Accurate knowledge of the state variables in a dynamical system is critical for effective control, diagnosis, and supervision, especially when direct measurements of all states are infeasible. This paper presents a novel approach to designing software sensors for nonlinear dynamical systems expressed in their most general form. Unlike traditional model-based observers that rely on explicit transformations or linearization, the proposed framework integrates neural networks with adaptive Sliding Mode Control (SMC) to design a robust state observer under a less restrictive set of conditions. The learning process is driven by available sensor measurements, which are used to correct the observer's state estimate. The training methodology leverages the system's governing equations as a physics-based constraint, enabling observer synthesis without access to ground-truth state trajectories. By employing a time-varying gain matrix dynamically adjusted by the neural network, the observer adapts in real-time to system changes, ensuring robustness against noise, external disturbances, and variations in system dynamics. Furthermore, we provide sufficient conditions to guarantee estimation error convergence, establishing a theoretical foundation for the observer's reliability. The methodology's effectiveness is validated through simulations on challenging examples, including systems with non-differentiable dynamics and varying observability conditions. These examples, which are often problematic for conventional techniques, serve to demonstrate the robustness and broad applicability of our approach. The results show rapid convergence and high accuracy, underscoring the method's potential for addressing complex state estimation challenges in real-world applications.