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

arXiv: https://arxiv.org/abs/2512.20956 · HTML: https://mlsanity.com/paper/2512.20956/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 15 of its 51 references that are in the map.
- First posted on arXiv 24 Dec 2025.

## Builds on (most cited first, 15 of 15)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Solving and Learning Nonlinear PDEs with Gaussian Processes](https://mlsanity.com/paper/2103.12959/) (2021)
- [Operator-valued Kernels for Learning from Functional Response Data](https://mlsanity.com/paper/1510.08231/) (2015)
- [Kernel Methods are Competitive for Operator Learning](https://mlsanity.com/paper/2304.13202/) (2023)
- [Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEs](https://mlsanity.com/paper/2305.04962/) (2023)
- [Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimation](https://mlsanity.com/paper/2005.11375/) (2020)
- [Do ideas have shape? Idea registration as the continuous limit of artificial neural networks](https://mlsanity.com/paper/2008.03920/) (2020)
- [Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dots](https://mlsanity.com/paper/2311.17007/) (2023)
- [Numerical methods for Mean field Games based on Gaussian Processes and Fourier Features](https://mlsanity.com/paper/2112.05414/) (2021)
- [Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processes](https://mlsanity.com/paper/2510.05568/) (2025)
- [Solving Roughly Forced Nonlinear PDEs via Misspecified Kernel Methods and Neural Networks](https://mlsanity.com/paper/2501.17110/) (2025)
- [Dreaming up scale invariance via inverse renormalization group](https://mlsanity.com/paper/2506.04016/) (2025)

## Abstract (arXiv)

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.