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Aquifer Thermal Energy Storage (ATES) offers sustainable, low carbon heating and cooling to the built environment. Optimising the design and operation of ATES installations requires numerical simulation of groundwater flow and heat transport in heterogeneous aquifers. These simulations are typically computationally expensive: high spatial resolution is required to accurately resolve pressure, flow and temperature fields; moreover, high temporal resolution may be necessary to control numerical diffusion and/or resolve frequent changes in injection flowrate and temperature. Simulations of (1) systems that utilize multiple well doublets, or (2) capture interactions between neighbouring systems, are particularly challenging. Multiple simulations may be required to quantify the impact of uncertain aquifer heterogeneity. Yet the time available for aquifer modelling in many commercial projects is very limited. Rapid but accurate approaches to simulate subsurface flow and heat transport in ATES and other shallow geothermal deployments are urgently required.Machine Learning (ML) offers a rapid alternative to conventional numerical simulation of complex subsurface flow and transport processes. Here we introduce the use of a transformer-based ML approach, on a purely data-driven basis, to significantly increase simulation efficiency whilst retaining its accuracy. The ML proxy is trained using ATES simulation outputs from the open-source Imperial College Finite Element Reservoir Simulator (IC-FERST), that uses dynamic mesh optimization to provide high solution accuracy at lower computational cost. The practical consequence here is that the mesh changes across solution snapshots recorded at successive time steps used for training. Conventional Convolutional Neural Network (CNN)-based models require a fixed mesh. Here, to provide a fast proxy, we implement atransformer-based model working on adaptive unstructured mesh, enabling a stronger capability in capturing long range changes in predictions. The model can take in the initial state of the reservoir in arbitrary input mesh, perform one-step prediction in non-physical latent space and recover the latent representation of the prediction back to physical space on any given query mesh, allowing the integration of adaptive mesh refinement adjusted to fit the predicted solution on unstructured graphs.Our results suggest a promising approach to rapid simulation of ATES, in which simulation time can be reduced significantly with a speed-up factor of over 6600 times.