Rethinking the ill-posedness of the spectral function reconstruction — Why is it fundamentally hard and how Artificial Neural Networks can help

Reconstructing hadron spectral functions through Euclidean correlation are of the important missions in lattice QCD calculations. However, a K\"allen--Lehmann(KL) representation, reconstruction is observed to be ill-posed practice. It usually ascribed fewer observation points compared number function. In this paper, by solving eigenvalue problem continuous KL convolution, we show analytically that ill-posedness inversion fundamental and it exists even for functions. We discussed how introduce regulators alleviate predicament, which include Artificial Neural Networks(ANNs) representations recently proposed Authors in~[Phys. Rev. D 106 (2022) L051502]. The uniqueness solutions using ANNs manifested validated numerically. Reconstructed different regularization schemes also demonstrated, together with their eigen-mode decomposition. observe components large eigenvalues can reliably reconstructed all methods, whereas those low need constrained regulators.