A novel neural network called the isomorphic mesh generator (iMG) is proposed to generate isomorphic meshes from point clouds containing noise and missing parts. Isomorphic meshes of arbitrary objects exhibit a unified mesh structure, despite objects belonging to different classes. This unified representation enables various modern deep neural networks (DNNs) to easily handle surface models without requiring additional pre-processing. Additionally, the unified mesh structure of isomorphic meshes enables the application of the same process to all isomorphic meshes, unlike general mesh models, where processes need to be tailored depending on their mesh structures. Therefore, the use of isomorphic meshes can ensure efficient memory usage and reduce calculation time. Apart from the point cloud of the target object used as input for the iMG, point clouds and mesh models need not be prepared in advance as training data because the iMG is a data-free method. Furthermore, the iMG outputs an isomorphic mesh obtained by mapping a reference mesh to a given input point cloud. To stably estimate the mapping function, a step-by-step mapping strategy is introduced. This strategy enables flexible deformation while simultaneously maintaining the structure of the reference mesh. Simulations and experiments conducted using a mobile phone have confirmed that the iMG reliably generates isomorphic meshes of given objects, even when the input point cloud includes noise and missing parts.