Solving Functional PDEs with Gaussian Processes and Applications to Functional Renormalization Group Equations

TL;DR

This paper introduces a Gaussian process operator learning framework for solving complex functional renormalization group equations, offering a flexible, discretization-independent approach that outperforms traditional approximations in physics applications.

Contribution

The paper presents a novel Gaussian process-based operator learning method for functional PDEs, enabling flexible, equation-agnostic solutions directly on function space with physical priors.

Findings

Achieves equal or better performance than local-potential approximation

Handles non-constant fields and complex configurations

Demonstrates effectiveness on Wetterich and Wilson--Polchinski equations

Abstract

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.