Integrally bladed disks, or blisks, are critical components within compressors of modern turbomachinery, and during operation can exhibit complex dynamic behaviors. Understanding and accurately modeling these dynamics is crucial for blisk design and safe operation. To model blisk dynamics, large finite-element models containing millions of degrees of freedom are often used. However, simulating blisk dynamics using these models can be computationally expensive and frequently infeasible. As such, physics-based reduced-order models are currently the preferred method for modeling blisk behavior and performing stress analyses. These models typically are used to run probabilistic analyses such as Monte Carlo simulations for capturing the effects of variabilities in blisk dynamics produced from unavoidable deviations in material properties and geometry, called mistuning. Mistuning is known to cause energy localization within the blisk, resulting in amplified response amplitudes and greater stresses than those present in a nominally cyclic-symmetric blisk. Because the uncertainties introduced by mistuning cannot be removed, it is of great interest to both understand the effects of mistuning and identify it in order to accurately predict forced responses during operation when designing a blisk. Additionally, to mitigate large response amplitudes, nonlinear damping mechanisms such as ring dampers or shrouds can be employed, which dissipate energy through friction contacts. For both linear and nonlinear systems, physics-based reduced-order models, however, often cannot incorporate experimental data to improve their accuracy and applicability to as-manufactured models and/or rely on experimental modal identification that can be challenging within the often unavoidable high-modal density regimes of blisk structures. To address these challenges, this work presents novel data-driven methods for 1) forced-response prediction of mistuned blisks with and without introducing friction nonlinearities, and 2) identifying the mistuning of as-manufactured blisks. All presented methodologies focus on sector-level, frequency-domain formulations to reduce necessary training data and avoid difficulties associated with time-domain data generation and experimental collection. First, a data-driven method for forced-response prediction is presented and demonstrated for a lumped mass model mimicking a mistuned blisk. This method is based on two artificial neural networks and a cyclic coupling procedure, and is applicable to linear mistuned cyclic structures in general. Next, a novel data-driven approach for mistuning identification is developed. Unlike previous methods, this approach does not require any modal analyses nor isolation of individual blades, and it removes all effects of forcing amplitude via a developed robust data processing, selection, and conditioning procedure. This approach is validated for a high-dimensional finite-element blisk model considering significant measurement noise similar to that observed experimentally. A first-of-its-kind physics-informed machine learning modeling approach for forced response prediction is developed that incorporates physical laws directly into a network architecture, still maintains a sector-level viewpoint, and only considers response data from easily experimentally measurable points along blade tips. With this approach, a series of numerical techniques are developed to significantly improve robustness and generalizability. Lastly, to predict forced responses of blisk systems containing contact nonlinearities, the original linear approach is extended using a harmonic-balance formulation. This includes the development of an efficient data selection and training scheme to further reduce the amount of necessary training data. Validation is shown for a nonlinear lumped mass model representative of a mistuned blisk with a friction ring damper and significant mistuning and nonlinear effects.