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At high redshift, due to both observational limitations and the variety of galaxy morphologies in the early universe, measuring galaxy structure can be challenging. Non-parametric measurements such as the CAS system have thus become an important tool due to both their model-independent nature and their utility as a straightforward computational process. 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 able to reproduce standard measures of non-parametric morphologies and shows the potential of employing neural networks to provide 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.