Neural Networks in Local Coordinates
Ahmed Abdeljawad · 2025
We explore the ability of deep ReLU neural networks to realize functions on manifolds. By establishing appropriate assumptions, we ensure that the coordinate charts can be exactly represented without error. Locally, we construct networks characterizing tooth functions on coordinate neighborhoods. To resolve mismatches arising from the manifold’s complex structure, we further develop a global tooth function defined over the entire manifold, effectively represented by a ReLU neural network.