  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.
