用高斯过程求解非微扰函数型重整化群方程，更灵活且能处理复杂场构型。
Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equations
- 基于高斯过程构建函数空间上的泛化表示，不依赖特定方程或离散化
- 在Wetterich和Wilson-Polchinski方程上表现优于局部势近似，支持非恒定场
- 可融合物理先验，适合研究瞬子等复杂场结构
我们提出一种算子学习框架，用于求解非微扰的函数型重整化群方程，这类方程是定义在泛函上的积分微分方程。所提方法利用高斯过程算子学习，在函数空间上构建灵活的泛函表示，独立于特定方程或离散化方式。该方法具有广泛适用性，同时可将物理先验融入先验均值或核函数设计中。我们在多个相关方程（如Wetterich方程和Wilson-Polchinski方程）上验证了其性能，结果表明其精度不低于甚至优于现有近似方法（如局部势近似），且灵活性显著提升。特别地，该方法可处理非恒定场，为研究瞬子等复杂场构型提供了可能。
原文摘要 · Abstract (English)
We present an operator learning framework for solving non-perturbative functional renormalization group equations, which are integro-differential equations defined on functionals. Our proposed approach uses Gaussian process operator learning to construct a flexible functional representation formulated directly on function space, making it independent of a particular equation or discretization. Our method is flexible, and can apply to a broad range of functional differential equations while still allowing for the incorporation of physical priors in either the prior mean or the kernel design. We demonstrate the performance of our method on several relevant equations, such as the Wetterich and Wilson--Polchinski equations, showing that it achieves equal or better performance than existing approximations such as the local-potential approximation, while being significantly more flexible. In particular, our method can handle non-constant fields, making it promising for the study of more complex field configurations, such as instantons.
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