无需训练即可快速求解各类微分方程，效率提升数个数量级。
MetaColloc: Optimization-Free PDE Solving via Meta-Learned Basis Functions
- 用元学习预先构建通用神经基函数库，测试时直接调用
- 求解仅需一次线性最小二乘或牛顿法，计算量极低
- 适合需要快速部署、多场景求解的科研与工程应用
传统机器学习求解偏微分方程（PDE）需为每类新方程重新训练网络，过程缓慢。本文提出MetaColloc，一种无需优化、无需数据的框架，彻底消除这一瓶颈。通过在多样化高斯随机场上对双分支神经网络进行元训练，离线生成一个通用的神经基函数字典。测试时冻结网络，仅通过组装配置矩阵并求解一次线性最小二乘问题即可获得解；对非线性PDE则采用牛顿-拉夫森方法实现快速二次收敛。在六组二维和三维PDE上的实验表明，该方法在光滑及非线性问题上均达到当前最优精度，且测试计算量降低多个数量级。进一步的频率分析揭示了函数逼近与算子稳定性在极高频率下的严重失配，为未来面向算子感知的元学习指明方向。
原文摘要 · Abstract (English)
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.
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