# Kernel Learning for Robust Dynamic Mode Decomposition: Linear and Nonlinear Disambiguation Optimization (LANDO)

arXiv: https://arxiv.org/abs/2106.01510 · HTML: https://mlsanity.com/paper/2106.01510/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Jun 2021.

## Built on by (most cited first, 22 of 22)

- [Stabilized Neural Ordinary Differential Equations for Long-Time Forecasting of Dynamical Systems](https://mlsanity.com/paper/2203.15706/) (2022)
- [Rigorous data-driven computation of spectral properties of Koopman operators for dynamical systems](https://mlsanity.com/paper/2111.14889/) (2021)
- [Residual Dynamic Mode Decomposition: Robust and verified Koopmanism](https://mlsanity.com/paper/2205.09779/) (2022)
- [Machine Learning for Partial Differential Equations](https://mlsanity.com/paper/2303.17078/) (2023)
- [Learning Dynamical Systems from Data: A Simple Cross-Validation Perspective, Part V: Sparse Kernel Flows for 132 Chaotic Dynamical Systems](https://mlsanity.com/paper/2301.10321/) (2023)
- [Estimating Koopman operators with sketching to provably learn large scale dynamical systems](https://mlsanity.com/paper/2306.04520/) (2023)
- [Interpreting and generalizing deep learning in physics-based problems with functional linear models](https://mlsanity.com/paper/2307.04569/) (2023)
- [Streaming Compression of Scientific Data via weak-SINDy](https://mlsanity.com/paper/2308.14962/) (2023)
- [Enhancing Predictive Capabilities in Data-Driven Dynamical Modeling with Automatic Differentiation: Koopman and Neural ODE Approaches](https://mlsanity.com/paper/2310.06790/) (2023)
- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Ensemble forecasts in reproducing kernel Hilbert space family](https://mlsanity.com/paper/2207.14653/) (2022)
- [A parametric framework for kernel-based dynamic mode decomposition using deep learning](https://mlsanity.com/paper/2409.16817/) (2024)
- [Featurizing Koopman Mode Decomposition For Robust Forecasting](https://mlsanity.com/paper/2312.09146/) (2023)
- [Inferring Kernel ε-Machines: Discovering Structure in Complex Systems](https://mlsanity.com/paper/2410.01076/) (2024)
- [Identifying Large-Scale Linear Parameter Varying Systems with Dynamic Mode Decomposition Methods](https://mlsanity.com/paper/2502.02336/) (2025)
- [Kernel Dynamic Mode Decomposition For Sparse Reconstruction of Closable Koopman Operators](https://mlsanity.com/paper/2505.06806/) (2025)
- [SINDy on slow manifolds](https://mlsanity.com/paper/2507.00747/) (2025)
- [Bayesian Transfer Operators in Reproducing Kernel Hilbert Spaces](https://mlsanity.com/paper/2509.22482/) (2025)
- [Sparse and nonparametric estimation of equations governing dynamical systems with applications to biology](https://mlsanity.com/paper/2511.00579/) (2025)
- [Fast dynamical similarity analysis](https://mlsanity.com/paper/2511.22828/) (2025)
- [A neural operator framework for data-driven discovery of stability and receptivity in physical systems](https://mlsanity.com/paper/2604.19465/) (2026)
- [A Quadratic Order Reduction -- Gaussian Process Ordinary Differential Equation framework for the inference of Large Continuous Dynamical Systems](https://mlsanity.com/paper/2606.13063/) (2026)

## Abstract (arXiv)

Research in modern data-driven dynamical systems is typically focused on the three key challenges of high dimensionality, unknown dynamics, and nonlinearity. The dynamic mode decomposition (DMD) has emerged as a cornerstone for modeling high-dimensional systems from data. However, the quality of the linear DMD model is known to be fragile with respect to strong nonlinearity, which contaminates the model estimate. In contrast, sparse identification of nonlinear dynamics (SINDy) learns fully nonlinear models, disambiguating the linear and nonlinear effects, but is restricted to low-dimensional systems. In this work, we present a kernel method that learns interpretable data-driven models for high-dimensional, nonlinear systems. Our method performs kernel regression on a sparse dictionary of samples that appreciably contribute to the underlying dynamics. We show that this kernel method efficiently handles high-dimensional data and is flexible enough to incorporate partial knowledge of system physics. It is possible to accurately recover the linear model contribution with this approach, disambiguating the effects of the implicitly defined nonlinear terms, resulting in a DMD-like model that is robust to strongly nonlinear dynamics. We demonstrate our approach on data from a wide range of nonlinear ordinary and partial differential equations that arise in the physical sciences. This framework can be used for many practical engineering tasks such as model order reduction, diagnostics, prediction, control, and discovery of governing laws.