Machine learning approaches for solving partial differential equations require learning mappings between function spaces. While convolutional or graph neural networks are constrained to discretized functions, neural operators present a promising milestone toward mapping functions directly. Despite impressive results they still face challenges with respect to the domain geometry and typically rely on some form of discretization. In order to alleviate such limitations, we present CORAL, a new method that leverages coordinate-based networks for solving PDEs on general geometries. CORAL is designed to remove constraints on the input mesh, making it applicable to any spatial sampling and geometry. Its ability extends to diverse problem domains, including PDE solving, spatio-temporal forecasting, and inverse problems like geometric design. CORAL demonstrates robust performance across multiple resolutions and performs well in both convex and non-convex domains, surpassing or performing on par with state-of-the-art models.

翻译：利用机器学习求解偏微分方程的方法需要学习函数空间之间的映射。虽然卷积神经网络或图神经网络受限于离散化函数，神经算子则为直接映射函数提供了一个有前景的里程碑。尽管取得了令人瞩目的成果，但它们在处理域几何形状时仍面临挑战，且通常依赖于某种形式的离散化。为了缓解这些局限，我们提出了CORAL，一种利用基于坐标的网络在一般几何形状上求解偏微分方程的新方法。CORAL旨在消除对输入网格的约束，使其适用于任意空间采样和几何形状。其能力扩展到不同的问题领域，包括偏微分方程求解、时空预测以及几何设计等逆问题。CORAL在多个分辨率下均展现出稳健的性能，并在凸域和非凸域中表现良好，超越或达到了与最先进模型相当的性能水平。