  Classical model reduction techniques project the governing equations onto a
linear subspace of the original state space. More recent data-driven techniques
use neural networks to enable nonlinear projections. Whilst those often enable
stronger compression, they may have redundant parameters and lead to suboptimal
latent dimensionality. To overcome these, we propose a multistep algorithm that
induces sparsity in the encoder-decoder networks for effective reduction in the
number of parameters and additional compression of the latent space. This
algorithm starts with sparsely initialized a network and training it using
linearized Bregman iterations. These iterations have been very successful in
computer vision and compressed sensing tasks, but have not yet been used for
reduced-order modelling. After the training, we further compress the latent
space dimensionality by using a form of proper orthogonal decomposition. Last,
we use a bias propagation technique to change the induced sparsity into an
effective reduction of parameters. We apply this algorithm to three
representative PDE models: 1D diffusion, 1D advection, and 2D
reaction-diffusion. Compared to conventional training methods like Adam, the
proposed method achieves similar accuracy with 30% less parameters and a
significantly smaller latent space.
