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

arXiv 2512.20956 ↗First posted 24 Dec 2025cs.LGcited by 0

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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.

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References · 15

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 527
  3. 2020Model Reduction and Neural Networks for Parametric PDEscited by 258
  4. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 214
  5. 2021Solving and Learning Nonlinear PDEs with Gaussian Processescited by 129
  6. 2015Operator-valued Kernels for Learning from Functional Response Datacited by 60
  7. 2023Kernel Methods are Competitive for Operator Learningcited by 50
  8. 2023Error Analysis of Kernel/GP Methods for Nonlinear and Parametric PDEscited by 21
  9. 2020Consistency of Empirical Bayes And Kernel Flow For Hierarchical Parameter Estimationcited by 18
  10. 2020Do ideas have shape? Idea registration as the continuous limit of artificial neural networkscited by 16
  11. 2023Codiscovering graphical structure and functional relationships within data: A Gaussian Process framework for connecting the dotscited by 8
  12. 2021Numerical methods for Mean field Games based on Gaussian Processes and Fourier Featurescited by 7
3 more
  1. 2025Bilevel optimization for learning hyperparameters: Application to solving PDEs and inverse problems with Gaussian processescited by 3
  2. 2025Solving Roughly Forced Nonlinear PDEs via Misspecified Kernel Methods and Neural Networkscited by 2
  3. 2025Dreaming up scale invariance via inverse renormalization groupcited by 1

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