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

Xianjin Yang, Matthieu Darcy, Matthew Hudes, Francis J. Alexander, Gregory Eyink, Houman Owhadi

TL;DR

This work presents a GP-based operator-learning framework that solves FRG flow equations directly in function space, thereby treating the infinite-dimensional functional problem in a discretization-agnostic manner. By projecting functionals onto a finite set of collocation fields and learning a GP surrogate, the method enforces the FRG PDE constraints on collocation points and evolves a finite-dimensional set of values via an ODE in the RG scale $\kappa$, enabling accurate reconstruction of the full functional at later scales. The authors demonstrate the approach on two FRG benchmarks—the Wetterich–Morris and Wilson–Polchinski equations—showing high accuracy on Gaussian models and competitive performance against standard LPA in lattice $\phi^4$ settings, with better handling of non-homogeneous fields. The framework offers a flexible, physics-informed, discretization-agnostic solver with potential for uncertainty quantification and rigorous error analysis, and it points toward extensions to more complex field configurations and higher-dimensional problems.

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.

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

TL;DR

This work presents a GP-based operator-learning framework that solves FRG flow equations directly in function space, thereby treating the infinite-dimensional functional problem in a discretization-agnostic manner. By projecting functionals onto a finite set of collocation fields and learning a GP surrogate, the method enforces the FRG PDE constraints on collocation points and evolves a finite-dimensional set of values via an ODE in the RG scale , enabling accurate reconstruction of the full functional at later scales. The authors demonstrate the approach on two FRG benchmarks—the Wetterich–Morris and Wilson–Polchinski equations—showing high accuracy on Gaussian models and competitive performance against standard LPA in lattice settings, with better handling of non-homogeneous fields. The framework offers a flexible, physics-informed, discretization-agnostic solver with potential for uncertainty quantification and rigorous error analysis, and it points toward extensions to more complex field configurations and higher-dimensional problems.

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.
Paper Structure (38 sections, 154 equations, 5 figures, 2 tables)

This paper contains 38 sections, 154 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Functional data and operator learning
  3. Functional renormalization group equations
  4. Numerical methods for FRG and functional PDEs
  5. Summary of contributions
  6. Notation.
  7. Outline of the rest of the paper
  8. Methodology
  9. Gaussian processes and kernel interpolation
  10. Kernel approximation of functionals
  11. A GP method for solving functional PDEs
  12. A functional RG example: the Wetterich--Morris equation
  13. Numerical results
  14. The Wilson--Polchinski equation: Gaussian-model
  15. Numerical setup
  16. ...and 23 more sections

Figures (5)

  • Figure 1: Commutative diagram for the functional learning setup.
  • Figure 2: Numerical results for Gaussian-model solution of the Wilson--Polchinski equation. Subfigures (A)--(C) show results at collocation points, while (D)--(F) present corresponding results at testing points. For visualization, only the first 3 of 1000 collocation points and the first 3 of 200 testing points are displayed. The relative $L^2$ error is evaluated for each scale variable $\kappa$.
  • Figure 3: Benchmark of the GP reconstruction versus standard LPA, lattice Monte Carlo, and the transfer matrix method. (A) Values of the potential $U_{\kappa_{\mathrm{IR}}}(\phi)$ on $[-5,5]$ of different methods. (B) Susceptibility $\chi(c)$ versus external field $c$ (log-log). (C) Susceptibility errors (relative to transfer matrix); for most values of $c$, the GP exhibits smaller errors than LPA, especially at small $c$. (D) Magnetization $m(c)$ (log-log). (E) Magnetization errors (relative to transfer matrix); in most of the $c$ range the GP is more accurate than LPA.
  • Figure 4: Numerical results of the Wetterich Gaussian model. Subfigures (A)--(C) show results at collocation points, while (D)--(F) present corresponding results at testing points. For visualization, only the first 3 of 1000 collocation points and the first 3 of 200 testing points are displayed. The relative $L^2$ error is evaluated for each scale variable $\kappa$.
  • Figure 5: Numerical results for the Wetterich flow with the $\phi^4$ bare action. Panels (A)--(B) illustrate three of the $N_\text{col}=1000$ training fields and the corresponding GP predictions, while panels (C)--(D) plot the surrogate and the LPA potential on constant-field inputs in $[0, 4]$. Panel (E) maps the pointwise relative difference across RG scales $\kappa\in[2,10]$ and field amplitudes $\phi\in[0,4]$.

Theorems & Definitions (1)

  • Remark 2.1