  The Kolmogorov $n$-width of the solution manifolds of transport-dominated
problems can decay slowly. As a result, it can be challenging to design
efficient and accurate reduced order models (ROMs) for such problems. To
address this issue, we propose a new learning-based projection method to
construct nonlinear adaptive ROMs for transport problems. The construction
follows the offline-online decomposition. In the offline stage, we train a
neural network to construct adaptive reduced basis dependent on time and model
parameters. In the online stage, we project the solution to the learned reduced
manifold. Inheriting the merits from both deep learning and the projection
method, the proposed method is more efficient than the conventional linear
projection-based methods, and may reduce the generalization error of a solely
learning-based ROM. Unlike some learning-based projection methods, the proposed
method does not need to take derivatives of the neural network in the online
stage.
