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Multimodal deep learning has been used to predict clinical endpoints and diagnoses from clinical routine data. However, these models suffer from scaling issues: they have to learn pairwise interactions between each piece of information in each data type, thereby escalating model complexity beyond manageable scales. This has so far precluded a widespread use of multimodal deep learning. Here, we present a new technical approach of "learnable synergies", in which the model only selects relevant interactions between data modalities and keeps an "internal memory" of relevant data. Our approach is easily scalable and naturally adapts to multimodal data inputs from clinical routine. We demonstrate this approach on three large multimodal datasets from radiology and ophthalmology and show that it outperforms state-of-the-art models in clinically relevant diagnosis tasks. Our new approach is transferable and will allow the application of multimodal deep learning to a broad set of clinically relevant problems.