Polytopic autoencoders with smooth clustering for reduced-order modeling of flows - Heiland, Jan; Kim, Yongho (Journal of computational physics, 2025)
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. •Autoencoders for nonlinear model reduction of PDEs based on smooth clustering and reconstruction in a polytope.•A realizable framework to integrate the training of smooth clustering algorithms with deep neural network optimization.•A proof of concept and comparison to standard approaches for two numerical flow simulations.•The highlighting of possible applications in low-dimensional LPV approximations of nonlinear systems.