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We propose a new neural network, called isomorphic mesh generator (iMG), which generates isomorphic meshes from point clouds containing noise and missing parts. Isomorphic meshes of arbitrary objects have a unified mesh structure even though the objects belong to different classes. This unified representation enables surface models to be handled by DNNs. Moreover, the unified mesh structure of isomorphic meshes enables the same process to be applied to all isomorphic meshes; although in the case of general mesh models, we need to consider the processes depending on their mesh structures. Therefore, the use of isomorphic meshes leads to efficient memory usage and calculation time compared with general mesh models. As iMG is a data-free method, preparing any point clouds as training data in advance is unnecessary, except a point cloud of the target object used as the input data of iMG. Additionally, iMG outputs an isomorphic mesh obtained by mapping a reference mesh to a given input point cloud. To estimate the mapping function stably, we introduce a step-by-step mapping strategy. This strategy achieves a flexible deformation while maintaining the structure of the reference mesh. From simulation and experiments using a mobile phone, we confirmed that iMG can generate isomorphic meshes of given objects reliably even when the input point cloud includes noise and missing parts.