# Symplectic model reduction of Hamiltonian systems using data-driven quadratic manifolds

arXiv: https://arxiv.org/abs/2305.15490 · HTML: https://mlsanity.com/paper/2305.15490/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 62 references that are in the map.
- First posted on arXiv 24 May 2023.

## Builds on (most cited first, 10 of 10)

- [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)
- [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)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Quadratic Approximation Manifold for Mitigating the Kolmogorov Barrier in Nonlinear Projection-Based Model Order Reduction](https://mlsanity.com/paper/2204.02462/) (2022)
- [Non-intrusive reduced order modeling of natural convection in porous media using convolutional autoencoders: comparison with linear subspace techniques](https://mlsanity.com/paper/2107.11460/) (2021)
- [Canonical and Noncanonical Hamiltonian Operator Inference](https://mlsanity.com/paper/2304.06262/) (2023)
- [The Neural Network shifted-Proper Orthogonal Decomposition: a Machine Learning Approach for Non-linear Reduction of Hyperbolic Equations](https://mlsanity.com/paper/2108.06558/) (2021)
- [Nonlinear proper orthogonal decomposition for convection-dominated flows](https://mlsanity.com/paper/2110.08295/) (2021)
- [Neural-Network-Augmented Projection-Based Model Order Reduction for Mitigating the Kolmogorov Barrier to Reducibility of CFD Models](https://mlsanity.com/paper/2212.08939/) (2022)

## Built on by (most cited first, 13 of 13)

- [A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics](https://mlsanity.com/paper/2410.18868/) (2024)
- [Data-Driven Identification of Quadratic Representations for Nonlinear Hamiltonian Systems using Weakly Symplectic Liftings](https://mlsanity.com/paper/2308.01084/) (2023)
- [Deep Learning for Structure-Preserving Universal Stable Koopman-Inspired Embeddings for Nonlinear Canonical Hamiltonian Dynamics](https://mlsanity.com/paper/2308.13835/) (2023)
- [Nonlinear embeddings for conserving Hamiltonians and other quantities with Neural Galerkin schemes](https://mlsanity.com/paper/2310.07485/) (2023)
- [Data-driven identification of latent port-Hamiltonian systems](https://mlsanity.com/paper/2408.08185/) (2024)
- [Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling](https://mlsanity.com/paper/2401.12476/) (2024)
- [Nonlinear energy-preserving model reduction with lifting transformations that quadratize the energy](https://mlsanity.com/paper/2503.02273/) (2025)
- [Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction](https://mlsanity.com/paper/2507.20463/) (2025)
- [Learning Hamiltonian Dynamics at Scale: A Differential-Geometric Approach](https://mlsanity.com/paper/2509.24627/) (2025)
- [Reduced-order modeling of Hamiltonian dynamics based on symplectic neural networks](https://mlsanity.com/paper/2508.11911/) (2025)
- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)
- [Structure-preserving Lift & Learn: Scientific machine learning for nonlinear conservative partial differential equations](https://mlsanity.com/paper/2507.00301/) (2025)
- [Learning symplectic model reduction based on an approximation theorem of symplectic embeddings](https://mlsanity.com/paper/2606.04623/) (2026)

## Abstract (arXiv)

This work presents two novel approaches for the symplectic model reduction of high-dimensional Hamiltonian systems using data-driven quadratic manifolds. Classical symplectic model reduction approaches employ linear symplectic subspaces for representing the high-dimensional system states in a reduced-dimensional coordinate system. While these approximations respect the symplectic nature of Hamiltonian systems, linear basis approximations can suffer from slowly decaying Kolmogorov N-width, especially in wave-type problems, which then requires a large basis size. We propose two different model reduction methods based on recently developed quadratic manifolds, each presenting its own advantages and limitations. The addition of quadratic terms to the state approximation, which sits at the heart of the proposed methodologies, enables us to better represent intrinsic low-dimensionality in the problem at hand. Both approaches are effective for issuing predictions in settings well outside the range of their training data while providing more accurate solutions than the linear symplectic reduced-order models.