Knowledge Integration in Differentiable Models: A Comparative Study of Data-Driven, Soft-Constrained, and Hard-Constrained Paradigms for Identification and Control of the Single Machine Infinite Bus System

Integrating domain knowledge into neural networks is a central challenge in scientific machine learning. Three paradigms have emerged -- data-driven (Neural Ordinary Differential Equations, NODEs), soft-constrained (Physics-Informed Neural Networks, PINNs), and hard-constrained (Differentiable Programming, DP) -- each encoding physical knowledge at different levels of structural commitment. However, how these strategies impact not only predictive accuracy but also downstream tasks such as control synthesis remains insufficiently understood. This paper presents a comparative study of NODEs, PINNs, and DP for dynamical system modeling, using the Single Machine Infinite Bus power system as a benchmark. We evaluate these paradigms across three tasks: trajectory prediction, parameter identification, and Linear Quadratic Regulator control synthesis. Our results yield three principal findings. First, knowledge representation determines generalization: NODE, which learns the system operator, enables robust extrapolation, whereas PINN, which approximates a solution map, restricts generalization to the training horizon. Second, hard-constrained formulations (DP) reduce learning to a low-dimensional physical parameter space, achieving faster and more reliable convergence than soft-constrained approaches. Third, knowledge fidelity propagates to control performance: DP produces controllers that closely match those obtained from true system parameters, while NODE provides a viable data-driven alternative by recovering control-relevant Jacobians with 3-4% relative error and yielding LQR gains within 0.36% of the ground truth. Based on these findings, we propose a practical decision framework for selecting knowledge integration strategies in neural modeling of dynamical systems.