In manufacturing applications, a large number of data can be collected by experimental studies and/or sensors. This collected data is vital to improving process efficiency, scheduling maintenance activities, and predicting target variables. This dissertation explores a wide range of numerical modeling techniques that use data for manufacturing applications. Ignorance of uncertainty and the physical principle of a system are shortcomings of the existing methods. Besides, different methods are proposed to overcome the shortcomings by incorporating uncertainty and physics-based knowledge. In the first part of this dissertation, artificial neural networks (ANNs) are applied to develop a functional relationship between input and target variables and process parameter optimization. The second part evaluates the robust response surface optimization (RRSO) to quantify different sources of uncertainty in numerical analysis. Additionally, a framework based on the Bayesian network (BN) approach is proposed to support decision-making. Due to various uncertainties, estimating interval and probability distribution are often more helpful than deterministic point value estimation. Thus, the Monte Carlo (MC) dropout-based interval prediction technique is explored in the third part of this dissertation. A conservative interval prediction technique for the linear and polynomial regression model is also developed using linear optimization. Applications of different data-driven methods in manufacturing are useful to analyze situations, gain insights, and make essential decisions. But, the prediction by data-driven methods may be physically inconsistent. Thus, in the fourth part of this dissertation, a physics-informed machine learning (PIML) technique is proposed to incorporate physics-based knowledge with collected data for improving prediction accuracy and generating physically consistent outcomes. Each numerical analysis section is presented with case studies that involve conventional or additive manufacturing applications. Based on various case studies carried out, it can be concluded that advanced numerical modeling methods are essential to be incorporated in manufacturing applications to gain advantages in the era of Industry 4.0 and Industry 5.0. Although the case study for the advanced numerical modeling proposed in this dissertation is only presented in manufacturing-related applications, the methods presented in this dissertation is not exhaustive to manufacturing application and can also be expanded to other data-driven engineering and system applications.