Measuring the shape, motion and physical properties of os- cillating fluids is critical for understanding the physics of fluidic systems, as well as optimizing and controlling such systems in real time. Conventional surface measurement techniques such as profile analysis or stereo reconstruction are not effective for mon- itoring fluids in industrial processes due to the presence of oc- cluding structures, extreme heat, and complex light interactions at the fluid surface. We propose a video-based method comprising forward and inverse transforms. The forward transform employs a physics-based fluid surface model combined with a ray-traced renderer to map shape and motion parameters to synthetic video frames. The inverse transform uses machine learning models to recover surface parameters from video. The inverse models are trained on synthetic data generated by the forward transform. We illustrate the method on an industrial 3D printer for which we recover the motion and surface of a molten aluminum alloy os- cillating inside a microscopic nozzle. The inverse transform is ill-posed, but can be regularized. We show that surface properties can be reliably inferred with either a suitably regularized non- parametric k-nearest neighbor regressor or a deep convolutional network whose results are less stable but faster to compute.