Polytopic Autoencoders with Smooth Clustering for Reduced-order
  Modelling of Flows
Jan Heiland Yongho Kim
With the advancement of neural networks, there has been a notable increase,
both in terms of quantity and variety, in research publications concerning the
application of autoencoders to reduced-order models. We propose a polytopic
autoencoder architecture that includes a lightweight nonlinear encoder, a
convex combination decoder, and a smooth clustering network. Supported by
several proofs, the model architecture ensures that all reconstructed states
lie within a polytope, accompanied by a metric indicating the quality of the
constructed polytopes, referred to as polytope error. Additionally, it offers a
minimal number of convex coordinates for polytopic linear-parameter varying
systems while achieving acceptable reconstruction errors compared to proper
orthogonal decomposition (POD). To validate our proposed model, we conduct
simulations involving two flow scenarios with the incompressible Navier-Stokes
equation. Numerical results demonstrate the guaranteed properties of the model,
low reconstruction errors compared to POD, and the improvement in error using a
clustering network.