# Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions

arXiv: https://arxiv.org/abs/2211.06786 · HTML: https://mlsanity.com/paper/2211.06786/

## Key facts

- Cited by 29 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 65 references that are in the map.
- First posted on arXiv 13 Nov 2022.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Sparse identification of nonlinear dynamics for model predictive control in the low-data limit](https://mlsanity.com/paper/1711.05501/) (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)
- [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)
- [Discovering Governing Equations from Partial Measurements with Deep Delay Autoencoders](https://mlsanity.com/paper/2201.05136/) (2022)
- [Deep convolutional recurrent autoencoders for learning low-dimensional feature dynamics of fluid systems](https://mlsanity.com/paper/1808.01346/) (2018)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations](https://mlsanity.com/paper/2103.06183/) (2021)
- [Multi-fidelity surrogate modeling using long short-term memory networks](https://mlsanity.com/paper/2208.03115/) (2022)
- [Learning normal form autoencoders for data-driven discovery of universal,parameter-dependent governing equations](https://mlsanity.com/paper/2106.05102/) (2021)
- [Real-time simulation of parameter-dependent fluid flows through deep learning-based reduced order models](https://mlsanity.com/paper/2106.05722/) (2021)
- [Direct computation of nonlinear mapping via normal form for reduced-order models of finite element nonlinear structures](https://mlsanity.com/paper/2009.12145/) (2020)
- [Model Order Reduction based on Direct Normal Form: Application to Large Finite Element MEMS Structures Featuring Internal Resonance](https://mlsanity.com/paper/2103.10545/) (2021)
- [High order direct parametrisation of invariant manifolds for model order reduction of finite element structures: application to large amplitude vibrations and uncovering of a folding point](https://mlsanity.com/paper/2109.10031/) (2021)
- [Learning Low-Dimensional Quadratic-Embeddings of High-Fidelity Nonlinear Dynamics using Deep Learning](https://mlsanity.com/paper/2111.12995/) (2021)

## Built on by (most cited first, 29 of 29)

- [Reduced Order Modeling with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2502.10930/) (2025)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Sparse identification of nonlinear dynamics and Koopman operators with Shallow Recurrent Decoder Networks](https://mlsanity.com/paper/2501.13329/) (2025)
- [Benchmarking sparse system identification with low-dimensional chaos](https://mlsanity.com/paper/2302.10787/) (2023)
- [Multi-fidelity reduced-order surrogate modeling](https://mlsanity.com/paper/2309.00325/) (2023)
- [Learning Nonlinear Projections for Reduced-Order Modeling of Dynamical Systems using Constrained Autoencoders](https://mlsanity.com/paper/2307.15288/) (2023)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [On latent dynamics learning in nonlinear reduced order modeling](https://mlsanity.com/paper/2408.15183/) (2024)
- [VENI, VINDy, VICI: a generative reduced-order modeling framework with uncertainty quantification](https://mlsanity.com/paper/2405.20905/) (2024)
- [Low-dimensional Data-based Surrogate Model of a Continuum-mechanical Musculoskeletal System Based on Non-intrusive Model Order Reduction](https://mlsanity.com/paper/2302.06528/) (2023)
- [Discover governing differential equations from evolving systems](https://mlsanity.com/paper/2301.07863/) (2023)
- [Explainable AI: Learning from the Learners](https://mlsanity.com/paper/2601.05525/) (2026)
- [Convolutional Autoencoders, Clustering and POD for Low-dimensional Parametrization of Navier-Stokes Equations](https://mlsanity.com/paper/2302.01278/) (2023)
- [Data-driven identification of latent port-Hamiltonian systems](https://mlsanity.com/paper/2408.08185/) (2024)
- [Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models](https://mlsanity.com/paper/2410.13376/) (2024)
- [Toward Adaptive Non-Intrusive Reduced-Order Models: Design and Challenges](https://mlsanity.com/paper/2602.11378/) (2026)
- [Interpretable and Efficient Data-driven Discovery and Control of Distributed Systems](https://mlsanity.com/paper/2411.04098/) (2024)
- [Recurrent Deep Kernel Learning of Dynamical Systems](https://mlsanity.com/paper/2405.19785/) (2024)
- [A POD-TANN approach for the multiscale modeling of materials and macroelement derivation in geomechanics](https://mlsanity.com/paper/2408.07165/) (2024)
- [Spatially-Enhanced Temporal Fusion Transformer: Interpretable Multi-Output Prediction for Parametric Dynamical Systems with Time-Varying Inputs](https://mlsanity.com/paper/2505.00473/) (2025)
- [KROM: Kernelized Reduced Order Modeling](https://mlsanity.com/paper/2603.00360/) (2026)
- [Nonlinear Model Order Reduction of Dynamical Systems in Process Engineering: Review and Comparison](https://mlsanity.com/paper/2506.12819/) (2025)
- [Latent Representation Learning of Multi-scale Thermophysics: Application to Dynamics in Shocked Porous Energetic Material](https://mlsanity.com/paper/2506.12996/) (2025)
- [SINDy on slow manifolds](https://mlsanity.com/paper/2507.00747/) (2025)
- [Parameter-Aware Ensemble SINDy for Interpretable Symbolic SGS Closure](https://mlsanity.com/paper/2508.14085/) (2025)
- [Data-driven stochastic reduced-order modeling of parametrized dynamical systems](https://mlsanity.com/paper/2601.10690/) (2026)
- [Latent Dynamics Graph Convolutional Networks for model order reduction of parameterized time-dependent PDEs](https://mlsanity.com/paper/2601.11259/) (2026)
- [Learning the Geometry of Admissible Hypotheses through Inductive Bias in Training Distributions](https://mlsanity.com/paper/2608.31028/) (2026)

## Abstract (arXiv)

Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.