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：机器学习方法求解偏微分方程需要学习函数空间之间的映射。卷积或图神经网络受限于离散化函数，而神经算子则直接为映射函数奠定了有前景的里程碑。尽管取得了显著成果，但在域几何方面仍面临挑战，且通常依赖某种形式的离散化。为缓解这些局限，我们提出CORAL，一种利用基于坐标的网络求解通用几何上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与几何形状。其能力涵盖广泛的问题领域，包括偏微分方程求解、时空预测以及几何设计等逆问题。CORAL在多分辨率下展现出稳健性能，在凸域与非凸域中均表现出色，超越或持平于最先进模型。