  In this paper, we propose a unified framework for identifying interpretable
nonlinear dynamical models that preserve physical properties. The proposed
approach integrates physical principles with black-box basis functions to
compensate for unmodeled dynamics, ensuring accuracy over long prediction
horizons and computational efficiency. Additionally, we introduce penalty terms
to enforce physical consistency and stability during training. We provide a
comprehensive analysis of theoretical properties related to multi-step
nonlinear system identification, establishing bounds on parameter estimation
errors and conditions for gradient stability and sparsity recovery. The
proposed framework demonstrates significant potential for improving model
accuracy and reliability in various engineering applications, making a
substantial step towards the effective use of combined off-white and sparse
black models in system identification. The effectiveness of the proposed
approach is demonstrated on a nonlinear system identification benchmark.
