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

2025-12-24

AI Summary

This work addresses the computational challenge of solving nonperturbative functional renormalization group (FRG) equations—nonlinear integro-differential equations defined on infinite-dimensional function spaces. We propose the first Gaussian process operator learning framework for FRG, directly modeling the Wetterich and Wilson–Polchinski equations in the functional space without spatial discretization or reliance on specific equation forms or the local potential approximation (LPA). Crucially, we generalize Gaussian processes to learn functional differential operators, enabling incorporation of physical priors—such as symmetry constraints and asymptotic behavior—into both the mean function and kernel design. Experiments demonstrate that our method achieves significantly higher accuracy than LPA on benchmark models and, for the first time, enables mesh-free, interpretable, end-to-end solution of functional PDEs for non-constant field configurations—including instantons—while preserving theoretical consistency and functional structure.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Non-convex Optimization
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

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.

Problem

• Solving non-perturbative functional renormalization group equations
• Developing a flexible operator learning framework for functional differential equations
• Handling non-constant fields for complex field configurations like instantons

Innovation

• Gaussian process operator learning for functional representation
• Flexible method independent of equation discretization
• Incorporates physical priors in prior mean or kernel design