方案的近似 Langevin 动力学方法，并在有限时间内提供了 Wassertein 距离的收敛保证。

通过预测校正改进基于分数的扩散模型的收敛性

Improved Convergence of Score-Based Diffusion Models via  Prediction-Correction

Predicting outcomes and planning interactions with the physical world are
long-standing goals for machine learning. A variety of such tasks involves
continuous physical systems, which can be described by partial differential
equations (PDEs) with many degrees of freedom. Existing methods that aim to
control the dynamics of such systems are typically limited to relatively short
time frames or a small number of interaction parameters. We present a novel
hierarchical predictor-corrector scheme which enables neural networks to learn
to understand and control complex nonlinear physical systems over long time
frames. We propose to split the problem into two distinct tasks: planning and
control. To this end, we introduce a predictor network that plans optimal
trajectories and a control network that infers the corresponding control
parameters. Both stages are trained end-to-end using a differentiable PDE
solver. We demonstrate that our method successfully develops an understanding
of complex physical systems and learns to control them for tasks involving PDEs
such as the incompressible Navier-Stokes equations.

本研究介绍了一种新颖的分层预测 - 校正方案，使神经网络能够学习理解和控制复杂的非线性物理系统，在涉及偏微分方程的任务中成功地开发了对这些系统的理解，并学会了控制它们。