Peridynamics is a nonlocal version of continuum mechanics theory able to incorporate singularities since it does not take into account spatial partial derivatives. As a consequence, it assumes long-range interactions among material particles and is able to describe the formation and the evolution of fractures. The discretization of such nonlocal model requires the use of raffinate numerical tools for approximating the solutions to the model. Due to the presence of a convolution product in the definition of the nonlocal operator, we propose a spectral collocation method based on the implementation of Fourier and Chebyshev polynomials to discretize the model. The choice can benefit of the FFT algorithm and allow us to deal efficiently with the imposition of non-periodic boundary conditions by a volume penalization technique. We prove the convergence of such methods in the framework of fractional Sobolev space and discuss numerically the stability of the scheme. Additionally, we investigate the qualitative aspects of the convolution kernel and of the nonlocality parameters by solving an inverse peridynamic problem by using a Physics-Informed Neural Network activated by suitable Radial Basis functions.