Abstract: Dynamic astrophysical phenomena are predominantly described by differential equations, yet our understanding of these systems is constrained by our incomplete grasp of non-linear physics and scarcity of of comprehensive datasets. As such, advancing techniques in solving non-linear inverse problems becomes pivotal to addressing numerous outstanding questions in the field. In particular, modeling hot galactic winds is difficult because of unknown structure for various physical terms, and the lack of \textit{any} kinematic observational data. Additionally, the flow equations contain singularities that lead to numerical instability, making parameter sweeps non-trivial. We leverage differentiable programming, which enables neural networks to be embedded as individual terms within the governing coupled ordinary differential equations (ODEs), and show that this method can adeptly discover hidden physics. We robustly discern the structure of a mass-loading function which captures the physical effects of cloud destruction and entrainment into the hot superwind. Within a supervised learning framework, we formulate our loss function anchored on the astrophysical entropy ($K \propto P/\rho^{5/3}$). Our results demonstrate the efficacy of this approach, even in the absence of kinematic data $v$. We then apply these models to real Chandra X-Ray observations of starburst galaxy M82, providing the first systematic description of mass-loading within the superwind. This work further highlights neural ODEs as a useful discovery tool with mechanistic interpretability in non-linear inverse problems. We make our code public at this GitHub repository.