There are many problems in geometric analysis where one wants to prove existence of a solution of an elliptic PDE on a compact manifold. The curvature of the geometry makes it difficult to apply classical PDE solvers and recently Physics-informed Neural Networks (PINNs) have been applied with some success to study many equations through numerical simulations, and I will list some of them. It would then be desriable to turn these simulations into a numerically verified proof. The equations are often much better behaved compared to some analysis problems on Euclidean domains where numerically verified proofs were carried out. But the curvature enters in a difficult way into linear estimates, which makes it difficult to run such proofs on closed manifolds, even if the equations are simple. I will compare a recent numerically verified proof on the sphere with ongoing work on a more complicated manifold and highlight where the challenges lie.
