This question comes after experimenting with learning the parameters of a nonlinear system $\dot{\mathbf{x}} = f(\mathbf{x}), f: \mathbb{R}^n \rightarrow \mathbb{R}^n$. If I take a system that has $f = \sum_{i=1}^k a_i\phi_i(\mathbf{x})$, where I know the $\phi_i$'s and try to learn the $a_i$'s from data (trajectories obtained from numerical integration of a known system or from measurements of a physical/biological system with well-studied dynamics) using nonlinear regression (neural network, trust-region methods, Levenberg-Marquadt, etc.) I cannot seem to exactly recover the true $a_i$'s. However, the learned $a_i$'s are doing a great job at predicting unseen initial conditions. From such numerical experiments, I'm reasoning that the solution to this nonlinear system is identical for two sets of parameters, the first being the set of parameters that was used for simulation, and the second being the set of parameters learned from the simulated data. If there is any error in my training data, it would come from the numerical integration. I wonder if numerical integration can bias the solution enough as to throw off the learning process.
My pointed question is this: Can we ever have that a nonlinear system gives identical solutions for two sets of parameters and when might this happen? Are there any known examples of this that are used for teaching/introduction to such behavior?