Understanding the physical mechanisms of the Madden-Julian Oscillation (MJO) and its evolution is a major concern within the climate community. Its main importance relies on its ability to act as a source of predictability within the intra-seasonal time-scale in tropical and extratropical regions, therefore filling the gap between weather and climate forecasts. However, most atmospheric general circulation models fail to correctly represent MJO’s evolution, and their prediction skills are still far from MJO’s theoretical predictability. In this work we infer low dimensional models of the MJO from data by applying a recently developed machine learning technique, the Sparse Identification of Non-linear Dynamics (SINDy). We use the daily-mean outgoing longwave radiation MJO index (OMI) as input data to infer bi-dimensional climatological models of the MJO, and analyse the inferred models during El Niño and La Niña years. This approach allows us to diagnose the MJO’s behaviour in OMI’s phase space. Our results show that MJO can be most frequently represented by a harmonic oscillator, which represents the MJO’s eastward propagation and characteristic period. Upon this basic oscillatory behaviour, we find that small non-linear corrections play a fundamental role in representing MJO’s non-uniform speed of propagation, explaining its acceleration over the Pacific Ocean region. Particularly, we find that MJO’s evolution is most frequently non-linear [linear] during El Niño [La Niña] years. Overall, our work shows that SINDy can robustly model MJO’s evolution as a linear oscillator with small non-linear corrections, contributing to understand the MJO’s dynamics and dependency on El Niño-Southern Oscillation.