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MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions

arXiv 2605.12368 ↗First posted 12 May 2026cs.LGcited by 0

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Abstract (arXiv)

Solving partial differential equations (PDEs) with machine learning typically requires training a new neural network for every new equation. This optimization is slow. We introduce MetaColloc. It is an optimization-free and data-free framework that removes this bottleneck completely. We decouple basis discovery from the solving process. We meta-train a dual-branch neural network on diverse Gaussian Random Fields. This offline process creates a universal dictionary of neural basis functions. At test time, we freeze the network. We solve the PDE by assembling a collocation matrix. We find the solution through a single linear least squares step. For non-linear PDEs, we apply the Newton-Raphson method to achieve fast quadratic convergence. Our experiments across six 2D and 3D PDEs show massive improvements. MetaColloc reaches state-of-the-art accuracy on smooth and non-linear problems. It also reduces test-time computation by several orders of magnitude. Finally, we provide a detailed frequency sweep analysis. This analysis reveals a critical mismatch between function approximation and operator stability at extremely high frequencies. This profound finding opens a clear path toward future operator-aware meta-learning.

Key facts. No citing papers in the map yet (data through 7 Oct 2026). Builds on 33 of its 53 references that are in the map. First posted on arXiv 12 May 2026.

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References · 33

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
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  5. 2024KAN: Kolmogorov-Arnold Networkscited by 634
  6. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  7. 2020On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networkscited by 279
  8. 2024Poseidon: Efficient Foundation Models for PDEscited by 186
  9. 2022Respecting causality is all you need for training physics-informed neural networkscited by 154
  10. 2024DiffusionPDE: Generative PDE-Solving Under Partial Observationcited by 100
  11. 2024Physics-Informed Diffusion Modelscited by 88
  12. 2020Local Extreme Learning Machines and Domain Decomposition for Solving Linear and Nonlinear Partial Differential Equationscited by 78
21 more
  1. 2024Latent Neural Operator for Solving Forward and Inverse PDE Problemscited by 76
  2. 2024Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEscited by 58
  3. 2024On conditional diffusion models for PDE simulationscited by 56
  4. 2024CViT: Continuous Vision Transformer for Operator Learningcited by 47
  5. 2024ConFIG: Towards Conflict-free Training of Physics Informed Neural Networkscited by 43
  6. 2024AROMA: Preserving Spatial Structure for Latent PDE Modeling with Local Neural Fieldscited by 38
  7. 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
  8. 2024Text2PDE: Latent Diffusion Models for Accessible Physics Simulationcited by 25
  9. 2024Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networkscited by 21
  10. 2024Dual Cone Gradient Descent for Training Physics-Informed Neural Networkscited by 21
  11. 2021Solving Partial Differential Equations with Point Source Based on Physics-Informed Neural Networkscited by 14
  12. 2023Solving High Frequency and Multi-Scale PDEs with Gaussian Processescited by 14
  13. 2023TransNet: Transferable Neural Networks for Partial Differential Equationscited by 13
  14. 2024PIG: Physics-Informed Gaussians as Adaptive Parametric Mesh Representationscited by 11
  15. 2024CL-DiffPhyCon: Closed-loop Diffusion Control of Complex Physical Systemscited by 8
  16. 2024Metamizer: a versatile neural optimizer for fast and accurate physics simulationscited by 7
  17. 2024Solving Partial Differential Equations with Random Feature Modelscited by 6
  18. 2023The Power of Linear Combinations: Learning with Random Convolutionscited by 4
  19. 2024RandNet-Parareal: a time-parallel PDE solver using Random Neural Networkscited by 3
  20. 2025PRDP: Progressively Refined Differentiable Physicscited by 3
  21. 2025A Multiple Transferable Neural Network Method with Domain Decomposition for Elliptic Interface Problemscited by 2

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