A Neural Galerkin Solver for Accurate Surface Reconstruction

To reconstruct meshes from the widely-available 3D point cloud data, implicit shape representation is among the primary choices as an intermediate form due to its superior representation power and robustness in topological optimizations. Although different parameterizations of the implicit fields have been explored to model the underlying geometry, there is no explicit mechanism to ensure the fitting tightness of the surface to the input. We present in response, NeuralGalerkin, a neural Galerkin-method-based solver designed for reconstructing highly-accurate surfaces from the input point clouds. NeuralGalerkin internally discretizes the target implicit field as a linear combination of a set of spatially-varying basis functions inferred by an adaptive sparse convolution neural network. It then solves differentiably for a variational problem that incorporates both positional and normal constraints from the data in closed form within a single forward pass, highly respecting the raw input points. The reconstructed surface extracted from the implicit interpolants is hence very accurate and incorporates useful inductive biases benefiting from the training data. Extensive evaluations on various datasets demonstrate our method's promising reconstruction performance and scalability.