# CD-ROM: Complemented Deep-Reduced Order Model

arXiv: https://arxiv.org/abs/2202.10746 · HTML: https://mlsanity.com/paper/2202.10746/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 77 references that are in the map.
- First posted on arXiv 22 Feb 2022.

## Builds on (most cited first, 25 of 25)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Invertible Residual Networks](https://mlsanity.com/paper/1811.00995/) (2018)
- [Stable Architectures for Deep Neural Networks](https://mlsanity.com/paper/1705.03341/) (2017)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Linearly-Recurrent Autoencoder Networks for Learning Dynamics](https://mlsanity.com/paper/1712.01378/) (2017)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Chaos as an interpretable benchmark for forecasting and data-driven modelling](https://mlsanity.com/paper/2110.05266/) (2021)
- [Multiscale Simulations of Complex Systems by Learning their Effective Dynamics](https://mlsanity.com/paper/2006.13431/) (2020)
- [Adaptive Checkpoint Adjoint Method for Gradient Estimation in Neural ODE](https://mlsanity.com/paper/2006.02493/) (2020)
- [Physics-informed Autoencoders for Lyapunov-stable Fluid Flow Prediction](https://mlsanity.com/paper/1905.10866/) (2019)
- [Physics-Informed Probabilistic Learning of Linear Embeddings of Non-linear Dynamics With Guaranteed Stability](https://mlsanity.com/paper/1906.03663/) (2019)
- [Deep Neural Networks for Nonlinear Model Order Reduction of Unsteady Flows](https://mlsanity.com/paper/2007.00936/) (2020)
- [Assessment of unsteady flow predictions using hybrid deep learning based reduced order models](https://mlsanity.com/paper/2009.04396/) (2020)
- [Physics-Informed Echo State Networks](https://mlsanity.com/paper/2011.02280/) (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, 8 of 8)

- [Mori-Zwanzig latent space Koopman closure for nonlinear autoencoder](https://mlsanity.com/paper/2310.10745/) (2023)
- [Interpretable learning of effective dynamics for multiscale systems](https://mlsanity.com/paper/2309.05812/) (2023)
- [NeurIPS 2024 ML4CFD Competition: Harnessing Machine Learning for Computational Fluid Dynamics in Airfoil Design](https://mlsanity.com/paper/2407.01641/) (2024)
- [ML4PhySim : Machine Learning for Physical Simulations Challenge (The airfoil design)](https://mlsanity.com/paper/2403.01623/) (2024)
- [Neural delay differential equations: learning non-Markovian closures for partially known dynamical systems](https://mlsanity.com/paper/2410.02843/) (2024)
- [NeurIPS 2024 ML4CFD Competition: Results and Retrospective Analysis](https://mlsanity.com/paper/2506.08516/) (2025)
- [UP-dROM : Uncertainty-Aware and Parametrised dynamic Reduced-Order Model, application to unsteady flows](https://mlsanity.com/paper/2503.23236/) (2025)
- [Efficient Adaptation of ROMs for Unsteady Flows Using Data Assimilation](https://mlsanity.com/paper/2602.23188/) (2026)

## Abstract (arXiv)

Model order reduction through the POD-Galerkin method can lead to dramatic gains in terms of computational efficiency in solving physical problems. However, the applicability of the method to non linear high-dimensional dynamical systems such as the Navier-Stokes equations has been shown to be limited, producing inaccurate and sometimes unstable models. This paper proposes a deep learning based closure modeling approach for classical POD-Galerkin reduced order models (ROM). The proposed approach is theoretically grounded, using neural networks to approximate well studied operators. In contrast with most previous works, the present CD-ROM approach is based on an interpretable continuous memory formulation, derived from simple hypotheses on the behavior of partially observed dynamical systems. The final corrected models can hence be simulated using most classical time stepping schemes. The capabilities of the CD-ROM approach are demonstrated on two classical examples from Computational Fluid Dynamics, as well as a parametric case, the Kuramoto-Sivashinsky equation.