# Operator inference with roll outs for learning reduced models from scarce and low-quality data

arXiv: https://arxiv.org/abs/2212.01418 · HTML: https://mlsanity.com/paper/2212.01418/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 85 references that are in the map.
- First posted on arXiv 2 Dec 2022.

## Builds on (most cited first, 28 of 28)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Machine learning accelerated computational fluid dynamics](https://mlsanity.com/paper/2102.01010/) (2021)
- [Solver-in-the-Loop: Learning from Differentiable Physics to Interact with Iterative PDE-Solvers](https://mlsanity.com/paper/2007.00016/) (2020)
- [Survey of multifidelity methods in uncertainty propagation, inference, and optimization](https://mlsanity.com/paper/1806.10761/) (2018)
- [A comprehensive deep learning-based approach to reduced order modeling of nonlinear time-dependent parametrized PDEs](https://mlsanity.com/paper/2001.04001/) (2020)
- [Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems](https://mlsanity.com/paper/1912.08177/) (2019)
- [Extracting Sparse High-Dimensional Dynamics from Limited Data](https://mlsanity.com/paper/1707.08528/) (2017)
- [Parameterized Neural Ordinary Differential Equations: Applications to Computational Physics Problems](https://mlsanity.com/paper/2010.14685/) (2020)
- [Data-driven reduced-order models via regularized operator inference for a single-injector combustion process](https://mlsanity.com/paper/2008.02862/) (2020)
- [Discovery of Nonlinear Dynamical Systems using a Runge-Kutta Inspired Dictionary-based Sparse Regression Approach](https://mlsanity.com/paper/2105.04869/) (2021)
- [Operator inference for non-intrusive model reduction of systems with non-polynomial nonlinear terms](https://mlsanity.com/paper/2002.09726/) (2020)
- [Non-intrusive reduced-order models for parametric partial differential equations via data-driven operator inference](https://mlsanity.com/paper/2110.07653/) (2021)
- [Learning physics-based reduced-order models for a single-injector combustion process](https://mlsanity.com/paper/1908.03620/) (2019)
- [Sampling low-dimensional Markovian dynamics for pre-asymptotically recovering reduced models from data with operator inference](https://mlsanity.com/paper/1908.11233/) (2019)
- [Bayesian operator inference for data-driven reduced-order modeling](https://mlsanity.com/paper/2204.10829/) (2022)
- [Bayesian differential programming for robust systems identification under uncertainty](https://mlsanity.com/paper/2004.06843/) (2020)
- [Physics-informed regularization and structure preservation for learning stable reduced models from data with operator inference](https://mlsanity.com/paper/2107.02597/) (2021)
- [Neural Ordinary Differential Equations for Nonlinear System Identification](https://mlsanity.com/paper/2203.00120/) (2022)
- [Kernel Analog Forecasting: Multiscale Test Problems](https://mlsanity.com/paper/2005.06623/) (2020)
- [Stability Domains for Quadratic-Bilinear Reduced-Order Models](https://mlsanity.com/paper/2009.02769/) (2020)
- [Learning low-dimensional dynamical-system models from noisy frequency-response data with Loewner rational interpolation](https://mlsanity.com/paper/1910.00110/) (2019)
- [Probabilistic error estimation for non-intrusive reduced models learned from data of systems governed by linear parabolic partial differential equations](https://mlsanity.com/paper/2005.05890/) (2020)
- [Vector Fitting for Matrix-valued Rational Approximation](https://mlsanity.com/paper/1503.00411/) (2015)
- [A non-intrusive reduced order modeling framework for quasi-geostrophic turbulence](https://mlsanity.com/paper/1906.11617/) (2019)
- [Operator inference of non-Markovian terms for learning reduced models from partially observed state trajectories](https://mlsanity.com/paper/2103.01362/) (2021)
- [Active operator inference for learning low-dimensional dynamical-system models from noisy data](https://mlsanity.com/paper/2107.09256/) (2021)
- [Neural ODEs with Irregular and Noisy Data](https://mlsanity.com/paper/2205.09479/) (2022)

## Built on by (most cited first, 11 of 11)

- [Guaranteed Stable Quadratic Models and their applications in SINDy and Operator Inference](https://mlsanity.com/paper/2308.13819/) (2023)
- [Scientific Machine Learning Based Reduced-Order Models for Plasma Turbulence Simulations](https://mlsanity.com/paper/2401.05972/) (2024)
- [Gaussian process learning of nonlinear dynamics](https://mlsanity.com/paper/2312.12193/) (2023)
- [Nested Operator Inference for Adaptive Data-Driven Learning of Reduced-order Models](https://mlsanity.com/paper/2508.11542/) (2025)
- [Operator Inference Aware Quadratic Manifolds with Isotropic Reduced Coordinates for Nonintrusive Model Reduction](https://mlsanity.com/paper/2507.20463/) (2025)
- [Learning Noise-Robust Stable Koopman Operator for Control with Hankel DMD](https://mlsanity.com/paper/2408.06607/) (2024)
- [A parametric framework for kernel-based dynamic mode decomposition using deep learning](https://mlsanity.com/paper/2409.16817/) (2024)
- [Causal Multi-fidelity Surrogate Forward and Inverse Models for ICF Implosions](https://mlsanity.com/paper/2509.05510/) (2025)
- [Inference of Continuous Linear Systems from Data with Guaranteed Stability](https://mlsanity.com/paper/2301.10060/) (2023)
- [Small-data Reduced Order Modeling of Chaotic Dynamics through SyCo-AE: Synthetically Constrained Autoencoders](https://mlsanity.com/paper/2305.08036/) (2023)
- [Stability-Certified Learning of Control Systems with Quadratic Nonlinearities](https://mlsanity.com/paper/2403.00646/) (2024)

## Abstract (arXiv)

Data-driven modeling has become a key building block in computational science and engineering. However, data that are available in science and engineering are typically scarce, often polluted with noise and affected by measurement errors and other perturbations, which makes learning the dynamics of systems challenging. In this work, we propose to combine data-driven modeling via operator inference with the dynamic training via roll outs of neural ordinary differential equations. Operator inference with roll outs inherits interpretability, scalability, and structure preservation of traditional operator inference while leveraging the dynamic training via roll outs over multiple time steps to increase stability and robustness for learning from low-quality and noisy data. Numerical experiments with data describing shallow water waves and surface quasi-geostrophic dynamics demonstrate that operator inference with roll outs provides predictive models from training trajectories even if data are sampled sparsely in time and polluted with noise of up to 10%.