Facial action unit (AU) intensity is an index to characterize human expressions. Accurate AU intensity estimation depends on three major elements: image representation, intensity estimator, and supervisory information. Most existing methods learn intensity estimator with fixed image representation, and rely on the availability of fully annotated supervisory information. In this paper, a novel general framework for AU intensity estimation is presented, which differs from traditional estimation methods in two aspects. First, rather than keeping image representation fixed, it simultaneously learns representation and intensity estimator to achieve an optimal solution. Second, it allows incorporating weak supervisory training signal from human knowledge (e.g. feature smoothness, label smoothness, label ranking, and positive label), which makes our model trainable even fully annotated information is not available. More specifically, human knowledge is represented as either soft or hard constraints which are encoded as regularization terms or equality/inequality constraints, respectively. On top of our novel framework, we additionally propose an efficient algorithm for optimization based on Alternating Direction Method of Multipliers (ADMM). Evaluations on two benchmark databases show that our method outperforms competing methods under different ratios of AU intensity annotations, especially for small ratios.