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

TL;DR

MetaColloc is a novel, optimization-free PDE solving framework that uses meta-learned basis functions, enabling rapid and accurate solutions across diverse PDEs without retraining.

Contribution

It introduces a universal neural basis function dictionary via meta-learning, eliminating the need for optimization during PDE solving.

Findings

Achieves state-of-the-art accuracy on multiple PDEs.

Reduces test-time computation by several orders of magnitude.

Reveals a frequency mismatch affecting operator stability at high frequencies.

Abstract

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.