Manifold-specific minimax rates for solving PDEs

Establish minimax optimal convergence rates for directly solving partial differential equations on compact d-dimensional Riemannian manifolds M^d that explicitly leverage the intrinsic manifold structure, and rigorously characterize these manifold-specific minimax rates (as opposed to rates derived in d-dimensional Euclidean domains).

Background

The paper proves convergence rates that match the classical minimax rate for Sobolev functions in d-dimensional Euclidean domains and argues that the rates depend only on the intrinsic dimension d of the manifold. However, the authors note that minimax optimal rates for solving PDEs directly on manifolds, explicitly exploiting manifold geometry, have not been formally established.

Clarifying such manifold-specific minimax rates would provide a principled benchmark for algorithms that solve PDEs on manifolds (including the proposed wPINN framework), and would formalize how intrinsic manifold dimension affects sample complexity and convergence independently of the ambient dimension.