用几何谱方法解决稀疏数据下的物理方程求解问题
DeepVekua: Geometric-Spectral Representation Learning for Physics-Informed Fields
- 将几何变换与谱分析结合，分离几何与物理学习
- 在对流扩散系统上性能提升100倍于传统谱方法
- 适合需要高精度物理建模的科学计算场景
我们提出DeepVekua，一种融合几何深度学习与谱分析的混合架构，用于在稀疏数据条件下求解偏微分方程（PDEs）。通过学习一个微分同胚坐标变换，将复杂几何映射到潜在调和空间，该方法在对流扩散系统上优于现有隐式表示方法。不同于依赖坐标的网络因频谱偏差表现不佳，DeepVekua将几何学习与物理学习分离，并以闭式解求得最优谱权重。实验表明其性能相较谱基线提升100倍。代码已开源：https://github.com/VladimerKhasia/vekuanet。
原文摘要 · Abstract (English)
We present DeepVekua, a hybrid architecture that unifies geometric deep learning with spectral analysis to solve partial differential equations (PDEs) in sparse data regimes. By learning a diffeomorphic coordinate transformation that maps complex geometries to a latent harmonic space, our method outperforms state-of-the-art implicit representations on advection-diffusion systems. Unlike standard coordinate-based networks which struggle with spectral bias, DeepVekua separates the learning of geometry from the learning of physics, solving for optimal spectral weights in closed form. We demonstrate a 100x improvement over spectral baselines. The code is available at https://github.com/VladimerKhasia/vekuanet.
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