  Discretization invariant learning aims at learning in the
infinite-dimensional function spaces with the capacity to process heterogeneous
discrete representations of functions as inputs and/or outputs of a learning
model. This paper proposes a novel deep learning framework based on integral
autoencoders (IAE-Net) for discretization invariant learning. The basic
building block of IAE-Net consists of an encoder and a decoder as integral
transforms with data-driven kernels, and a fully connected neural network
between the encoder and decoder. This basic building block is applied in
parallel in a wide multi-channel structure, which are repeatedly composed to
form a deep and densely connected neural network with skip connections as
IAE-Net. IAE-Net is trained with randomized data augmentation that generates
training data with heterogeneous structures to facilitate the performance of
discretization invariant learning. The proposed IAE-Net is tested with various
applications in predictive data science, solving forward and inverse problems
in scientific computing, and signal/image processing. Compared with
alternatives in the literature, IAE-Net achieves state-of-the-art performance
in existing applications and creates a wide range of new applications.
