Parametric Taylor series based latent dynamics identification neural networks

Abstract

Numerical solving parameterised partial differential equations (P-PDEs) is highly practical yet computationally expensive, driving the development of reduced-order models (ROMs). Recently, methods that combine latent space identification techniques with deep learning algorithms (e.g., autoencoders) have shown great potential in describing the dynamical system in the lower dimensional latent space, for example, LaSDI, gLaSDI and GPLaSDI. In this paper, a new parametric latent identification of nonlinear dynamics neural networks, P-TLDINets, is introduced, which relies on a novel neural network structure based on Taylor series expansion and ResNets to learn the ODEs that govern the reduced space dynamics. During the training process, Taylor series-based Latent Dynamic Neural Networks (TLDNets) and identified equations are trained simultaneously to generate a smoother latent space. In order to facilitate the parameterised study, a $k$-nearest neighbours (KNN) method based on an inverse distance weighting (IDW) interpolation scheme is introduced to predict the identified ODE coefficients using local information. Compared to other latent dynamics identification methods based on autoencoders, P-TLDINets remain the interpretability of the model. Additionally, it circumvents the building of explicit autoencoders, avoids dependency on specific grids, and features a more lightweight structure, which is easy to train with high generalisation capability and accuracy. Also, it is capable of using different scales of meshes. P-TLDINets improve training speeds nearly hundred times compared to GPLaSDI and gLaSDI, maintaining an $L_2$ error below $2\%$ compared to high-fidelity models.