Beyond the Kolmogorov barrier: a learnable weighted hybrid autoencoder for model order reduction

Representation learning for high-dimensional, complex physical systems aims to identify a low-dimensional intrinsic latent space, which is crucial for reduced-order modelling and modal analysis. To overcome the well-known Kolmogorov barrier, deep autoencoders (AEs) have been introduced in recent years, but they often suffer from poor convergence behaviour as the rank of the latent space increases. To address this issue, we propose the learnable weighted hybrid AE , a hybrid approach that combines the strengths of singular value decomposition (SVD) with deep AEs through a learnable weighted framework. We find that the introduction of learnable weighting parameters is essential—without them, the resulting model would either collapse into a standard proper orthogonal decomposition or fail to exhibit the desired convergence behaviour. Interestingly, we empirically find that our trained model has a sharpness thousands of times smaller compared to other models, which in turn enhances its robustness to input noise. Our experiments on classical chaotic PDE systems, including the one-dimensional Kuramoto–Sivashinsky and forced isotropic turbulence datasets, demonstrate that our approach significantly improves generalization performance compared to several competing frameworks. In addition, when combined with time series modelling techniques (e.g. Koopman operator, long-short-term memory), the proposed technique offers significant improvements for surrogate modelling of high-dimensional multiscale PDE systems.