Over the past few years, machine learning (ML) has gained increasing importance in engineering disciplines. In computational mechanics, ML techniques are often employed to construct surrogate models ensuring both physical consistency and fast evaluation. These models are particularly relevant for multi-query tasks such as optimization, uncertainty quantification, and inverse analysis, which are often computationally intractable when relying solely on conventional high-fidelity simulations. Beyond computational efficiency, real-time-capable surrogates play a crucial role in the development of digital twins, for example, enabling continuous monitoring and predictive maintenance of engineering structures. Despite promising results, classical data-driven approaches remain fundamentally limited by their reliance on large, high-quality datasets. In many practical scenarios, generating such data via finite element method (FEM) simulations is prohibitively expensive, while experimental data acquisition may be infeasible. Physics-based ML addresses this challenge by reducing the dependence on training data through the incorporation of physics-based regularization into the learning process. Building on previous work, this contribution aims to systematically investigate physics-based ML approaches for computational (contact) mechanics, ranging from classical physics-informed neural networks (PINNs) to energy-based and variational formulations. Primarily, we focus on the recently proposed energy-based PINN (EPINN) framework for quasi-static nonlinear mechanical problems involving nonsmooth contact phenomena. While EPINNs have been shown to solve such problems in a single step without load stepping, as commonly used in traditional numerical methods, we target the recovery of the complete nonlinear load path rather than only the final equilibrium configuration. Therefore, we propose an efficient incremental solution strategy that decomposes the full nonlinear optimization problem into a sequence of subproblems with reduced nonlinearity. Numerical results are validated with FEM reference solutions, demonstrating both accuracy and comparable performance to established numerical methods. Since EPINNs are restricted to conservative systems without dissipative effects such as damping, friction, or plasticity, we also briefly discuss ongoing work on variational PINNs and share preliminary findings that provide a framework for more general mechanical settings.