We investigate the ill-posedness of the inverse biosensor problem when the biosensor signals are corrupted by noise. To solve the problem, we employ feed-forward and convolutional neural networks. Computational experiments were performed with different levels of additive and multiplicative noises for the batch and flow injection analysis modes of the biosensor. Obtained results show that the largest errors of recovered concentrations are located on the edges of the training domain. We have found that the inverse problem is less ill-posed in the flow injection analysis mode and concentrations can be reliably recovered for higher levels of noise compared to the batch mode. This finding is confirmed by the application of the DIRECT global optimization method to the considered inverse biosensor problem.