Abstract: Many models in machine learning and PDE-based inverse problems exhibit intrinsic spectral properties, which have been used to explain the generalization capacity and the ill-posedness of such problems. In this talk, we discuss weighted training for computational learning and inversion with noisy data. The highlight of the proposed framework is that we allow weighting in both the parameter space and the data space. The weighting scheme encodes both a priori knowledge of the object to be learned and a strategy to weight the contribution of training data in the loss function. We demonstrate that appropriate weighting from prior knowledge can improve the generalization capability of the learned model in both machine learning and PDE-based inverse problems.