<p>Despite the active application of neural network (NN) techniques to topology optimization, many studies focus on its advantages, so that in-depth studies on the limitations of NN applications are rare. This study presents a mesh-free compliant mechanism design approach utilizing a topology optimization algorithm that integrates NN representations of variables with the adjoint method and multi-design objectives along with structural compliance. Additionally, the limitations of the proposed method from the perspective of robustness were studied. The energy-based approach of physics-informed NNs (PINNs) is employed to train NNs, obtaining state and adjoint variables, ensuring to satisfy partial differential equation constraints and corresponding adjoint equations. A loss function is formulated based on the Lagrangian of the optimization problem to update density variables. This approach unifies the solution of governing equations, adjoint equations, sensitivity analysis, and variable updates into a single NN training process utilizing automatic differentiation, eliminating the need for prior knowledge of sensitivity formulations or the objective function. The independent representation of design, state, and adjoint variables through NNs facilitates the implementation of multi-objective topology optimization for compliant mechanisms, addressing both output displacement to maximize flexibility and minimize structural compliance. The effectiveness of the proposed method is demonstrated through two- and three-dimensional numerical examples, such as force inverter and gripper design. The limitations were also investigated through case studies on various hyperparameters influencing the optimization results.</p>