将SPDE与SGM关联起来
Connecting SPDE to SGMs
AI总结：
该研究针对从SGM相关福克-普朗克方程推导的SPDE，修改原方程并引入噪声，用变分方法和新型空间证明了解的存在性与唯一性。
摘要：
本文研究了从与基于分数的生成模型（Score-based Generative Models，SGMs）相关的福克-普朗克方程推导而来的随机偏微分方程（Stochastic Partial Differential Equation，SPDE）。我们修改了标准福克-普朗克方程以更好地表示实际SGMs，并引入噪声来缓解潜在的离散化问题。主要目标是证明该SPDE解的存在性与唯一性。由于含时算子和无界域的存在，这一方面需要仔细考量。为克服这些障碍，我们采用变分方法，并引入了一个受奥恩斯坦-乌伦贝克算子启发的新型空间。通过证明该空间及其子空间满足必要假设，研究者确立了给定SPDE的解的存在性。
英文摘要：
This paper investigates a Stochastic Partial Differential Equation (SPDE) derived from the Fokker-Planck equation associated with Score-based Generative Models. We modify the standard Fokker-Planck equation to better represent practical SGMs and introduce noise to mitigate potential discretization issues. The primary goal is to prove the existence and uniqueness of solutions for this SPDE. This aspect requires careful consideration due to the time-dependent operator and unbounded domain. To overcome these hurdles, we employ a variational approach and introduce a novel space inspired by Ornstein-Uhlenbeck operators. By demonstrating that this space and its subspace satisfy the necessary assumptions, they establish the existence of a solution for the given SPDE.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
