In this study, we applied physics-informed machine learning (ML) to model slot-die coating flows. Slot-die coating is a versatile method for manufacturing various functional films. Understanding the physics of the coating fluid flow is critical to the safe process operation and production of defect-free film products. Analyzing slot-die coating flows is not a trivial task due to the presence of free surfaces. Previous studies have used the finite element method or finite volume method to tackle the problem. However, these methods require a mesh generation procedure, and the obtained solution may depend on the choice of the mesh. Also, formulating inverse problems to obtain unknown system parameters is not straightforward under these frameworks. We attempt to address these difficulties by using the physics-informed ML. The objective of the proposed method is to find the optimal parameters of the neural net that serves as a surrogate for the velocity and pressure fields of the coating flow. To handle the free surface problem, we use a specialized network structure inspired by the iso-parametric mapping used in the finite element method. We demonstrate how the mapping is learned by the network, and the location of the free surface is determined as the optimization proceeds. The optimal parameters of the network are obtained by optimizing the loss function that consists of the loss terms associated with the governing equation and the boundary conditions. Each term is multiplied by a weight that determines the relative importance during the optimization, and choosing the right values for the weights is a critical factor to successful optimization. In this study, we adopted a dynamic weight-adjusting scheme following the previous study. The capability of the proposed method for solving inverse problems is also demonstrated by obtaining the unknown viscosity of the coating fluid, given the coating flow data.