Physics-informed learning has emerged as a powerful paradigm for system identification, enabling data-driven models to capture complex nonlinear dynamics while respecting underlying physical structure. In this work, we build upon our recent work in learning nonlinear port-Hamiltonian (pH) systems from input–state–output data \cite{Cherifi2025} and investigate how model performance scales with available learning resources. We first present a unified framework for identifying nonlinear pH systems using neural networks as structured function approximators. By embedding the port-Hamiltonian formalism structure into the learning architecture, the proposed approach preserves passivity and energy-based properties while leveraging the universal approximation capabilities of modern neural networks. In addition, we have shown that incorporating prior knowledge about the underlying physical structure into the learning process constrains the hypothesis space, improves data efficiency, and yields models that are more accurate, physically consistent and reliable for long-term prediction than purely data-driven approaches. Building on this foundation, we then study the scalability of such physics-informed models through the lens of neural scaling laws \cite{Roschkowski2025}. While machine learning models in natural language processing \cite{Hestness2017} are known to improve with increased data, model size, and computational budget , the quantitative relationship between these resources and identification accuracy remains poorly understood in dynamical systems settings. We empirically verify neural scaling laws for system identification across a range of architectures, including standard input-affine models and physics-informed port-Hamiltonian representations. By training thousands of models across multiple system architectures and evaluating their performance using standardized metrics, we empirically determine scaling relationships that quantify how improvements in resources translate to gains in accuracy. The resulting scaling laws provide practical guidance for forecasting achievable accuracy, selecting model architectures, and designing data acquisition strategies. Overall, this work connects structure-preserving system identification with neural scaling theory, offering a new perspective into the efficient and reliable learning of nonlinear dynamical systems.