Physical unclonable function (PUF) is a promising lightweight hardware security primitive for resource-constrained systems. It can generate a large number of challenge-response pairs (CRPs) for device authentication based on process variations. However, attackers can collect the CRPs to build a machine learning (ML) model with high prediction accuracy for the PUF. Recently, a lot of ML-resistant PUF structures have been proposed, e.g., a multiplexer-based PUF (MPUF) was introduced to resist ML attacks and its two variants (rMPUF and cMPUF) were further proposed to resist reliability-based and cryptanalysis modeling attacks, respectively. In this article, we propose a general framework for ML attacks on strong PUFs, then based on the framework, we present two novel modeling attacks, named logical approximation and global approximation, that use artificial neural network (ANN) to characterize the nonlinear structure of MPUF, rMPUF, cMPUF, and XOR Arbiter PUF. The logical approximation method uses linear functions to approximate logical operations and builds a precise soft model based on the combination of logical gates in the PUF. The global approximation method uses the function sinc with filtering characteristics to fit the mapping relationship between the challenge and response. The experimental results show that the proposed two approximation attacks can successfully model the (n, k)-MPUF (k = 3,4), (n, k)-rMPUF (k = 2,3), cMPUF (k = 4,5), and l-XOR Arbiter PUF (l = 3, 4, 5) (n = 32, 64) with the average accuracies of 96.85%, 95.33%, 94.52%, and 96.26%, respectively.