This paper reviews some of the challenges posed by the huge growth of experimental data generated by the new generation of large-scale experiments at UK national facilities at the Rutherford Appleton Laboratory site at Harwell near Oxford. Such "Big Scientific Data" comes from the Diamond Light Source and Electron Microscopy Facilities, the ISIS Neutron and Muon Facility, and the UK's Central Laser Facility. Increasingly, scientists are now needing to use advanced machine learning and other AI technologies both to automate parts of the data pipeline and also to help find new scientific discoveries in the analysis of their data. For commercially important applications, such as object recognition, natural language processing and automatic translation, deep learning has made dramatic breakthroughs. Google's DeepMind has now also used deep learning technology to develop their AlphaFold tool to make predictions for protein folding. Remarkably, they have been able to achieve some spectacular results for this specific scientific problem. Can deep learning be similarly transformative for other scientific problems? After a brief review of some initial applications of machine learning at the Rutherford Appleton Laboratory, we focus on challenges and opportunities for AI in advancing materials science. Finally, we discuss the importance of developing some realistic machine learning benchmarks using Big Scientific Data coming from a number of different scientific domains. We conclude with some initial examples of our "SciML" benchmark suite and of the research challenges these benchmarks will enable.