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Deep learning for universal linear embeddings of nonlinear dynamics

Bethany LuschJ. KutzS. Brunton

article2017Nature Communications1,740 citations

Develops a deep autoencoder framework that discovers interpretable Koopman eigenfunctions from data, enabling linear prediction and control of strongly nonlinear dynamical systems with both discrete and continuous spectra.

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Real-world engineering, physical, and biological systems are predominantly nonlinear and often lack known governing equations. While linear systems possess well-established methods for accurate prediction and automated control, representing complex nonlinear dynamics within a linear framework remains a major challenge. Koopman operator theory provides a mathematical foundation to globally transform nonlinear systems into linear ones via coordinate functions known as eigenfunctions. However, discovering these eigenfunctions has historically required difficult manual derivations, produced uninterpretable black-box models, or failed entirely when dealing with systems exhibiting continuous frequency spectra.

The article demonstrates a deep learning framework designed to discover interpretable, low-dimensional coordinate representations that globally linearize strongly nonlinear dynamics. The primary objective is to construct parsimonious linear models from trajectory data while handling both discrete and continuous eigenvalue spectra.

To achieve this, the authors designed a modified deep auto-encoder network trained on simulated trajectory datasets ranging from 5,000 to 20,000 samples. The architecture encodes high-dimensional state data into low-dimensional latent coordinates, advances those coordinates using a linear dynamics matrix, and decodes them back to the original state space. A specialized auxiliary network parameterizes continuously shifting frequencies and damping rates across the state space. The models were evaluated across three classic benchmark systems: a canonical two-state nonlinear system with discrete eigenvalues, a nonlinear pendulum exhibiting a continuous frequency spectrum, and a high-dimensional fluid flow past a cylinder at Reynolds number 100.

The key findings confirm that the deep learning architecture accurately linearizes complex dynamics while keeping model dimensionality minimal. First, the framework achieved very low test errors, on the order of 10⁻⁷ to 10⁻⁶ across all benchmarks, confirming precise reconstruction and long-term state prediction. Second, for the continuous-spectrum pendulum, the auxiliary network captured the energy-dependent frequency shift using only a single pair of conjugate eigenfunctions, avoiding the cumbersome infinite harmonic expansions required by conventional techniques. Third, the model successfully mapped high-dimensional fluid vortex shedding onto an interpretable two-to-three coordinate manifold, accurately predicting transient trajectories that started far off the natural limit cycle.

These results demonstrate that nonlinear dynamical modeling can be made both computationally tractable and physically interpretable without sacrificing accuracy. Enabling linear representations of complex dynamics significantly reduces the computational overhead and operational risk associated with nonlinear predictive modeling, sensor estimation, and feedback control design in applied engineering settings.

Organizations developing data-driven modeling and control architectures should consider adopting auto-encoder frameworks with auxiliary parameter networks when modeling oscillatory or continuous-spectrum dynamics. Future research should prioritize developing automated techniques to detect latent coordinate dimensions and extending the architecture to higher-dimensional systems such as turbulent fluid flow, neuroscience, and epidemiology.

Confidence in these findings is high for low-dimensional systems governed by smooth attractors. However, the reader should note key limitations: deep learning models act primarily as sophisticated interpolation tools with limited capacity to extrapolate beyond their training data distribution. Consequently, applying this approach requires sufficient volume and diversity in training data, particularly regarding transient states, as well as upfront manual tuning of the latent subspace dimension.

University of Washingtonauto encodersdeep learninginterpretabilityuniversal linear embeddingsnonlinear dynamicsdeep learning
arXiv: 1712.09707BethanyL/DeepKoopman
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Abstract

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.

Topics

auto encodersdeep learninginterpretabilitylow-dimensional manifoldsconvolutional neural networksstochastic dynamics

Keywords

universal linear embeddingsnonlinear dynamicsdeep learningKoopman embeddingsdynamic mode decompositionmultilayer feedforward networksuniversal approximationHamiltonian systemspredictive controlmodel reduction

Table of Contents

  • 1 Introduction
  • 2 Data-driven dynamical systems
  • 3 Deep learning to identify Koopman eigenfunctions
  • 4 Results
  • 5 Discussion
  • References

Knowls

  1. Knowl 1 — Deep Autoencoder Architecture for Linear Koopman Embedding

    model/method

    To globally linearize a discrete-time nonlinear dynamical system xk+1=F(xk)x_{k+1} = F(x_k) where xagkag∈agRnx ag{k} ag{\in} ag{\mathbb{R}^n}, the deep Koopman framework uses an autoencoder structure coupled with a linear latent transformation.

    The framework consists of:

    1. An encoder neural network φ:Rn→Rp\varphi: \mathbb{R}^n \to \mathbb{R}^p mapping physical state coordinates xx to intrinsic Koopman coordinates y=φ(x)y = \varphi(x).
    2. A decoder neural network φ−1:Rp→Rn\varphi^{-1}: \mathbb{R}^p \to \mathbb{R}^n that reconstructs the physical state from the latent variables, x^=φ−1(y)\hat{x} = \varphi^{-1}(y).
    3. A discrete-time linear dynamical operator represented by a matrix K∈Rp×pK \in \mathbb{R}^{p \times p} advancing the intrinsic coordinates forward in time: yk+1=Kyky_{k+1} = K y_k.

    The latent dimension pp is chosen to match the expected low-rank intrinsic dynamics of the system, enabling both global linearization of trajectories in the latent space and exact inversion back to the original physical state space.

