At high redshift, observational limitations reduce the effectiveness of visual morphological classifications. Quantitative measures of galaxy structure provide an alternative, with model-independent approaches being preferred, due to the variety of galaxy morphologies in the early universe. Non-parametric measurements, such as the CAS system, have therefore become an important tool.Recently, convolutional neural networks (CNNs) have been shown to be adept at image analysis, and are beginning to supersede traditional measurements of visual morphology and model-based structural parameters. In this work, we take a further step by extending CNNs to measure well known non-parametric structural quantities: concentration ($C$) and asymmetry ($A$). We train CNNs to predict $C$ and $A$ from individual images of $\sim 150,000$ galaxies at $0 < z < 7$ in the CANDELS fields, using Bayesian hyperparameter optimisation to select suitable network architectures. Our resulting networks accurately reproduce measurements compared with standard algorithms. Furthermore, using simulated images, we show that our networks are more stable than the standard algorithms at low signal-to-noise. While both approaches suffer from similar systematic biases with redshift, these remain small out to $z \sim 7$. Once trained, measurements with our networks are $> 10^3$ times faster than previous methods. Our approach is thus not only able to reproduce standard measurements of non-parametric quantities, but gives superior results in substantially less time. This will be vital for making best use of the large and complex datasets provided by upcoming galaxy surveys, such as Euclid and Rubin-LSST.