Labyrinth weirs provide an economic option for flow control structures in a variety of applications, including as spillways at dams. The cycles of labyrinth weirs are typically placed in a linear configuration. However, numerous projects place labyrinth cycles along an arc to take advantage of reservoir conditions and dam alignment, and to reduce construction costs such as narrowing the spillway chute. Practitioners must optimize more than 10 geometric variables when developing a head–discharge relationship. This is typically done using the following tools: empirical relationships, numerical modeling, and physical modeling. This study applied a new tool, machine learning, to the analysis of the geometrically complex arced labyrinth weirs. In this work, both neural networks (NN) and random forests (RF) were employed to estimate the discharge coefficient for this specific type of weir with the results of physical modeling experiments used for training. Machine learning results are critiqued in terms of accuracy, robustness, interpolation, applicability, and new insights into the hydraulic performance of arced labyrinth weirs. Results demonstrate that NN and RF algorithms can be used as a unique expression for curve fitting, although neural networks outperformed random forest when interpolating among the tested geometries.