SharpNet: Enhancing MLPs to Represent Functions with Controlled Non-differentiability

Multi-layer perceptrons (MLPs) are a standard tool for learning and function approximation, but they inherently yield outputs that are globally smooth. As a result, they struggle to represent functions that are continuous yet deliberately non-differentiable (i.e., with prescribed $C^0$ sharp features) without relying on ad hoc post-processing. We present SharpNet, a modified MLP architecture capable of encoding functions with user-defined sharp features by enriching the network with an auxiliary feature function, which is defined as the solution to a Poisson equation with jump Neumann boundary conditions. It is evaluated via an efficient local integral that is fully differentiable with respect to the feature locations, enabling our method to jointly optimize both the feature locations and the MLP parameters to recover the target functions/models. The $C^0$-continuity of SharpNet is precisely controllable, ensuring $C^0$-continuity at the feature locations and smoothness elsewhere. We validate SharpNet on 2D problems and 3D CAD model reconstruction, and compare it against several state-of-the-art baselines. In both types of tasks, SharpNet accurately recovers sharp edges and corners while maintaining smooth behavior away from those features, whereas existing methods tend to smooth out gradient discontinuities. Both qualitative and quantitative evaluations highlight the benefits of our approach.