Data denoising and derivative estimation for data-driven modeling of nonlinear dynamical systems

Data-driven modeling of nonlinear dynamical systems is often hampered by measurement noise. We propose a denoising framework, called RKSDS-INR (Runge-Kutta and Second-Order Derivative Smoothness Based Implicit Neural Representation), that represents the state trajectory with an implicit neural representation (INR) fitted directly to noisy observations. Runge-Kutta integration and second-order derivative smoothness are imposed as constraints to ensure that the reconstructed state is a trajectory of a dynamical system that remains close to the original data. The trained INR yields a clean, continuous trajectory and provides accurate first-order derivatives via automatic differentiation. These denoised states and derivatives are then supplied to sparse identification of nonlinear dynamics to recover the governing equations. The experiments demonstrate effective noise suppression, precise derivative estimation, and reliable system identification.