A data-driven view of exterior calculus

Source abstract

Classical computational tools for differential forms assume symbolic expressions, mesh discretizations, or smoothness close to [Formula: see text]. In data-driven settings one often has only oracle access: the ability to evaluate a form at arbitrary points and tangent vectors, with no closed-form expression and no mesh. Building on a flux-based reformulation of the exterior derivative, recently developed by the author with Fadel and Sá Earp, in which the exterior derivative is recovered as a limit of normalized boundary integrals over shrinking blocks, we describe a mesh-free, symbol-free numerical scheme for exterior differentiation, valid under regularity assumptions weaker than [Formula: see text]. We demonstrate the method in [Formula: see text], [Formula: see text] and on the sphere [Formula: see text], verifying Stokes’ theorem, the identity [Formula: see text], and Gauss–Bonnet purely from pointwise samples, and detecting de Rham cohomology (closed-but-not-exact forms) from data. Companion Jupyter notebooks accompany the paper. This paper is based on a tutorial delivered at DANGER: Data, Numbers, and Geometry (Banff, April 2026).