We propose an efficient data-driven enrichment approach to adaptively enhance the expressivity of subspaces for elastic deformations with novel collisions. In general, subspace integration method (also known as model reduction) for elastic deformations can greatly increase simulation speed. However, when the deformations are beyond the expressivity of subspaces such as novel external collisions, obvious artifacts will appear. First, we construct a positionbased database of subspaces through full-space collided simulations. We then select small sets of basis vectors to enrich existing subspaces for incoming collided deformations. We also demonstrate that cubature can easily be exploited by our subspace database, and we propose a novel post-processing scheme for refining the cubature weights for more accurate and faster deformations. Our method can achieve well approximated full-space deformations when novel collisions occur. From our experiment results, we further show that our method is applicable to large deformations and large-steps in real-time.