In this work, we developed a robust radial point interpolation approach enhanced with neural network solvers (RPIM-NNS) to deal with highly nonlinear solid mechanics problems. In the proposed method, an energy loss is formulated under the RPIM and solved by gradient descendant algorithms. By using the radial basis functions (RBFs) for constructing displacements, the proposed method offers great robustness for uneven node distributions. (2) Nodes are placed beyond the domain boundary and thereby improve the computational accuracy for both Dirichlet and Neumann boundary conditions. (3) The proposed method incorporates strain energy in an integral manner inside the loss function to guarantee solution stability. (4) The well-developed gradient descent techniques in machine learning are employed to solve solid mechanics problems, resulting in robustness and simplicity in managing material and geometrical nonlinearities. (5) The proposed method is compatible with parallel computing techniques. The stability and robustness of the RPIM-NNS are shown by coping with challenging nonlinear problems, including Cook's membrane and 3D twisting rubber problems. Consequently, the proposed RPIM-NN is effective and shows great potential for nonlinear mechanics modelling.