We introduce a general framework for minimizing energy functionals within parametric functional spaces, offering an alternative to traditional adaptive methods for solving PDEs with solutions that exhibit localized features, such as singularities, sharp fronts, and multi-scale phenomena. Our approach is based on parametric functional spaces formed by a collection of free patches obtained by tensor-product of one-dimensional patches. These patches can move freely across the domain, allowing for dynamic adaptation to the local behaviour of the solution. We specifically focus on free-knot B-splines and shallow neural networks, both of which naturally leverage the tensor-product structure to balance flexibility and computational efficiency. A key strength of our framework is its versatility, as it encompasses a broad range of functional spaces, including sparse grids. We present elements of the numerical analysis of our approach, highlighting its theoretical foundations and practical advantages over conventional techniques. Several representative numerical experiments are provided to demonstrate the versatility and effectiveness of our method.