Accurate compensation of brain shift is critical for maintaining the reliability of neuronavigation during neurosurgery. While keypoint-based registration methods offer robustness to large deformations and topological changes, they typically rely on simple geometric interpolators that ignore tissue biomechanics to create dense displacement fields. In this work, we propose a novel deep learning framework that estimates dense, physically plausible brain deformations from sparse matched keypoints. We first generate a large dataset of synthetic brain deformations using biomechanical simulations. Then, a residual 3D U-Net is trained to refine standard interpolation estimates into biomechanically guided deformations. Experiments on a large set of simulated displacement fields demonstrate that our method significantly outperforms classical interpolators, reducing by half the mean square error while introducing negligible computational overhead at inference time. Code available at: \href{https://github.com/tiago-assis/Deep-Biomechanical-Interpolator}{https://github.com/tiago-assis/Deep-Biomechanical-Interpolator}.