Nowadays, artificial intelligence systems seem to be urgent and breaking-through tools for efficient traffic management in modern urban road networks. However, these tools are highly sensitive to accurate travel demand data when predicting traffic congestion. Despite the fact that researchers have already developed numerous approaches to dealing with travel demand estimation, in many cases there is still a lack of ability to achieve the required accuracy level. The present paper focuses on the travel demand search task, formulated as an inverse traffic assignment problem. We developed multi-output regression models to estimate travel demand values for a road network with multiple origin-destination pairs. We used solutions to the non-linear optimization problem of equilibrium traffic assignment under different demand patterns to generate data for training and test sets. Our computational study gave several practical recommendations on the best scenarios for the implementation of such a tool as a multi-output regression for travel demand estimation.