The DOI matches its Crossref record; 12 of the 12 references it cites were found in a registry or an index.
Deep learning for universal linear embeddings of nonlinear dynamics
Open accessAbstract
Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear has the potential to enable nonlinear prediction, estimation, and control using linear theory. The Koopman operator is a leading data-driven embedding, and its eigenfunctions provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven challenging. This work leverages deep learning to discover representations of Koopman eigenfunctions from data. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold. We identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems with continuous spectra. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding, while connecting our models to decades of asymptotics. Thus, we benefit from the power of deep learning, while retaining the physical interpretability of Koopman embeddings.
Overview
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
The claims and statistics of this paper haven’t been checked yet.
Run the check — free, no account needed. It reads the open-access text and checks the sources it cites.
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
The self-check needs the paper’s full text.
Citations
Cites
ImageNet classification with deep convolutional neural networks
Krizhevsky, A., Sutskever, I., & Hinton, G. E. (2017) · Communications of the ACM · doi:10.1145/3065386 · Cited by 112,155
Adam: A Method for Stochastic Optimization
Kingma, D. P., & Ba, J. (2014) · arXiv · doi:10.48550/arxiv.1412.6980 · Cited by 98,946
LeCun, Y., Bengio, Y., & Hinton, G. (2015) · Nature · doi:10.1038/nature14539 · Cited by 84,126
Multilayer feedforward networks are universal approximators
Hornik, K., Stinchcombe, M., & White, H. (1989) · Neural Networks · doi:10.1016/0893-6080(89)90020-8 · Cited by 21,728
Approximation by superpositions of a sigmoidal function
Cybenko, G. (1989) · Mathematics of Control, Signals, and Systems · doi:10.1007/bf02551274 · Cited by 13,883
Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex
Hubel, D. H., & Wiesel, T. N. (1962) · The Journal of Physiology · doi:10.1113/jphysiol.1962.sp006837 · Cited by 13,680
TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems
Abadi, M., Agarwal, A., Barham, P., Brevdo, E., Chen, Z., Citro, C., Corrado, G. S., Davis, A., Dean, J., Devin, M., et al. (2016) · arXiv · doi:10.48550/arxiv.1603.04467 · Cited by 9,408
How transferable are features in deep neural networks?
Yosinski, J., Clune, J., Bengio, Y., & Lipson, H. (2014) · arXiv · doi:10.48550/arxiv.1411.1792 · Cited by 6,061
Fukushima, K. (1980) · Biological Cybernetics · doi:10.1007/bf00344251 · Cited by 5,759
Automated reverse engineering of nonlinear dynamical systems
Bongard, J., & Lipson, H. (2007) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.0609476104 · Cited by 792
Cited by
Physics-informed machine learning
Karniadakis, G. E., Kevrekidis, I. G., Lu, L., Perdikaris, P., Wang, S., & Yang, L. (2021) · Nature Reviews Physics · doi:10.1038/s42254-021-00314-5 · Cited by 8,718
Machine Learning for Fluid Mechanics
Brunton, S. L., Noack, B. R., & Koumoutsakos, P. (2020) · Annual Review of Fluid Mechanics · doi:10.1146/annurev-fluid-010719-060214 · Cited by 3,063
Self-citationExplainable machine learning for scientific insights and discoveries
Roscher, R., Bohn, B., Duarte, M. F., & Garcke, J. (2020) · IEEE Access · doi:10.1109/access.2020.2976199 · Cited by 1,040
Data-driven discovery of coordinates and governing equations
Champion, K., Lusch, B., Kutz, J. N., & Brunton, S. L. (2019) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.1906995116 · Cited by 909
Self-citationDynamic Mode Decomposition and Its Variants
Schmid, P. J. (2022) · Annual Review of Fluid Mechanics · doi:10.1146/annurev-fluid-030121-015835 · Cited by 640
Metadata
- Authors
- Lusch, B., Kutz, J. N., & Brunton, S. L.
- Venue
- Nature Communications
- Published
- 2018-11-23
- Volume
- 9
- Issue
- 1
- Pages
- 4950-4950
- ISSN
- 2041-1723 (ISSN-L)
- OpenAlex
- W2777417212
- Cited by
- 1,424
- References
- 12
- FWCI
- 48.45
- Type
- journal article
- Language
- en
- Open access
- gold
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Deep learning for universal linear embeddings of nonlinear dynamics. https://app.cochat.ai/papers/10.1038/s41467-018-07210-0
- The paper
- Lusch, B., Kutz, J. N., & Brunton, S. L. (2018). Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications. https://doi.org/10.1038/s41467-018-07210-0
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
