This paper introduces an intrinsic dimension estimation autoencoder (IDEA) that identifies the underlying intrinsic dimension of diverse datasets with samples on linear or nonlinear manifolds. In addition to estimating the intrinsic dimension, IDEA can reconstruct the original dataset by projecting it onto a latent space constructed using a reweighted double CancelOut layer. A key contribution is the introduction of a projected reconstruction loss term that continuously evaluates the reconstruction quality when additional latent dimensions are removed, thereby guiding model learning. We first validate the robustness of IDEA by evaluating its performance on a set of theoretical benchmarks. These experiments allow us to test its reconstruction capabilities and compare its performance with state-of-the-art intrinsic dimension estimators. The benchmarks demonstrate the high accuracy and versatility of our approach. We then apply the model to data generated from a numerical solution of a vertically decomposed one-dimensional free surface flow following pointwise discretization of the vertical velocity profile in the horizontal, vertical, and time directions. After successfully estimating the intrinsic dimension of the dataset, IDEA reconstructs the original solution by operating directly within the projection space identified by the network.
