Particle accelerators are enabling tools for scientific exploration and discovery in various disciplines. However, finding optimised operation points for these complex machines is a challenging task due to the large number of parameters involved and the underlying non-linear dynamics. Here, we introduce two families of data-driven surrogate models, based on deep and invertible neural networks, that can replace the expensive physics computer models. These models are employed in multi-objective optimisations to find Pareto optimal operation points for two fundamentally different types of particle accelerators. Our approach reduces the time-to-solution for a multi-objective accelerator optimisation up to a factor of 640 and the computational cost up to 98%. The framework established here should pave the way for future online and real-time multi-objective optimisation of particle accelerators.