A physically motivated method for surface reconstruction is proposed that can recover smooth surfaces from noisy and sparse data sets. No orientation information is required. By a new technique based on regularized-membrane potentials the input sample points are aggregated, leading to improved noise tolerability and outlier removal, without sacrificing much with respect to detail (feature) recovery. After aggregating the sample points on a volumetric grid, a novel, iterative algorithm is used to classify grid points as exterior or interior to the surface. This algorithm relies on intrinsic properties of the smooth scalar field on the grid which emerges after the aggregation step. Second, a mesh-smoothing paradigm based on a mass-spring system is introduced. By enhancing this system with a bending-energy minimizing term we ensure that the final triangulated surface is smoother than piecewise linear. In terms of speed and flexibility, the method compares favorably with respect to previous approaches. Most parts of the method are implemented on modern graphics processing units (GPUs). Results in a wide variety of settings are presented, ranging from surface reconstruction on noise-free point clouds to grayscale image segmentation.