The number and complexity of artificial intelligence (AI) applications is growing relentlessly. As a result, even with the many algorithmic and mathematical advances experienced over past decades as well as the impressive energy efficiency and computational capacity of current hardware accelerators, training the most powerful and popular deep neural networks comes at very high economic and environmental costs. Recognising that additional optimisations of conventional neural network training is very difficult, this work takes a radically different approach by proposing GreenLightningAI, a new AI system design consisting of a linear model that is capable of emulating the behaviour of deep neural networks by subsetting the model for each particular sample. The new AI system stores the information required to select the system subset for a given sample (referred to as structural information) separately from the linear model parameters (referred to as quantitative knowledge). In this paper we present a proof of concept, showing that the structural information stabilises far earlier than the quantitative knowledge. Additionally, we show experimentally that the structural information can be kept unmodified when re-training the AI system with new samples while still achieving a validation accuracy similar to that obtained when re-training a neural network with similar size. Since the proposed AI system is based on a linear model, multiple copies of the model, trained with different datasets, can be easily combined. This enables faster and greener (re)-training algorithms, including incremental re-training and federated incremental re-training.