Neural networks (NNs) are currently changing the computational paradigm on how to combine data with mathematical laws in physics and engineering in a profound way, tackling challenging inverse and ill-posed problems not solvable with traditional methods. However, quantifying errors and uncertainties in NN-based inference is more complicated than in traditional methods. In this talk, we will present a comprehensive framework that includes uncertainty modeling, new and existing solution methods, as well as Information bottleneck based uncertainty quantification for neural function regression and neural operator learning.