Model Order Reduction (MOR) methods have been widely used to reduce the computational cost of studies involving parametrized Partial Differential Equations (PDEs). However, classical approaches are based on the assumption that the solution manifold can be well approximated by a low dimensional linear subspace. This assumption is not valid for problems exhibiting slowly decaying Kolmogorov \textit{n}-width, such as advection-dominated problems or wave propagation problems [1][2][3]. In this talk, we present an efficient data-weak non-linear interpolation technique based on a parametrized mapping obtained by solving a penalized optimization problem in bounded domains. A registration problem is solved so that a Lagrangian approach can be used to improve the representation of the solution manifold. A mapping is defined using FreeForm Deformation (FFD) and is parametrized by control points. The displacement of control points is tuned via the minimization of an objective function. Implementation using the JAX library allows the use of a fast L-BFGS solver. We present results concerning the non linear interpolation of stationary solution fields. Subsequently, the methodology is applied to unsteady problems combining the registration approach and Dynamic Mode Decomposition (DMD) [4]. [1] Iollo A., Lombardi D., Advection Modes by Optimal Mass Transfer., [Research Report], pp.33., 2012 [2] Taddei T., A registration method for model order reduction: data compression and geometry reduction., SIAM Journal on Scientific Computing (SISC), 2020. [3] Cucchiara S., Iollo A., Taddei T, Telib H., Model order reduction by convex displacement interpolation., Journal of Computational Physics, Vol. 514, pp. 113230, 2024 [4] Schmid Peter J., Dynamic mode decomposition of numerical and experimental data., Journal of Fluid Mechanics, Vol. 656, pp. 5-28, 2010