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.

翻译：随着神经网络的进步，将自编码器应用于降阶模型的研究论文在数量与多样性上均显著增长。我们提出一种多面体自编码器架构，包含轻量非线性编码器、凸组合解码器及光滑聚类网络。多项数学证明支撑了该模型架构，确保所有重构状态均位于多面体内部，并附带衡量所构建多面体质量的指标——即多面体误差。此外，针对多面体线性参数变化系统，该模型在实现可接受重构误差（相较于本征正交分解法（POD））的同时，仅需最少数量的凸坐标。为验证所提模型，我们利用不可压缩纳维-斯托克斯方程模拟两种流动场景。数值结果证明了模型的保证性质、相较于POD的低重构误差，以及采用聚类网络后误差的改善。