用新方法让神经算子高效模拟奇异随机偏微分方程
Wiener Chaos Expansion based Neural Operator for Singular Stochastic Partial Differential Equations
- 引入特征调制机制，精准捕捉解与光滑余项的关系
- 在Φ⁴₂模型上相对L2误差更低，且无需重整化因子
- 首次实现动态Φ⁴₃模型的数据驱动仿真，适合量子场论研究
本文探讨了近期提出的基于维纳混沌展开（WCE）的神经算子（NO）在奇异随机偏微分方程中的应用，例如近期研究中模拟的动态Φ⁴₂模型。不同于以往将威克-埃尔米特特征直接嵌入主干模型的做法，我们采用特征级线性调制（FiLM）来准确捕捉奇异SPDE解与其光滑余项之间的依赖关系。所提出的WCE-FiLM-NO在Φ⁴₂模型上表现出色，以相对L2损失、分布外L2损失和自相关得分衡量均优于现有方法，且无需重整化因子。此外，我们还展示了模拟Φ⁴₃数据的潜力，该模型更贴近统计量子场论中的实际科学实践。据我们所知，这是首个针对动态Φ⁴₃模型构建高效数据驱动代理模型的工作。
原文摘要 · Abstract (English)
In this paper, we explore how our recently developed Wiener Chaos Expansion (WCE)-based neural operator (NO) can be applied to singular stochastic partial differential equations, e.g., the dynamic $\boldsymbolΦ^4_2$ model simulated in the recent works. Unlike the previous WCE-NO which solves SPDEs by simply inserting Wick-Hermite features into the backbone NO model, we leverage feature-wise linear modulation (FiLM) to appropriately capture the dependency between the solution of singular SPDE and its smooth remainder. The resulting WCE-FiLM-NO shows excellent performance on $\boldsymbolΦ^4_2$, as measured by relative $L_2$ loss, out-of-distribution $L_2$ loss, and autocorrelation score; all without the help of renormalisation factor. In addition, we also show the potential of simulating $\boldsymbolΦ^4_3$ data, which is more aligned with real scientific practice in statistical quantum field theory. To the best of our knowledge, this is among the first works to develop an efficient data-driven surrogate for the dynamical $\boldsymbolΦ^4_3$ model.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。