Topological deep learning

Topological deep learning (TDL)^{[1][2][3][4][5][6]} is a research field that extends deep learning to handle complex, non-Euclidean data structures. Traditional deep learning models, such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs), excel in processing data on regular grids and sequences. However, scientific and real-world data often exhibit more intricate data domains encountered in scientific computations , including point clouds, meshes, time series, scalar fields graphs, or general topological spaces like simplicial complexes and CW complexes.^[7] TDL addresses this by incorporating topological concepts to process data with higher-order relationships, such as interactions among multiple entities and complex hierarchies. This approach leverages structures like simplicial complexes and hypergraphs to capture global dependencies and qualitative spatial properties, offering a more nuanced representation of data. TDL also encompasses methods from computational and algebraic topology that permit studying properties of neural networks and their training process, such as their predictive performance or generalization properties.^{[8][9][10][11][12][13][14]} The mathematical foundations of TDL are algebraic topology, differential topology, and geometric topology. Therefore, TDL can be generalized for data on differentiable manifolds, knots, links, tangles, curves, etc.