# POD-DL-ROM: enhancing deep learning-based reduced order models for nonlinear parametrized PDEs by proper orthogonal decomposition

arXiv: https://arxiv.org/abs/2101.11845 · HTML: https://mlsanity.com/paper/2101.11845/

## Key facts

- Cited by 79 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 60 references that are in the map.
- First posted on arXiv 28 Jan 2021.

## Builds on (most cited first, 17 of 17)

- [BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding](https://mlsanity.com/paper/1810.04805/) (2018)
- [How transferable are features in deep neural networks?](https://mlsanity.com/paper/1411.1792/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [A fast and accurate physics-informed neural network reduced order model with shallow masked autoencoder](https://mlsanity.com/paper/2009.11990/) (2020)
- [A Theoretical Analysis of Deep Neural Networks and Parametric PDEs](https://mlsanity.com/paper/1904.00377/) (2019)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Deep learning-based reduced order models in cardiac electrophysiology](https://mlsanity.com/paper/2006.03040/) (2020)
- [DR-RNN: A deep residual recurrent neural network for model reduction](https://mlsanity.com/paper/1709.00939/) (2017)
- [Physics-informed deep generative models](https://mlsanity.com/paper/1812.03511/) (2018)
- [A deep learning framework for solution and discovery in solid mechanics](https://mlsanity.com/paper/2003.02751/) (2020)
- [Reduced order modeling of subsurface multiphase flow models using deep residual recurrent neural networks](https://mlsanity.com/paper/1810.10422/) (2018)

## Built on by (most cited first, 30 of 79)

- [Predicting Physics in Mesh-reduced Space with Temporal Attention](https://mlsanity.com/paper/2201.09113/) (2022)
- [Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics](https://mlsanity.com/paper/2205.03990/) (2022)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [Deep Learning Methods for Partial Differential Equations and Related Parameter Identification Problems](https://mlsanity.com/paper/2212.03130/) (2022)
- [Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions](https://mlsanity.com/paper/2211.06786/) (2022)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [Reduced Order Modeling with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2502.10930/) (2025)
- [Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems](https://mlsanity.com/paper/2210.03008/) (2022)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Efficient PDE-Constrained optimization under high-dimensional uncertainty using derivative-informed neural operators](https://mlsanity.com/paper/2305.20053/) (2023)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networks](https://mlsanity.com/paper/2310.04867/) (2023)
- [Deep Learning-based surrogate models for parametrized PDEs: handling geometric variability through graph neural networks](https://mlsanity.com/paper/2308.01602/) (2023)
- [Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks](https://mlsanity.com/paper/2112.07096/) (2021)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds](https://mlsanity.com/paper/2305.15490/) (2023)
- [Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models](https://mlsanity.com/paper/2106.05722/) (2021)
- [Long-time prediction of nonlinear parametrized dynamical systems by deep learning-based reduced order models](https://mlsanity.com/paper/2201.10215/) (2022)
- [Residual-Based Error Corrector Operator to Enhance Accuracy and Reliability of Neural Operator Surrogates of Nonlinear Variational Boundary-Value Problems](https://mlsanity.com/paper/2306.12047/) (2023)
- [A practical existence theorem for reduced order models based on convolutional autoencoders](https://mlsanity.com/paper/2402.00435/) (2024)
- [Deep-HyROMnet: A deep learning-based operator approximation for hyper-reduction of nonlinear parametrized PDEs](https://mlsanity.com/paper/2202.02658/) (2022)
- [Non-linear manifold ROM with Convolutional Autoencoders and Reduced Over-Collocation method](https://mlsanity.com/paper/2203.00360/) (2022)
- [Adaptive learning of effective dynamics: Adaptive real-time, online modeling for complex systems](https://mlsanity.com/paper/2304.01732/) (2023)
- [PTPI-DL-ROMs: pre-trained physics-informed deep learning-based reduced order models for nonlinear parametrized PDEs](https://mlsanity.com/paper/2405.08558/) (2024)
- [Real-time optimal control of high-dimensional parametrized systems by deep learning-based reduced order models](https://mlsanity.com/paper/2409.05709/) (2024)
- [Verification and Validation for Trustworthy Scientific Machine Learning](https://mlsanity.com/paper/2502.15496/) (2025)
- [Efficient Deconvolution in Populational Inverse Problems](https://mlsanity.com/paper/2505.19841/) (2025)
- [On the under-reaching phenomenon in message-passing neural PDE solvers: revisiting the CFL condition](https://mlsanity.com/paper/2507.08861/) (2025)

## Abstract (arXiv)

Deep learning-based reduced order models (DL-ROMs) have been recently proposed to overcome common limitations shared by conventional reduced order models (ROMs) - built, e.g., through proper orthogonal decomposition (POD) - when applied to nonlinear time-dependent parametrized partial differential equations (PDEs). These might be related to (i) the need to deal with projections onto high dimensional linear approximating trial manifolds, (ii) expensive hyper-reduction strategies, or (iii) the intrinsic difficulty to handle physical complexity with a linear superimposition of modes. All these aspects are avoided when employing DL-ROMs, which learn in a non-intrusive way both the nonlinear trial manifold and the reduced dynamics, by relying on deep (e.g., feedforward, convolutional, autoencoder) neural networks. Although extremely efficient at testing time, when evaluating the PDE solution for any new testing-parameter instance, DL-ROMs require an expensive training stage, because of the extremely large number of network parameters to be estimated. In this paper we propose a possible way to avoid an expensive training stage of DL-ROMs, by (i) performing a prior dimensionality reduction through POD, and (ii) relying on a multi-fidelity pretraining stage, where different physical models can be efficiently combined. The proposed POD-DL-ROM is tested on several (both scalar and vector, linear and nonlinear) time-dependent parametrized PDEs (such as, e.g., linear advection-diffusion-reaction, nonlinear diffusion-reaction, nonlinear elastodynamics, and Navier-Stokes equations) to show the generality of this approach and its remarkable computational savings.