# Parametric Taylor series based latent dynamics identification neural networks

arXiv: https://arxiv.org/abs/2410.04193 · HTML: https://mlsanity.com/paper/2410.04193/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 59 references that are in the map.
- First posted on arXiv 5 Oct 2024.

## Builds on (most cited first, 13 of 13)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Resurrecting Recurrent Neural Networks for Long Sequences](https://mlsanity.com/paper/2303.06349/) (2023)
- [Dynamic mode decomposition with control](https://mlsanity.com/paper/1409.6358/) (2014)
- [β-Variational autoencoders and transformers for reduced-order modelling of fluid flows](https://mlsanity.com/paper/2304.03571/) (2023)
- [Machine Learning for the Geosciences: Challenges and Opportunities](https://mlsanity.com/paper/1711.04708/) (2017)
- [Physics-Informed Multi-LSTM Networks for Metamodeling of Nonlinear Structures](https://mlsanity.com/paper/2002.10253/) (2020)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [GPLaSDI: Gaussian Process-based Interpretable Latent Space Dynamics Identification through Deep Autoencoder](https://mlsanity.com/paper/2308.05882/) (2023)
- [Grasping Extreme Aerodynamics on a Low-Dimensional Manifold](https://mlsanity.com/paper/2305.08024/) (2023)
- [Physics Guided Machine Learning for Variational Multiscale Reduced Order Modeling](https://mlsanity.com/paper/2205.12419/) (2022)
- [Reduced Simulations for High-Energy Physics, a Middle Ground for Data-Driven Physics Research](https://mlsanity.com/paper/2309.03780/) (2023)

## Built on by (most cited first, 1 of 1)

- [Learning PDE solution operators with variable initial conditions via Latent Dynamics Networks](https://mlsanity.com/paper/2610.08475/) (2026)

## Abstract (arXiv)

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 L2 error below 2% compared to high-fidelity models.