# Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models

arXiv: https://arxiv.org/abs/2201.10215 · HTML: https://mlsanity.com/paper/2201.10215/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 63 references that are in the map.
- First posted on arXiv 25 Jan 2022.

## Builds on (most cited first, 15 of 15)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Error bounds for approximations with deep ReLU networks](https://mlsanity.com/paper/1610.01145/) (2016)
- [Unsupervised Learning of Video Representations using LSTMs](https://mlsanity.com/paper/1502.04681/) (2015)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Dissecting Neural ODEs](https://mlsanity.com/paper/2002.08071/) (2020)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition](https://mlsanity.com/paper/2101.11845/) (2021)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [Data-driven discovery of PDEs in complex datasets](https://mlsanity.com/paper/1808.10788/) (2018)
- [Deep Learning with Long Short-Term Memory for Time Series Prediction](https://mlsanity.com/paper/1810.10161/) (2018)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [Deep learning-based reduced order models in cardiac electrophysiology](https://mlsanity.com/paper/2006.03040/) (2020)
- [Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models](https://mlsanity.com/paper/2106.05722/) (2021)

## Built on by (most cited first, 10 of 10)

- [Reduced Order Modeling with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2502.10930/) (2025)
- [Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks](https://mlsanity.com/paper/2308.01602/) (2023)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [textitFastSVD-ML-ROM: A Reduced-Order Modeling Framework based on Machine Learning for Real-Time Applications](https://mlsanity.com/paper/2207.11842/) (2022)
- [PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs](https://mlsanity.com/paper/2405.08558/) (2024)
- [Branched Latent Neural Maps](https://mlsanity.com/paper/2308.02599/) (2023)
- [Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models](https://mlsanity.com/paper/2410.13376/) (2024)
- [Deep Convolutional Architectures for Extrapolative Forecast in Time-dependent Flow Problems](https://mlsanity.com/paper/2209.09651/) (2022)
- [A two stages Deep Learning Architecture for Model Reduction of Parametric Time-Dependent Problems](https://mlsanity.com/paper/2301.09926/) (2023)
- [Convolutional Symmetric AutoEncoders: enhancing latent stability via differential geometry](https://mlsanity.com/paper/2607.00669/) (2026)

## Abstract (arXiv)

Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional ROMs - built, e.g., exclusively through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized PDEs. In particular, POD-DL-ROMs can achieve extreme efficiency in the training stage and faster than real-time performances at testing, thanks to a prior dimensionality reduction through POD and a DL-based prediction framework. Nonetheless, they share with conventional ROMs poor performances regarding time extrapolation tasks. This work aims at taking a further step towards the use of DL algorithms for the efficient numerical approximation of parametrized PDEs by introducing the μ t-POD-LSTM-ROM framework. This novel technique extends the POD-DL-ROM framework by adding a two-fold architecture taking advantage of long short-term memory (LSTM) cells, ultimately allowing long-term prediction of complex systems' evolution, with respect to the training window, for unseen input parameter values. Numerical results show that this recurrent architecture enables the extrapolation for time windows up to 15 times larger than the training time domain, and achieves better testing time performances with respect to the already lightning-fast POD-DL-ROMs.