Compact photonic elements that control both the diffraction and interference of light offer superior performance at ultra-compact dimensions. Unlike conventional optical structures, these diffractive optical elements can provide simultaneous control of spectral and spatial profiles of light. However, the inverse design of such a diffractive optical element is time-consuming with current algorithms, and the designs generally lack experimental validation. Here, we develop a neural network model to experimentally design and validate SpliCons; a special type of diffractive optical element that can achieve spectral splitting and simultaneous concentration of broadband light. We use neural networks to exploit nonlinear operations that result from wavefront reconstruction through a phase plate. Our results show that the neural network model yields enhanced spectral splitting performance for phase plates with quantitative assessment compared to phase plates that are optimized via the local search optimization algorithm. The capabilities of the phase plates optimized via the neural network are experimentally validated by comparing the intensity distribution at the output plane. Once the neural networks are trained, we manage to design SpliCons with 96.6% +/- 2.3% accuracy within 2 s, which is orders of magnitude faster than iterative search algorithms. We openly share the fast and efficient framework that we develop in order to contribute to the design and implementation of diffractive optical elements that can lead to transformative effects in microscopy, spectroscopy, and solar energy applications.