In this work we present a novel approach for computing correspondences between non-rigid objects, by exploiting a reduced representation of deformation fields. Different from existing works that represent deformation fields by training a generalpurpose neural network, we advocate for an approximation based on mesh-free methods. By letting the network learn deformation parameters at a sparse set of positions in space (nodes), we reconstruct the continuous deformation field in a closed-form with guaranteed smoothness. With this reduction in degrees of freedom, we show significant improvement in terms of data-efficiency thus enabling limited supervision. Furthermore, our approximation provides direct access to first-order derivatives of deformation fields, which facilitates enforcing desirable regularization effectively. Our resulting model has high expressive power and is able to capture complex deformations. We illustrate its effectiveness through stateof-the-art results across multiple deformable shape matching benchmarks. Our code and data are publicly available at: https://github.com/Sentient07/ DeformationBasis.