  2. Knowl 2 — Auxiliary Network Parameterization for Continuous Koopman Spectrum

    model/method

    Nonlinear systems exhibiting continuous frequency spectra (such as nonlinear oscillators whose oscillation period depends on energy) cannot be represented compactly by a fixed finite set of Koopman eigenfunctions, which would otherwise require an infinite series expansion of harmonic frequencies. To achieve a low-dimensional embedding, the linear dynamics matrix KK is parameterized by continuous eigenvalues λ=Λ(y)\lambda = \Lambda(y) output by an auxiliary neural network.

    For each complex conjugate pair of eigenvalues λ±=μ±iω\lambda_\pm = \mu \pm i\omega, the continuous-time eigenvalues are parameterized as a function of the radius in latent coordinates, R2=yj2+yj+12R^2 = y_j^2 + y_{j+1}^2, enforcing circular symmetry in phase space. The discrete-time matrix KK is structured into 2×22 \times 2 Jordan blocks B(μ,ω)B(\mu, \omega) for each complex conjugate eigenvalue pair, given a discrete sampling interval Δt\Delta t:

    B(μ,ω)=exp⁡(μΔt)[cos⁡(ωΔt)−sin⁡(ωΔt)sin⁡(ωΔt)cos⁡(ωΔt)]B(\mu, \omega) = \exp(\mu \Delta t) \begin{bmatrix} \cos(\omega \Delta t) & -\sin(\omega \Delta t) \\ \sin(\omega \Delta t) & \cos(\omega \Delta t) \end{bmatrix}

    For each real eigenvalue λ\lambda, the auxiliary network maps the corresponding coordinate yjy_j directly to λ\lambda. Along a trajectory where eigenvalues vary over time, the mm-step advance operator is given by the product of state-dependent matrices: Km=K(λ1)K(λ2)⋯K(λm)K^m = K(\lambda_1) K(\lambda_2) \cdots K(\lambda_m).

  3. Knowl 3 — Multi-Objective Loss Formulation for Deep Koopman Networks

    equation

    The deep Koopman autoencoder is trained using a composite loss function comprising reconstruction error, multi-step prediction error in the physical state space, multi-step linearity in the latent space, maximum absolute error penalty (L∞L_\infty), and L2L_2 weight regularization:

    L=α1(Lrecon+Lpred)+Llin+α2L∞+α3∥W∥22\mathcal{L} = \alpha_1 (\mathcal{L}_{\text{recon}} + \mathcal{L}_{\text{pred}}) + \mathcal{L}_{\text{lin}} + \alpha_2 \mathcal{L}_\infty + \alpha_3 \|W\|_2^2

    where the individual loss components are defined as:

    Lrecon=∥x1−φ−1(φ(x1))∥MSE\mathcal{L}_{\text{recon}} = \|x_1 - \varphi^{-1}(\varphi(x_1))\|_{\text{MSE}}

    Lpred=1Sp∑m=1Sp∥xm+1−φ−1(Kmφ(x1))∥MSE\mathcal{L}_{\text{pred}} = \frac{1}{S_p} \sum_{m=1}^{S_p} \|x_{m+1} - \varphi^{-1}(K^m \varphi(x_1))\|_{\text{MSE}}

    Llin=1T−1∑m=1T−1∥φ(xm+1)−Kmφ(x1)∥MSE\mathcal{L}_{\text{lin}} = \frac{1}{T-1} \sum_{m=1}^{T-1} \|\varphi(x_{m+1}) - K^m \varphi(x_1)\|_{\text{MSE}}

    L∞=∥x1−φ−1(φ(x1))∥∞+∥x2−φ−1(Kφ(x1))∥∞\mathcal{L}_\infty = \|x_1 - \varphi^{-1}(\varphi(x_1))\|_\infty + \|x_2 - \varphi^{-1}(K \varphi(x_1))\|_\infty

    Here, x1,x2,…,xTx_1, x_2, \dots, x_T is a sampled trajectory of length TT; MSE\text{MSE} denotes the mean squared error averaged over dimensions and batch samples; SpS_p is the prediction horizon hyperparameter; WW denotes all network weight matrices; α1,α2,α3\alpha_1, \alpha_2, \alpha_3 are weighting hyperparameters; and Km=∏j=1mK(λj)K^m = \prod_{j=1}^m K(\lambda_j) accounts for state-varying eigenvalues λj=Λ(yj)\lambda_j = \Lambda(y_j).

  4. Knowl 4 — Koopman Reconstruction and Prediction Errors Across Benchmark Systems

    data/table

    The performance of the deep Koopman autoencoder framework was evaluated on four dynamical system benchmarks: a 2D system with a discrete spectrum, a nonlinear pendulum exhibiting a continuous spectrum, and a 3D fluid cylinder wake model restricted to the slow manifold (Fluid flow 1) and including off-manifold transients (Fluid flow 2).

    Dataset Discrete spectrum Pendulum Fluid flow 1 Fluid flow 2
    Training Error 1.4×10−71.4 \times 10^{-7} 8.5×10−88.5 \times 10^{-8} 5.4×10−75.4 \times 10^{-7} 2.8×10−62.8 \times 10^{-6}
    Validation Error 1.4×10−71.4 \times 10^{-7} 9.4×10−89.4 \times 10^{-8} 5.4×10−75.4 \times 10^{-7} 2.9×10−62.9 \times 10^{-6}
    Test Error 1.5×10−71.5 \times 10^{-7} 1.1×10−71.1 \times 10^{-7} 5.5×10−75.5 \times 10^{-7} 2.9×10−62.9 \times 10^{-6}

    The reported metrics are mean squared errors averaged across trajectory steps and state dimensions. The close agreement between training, validation, and test errors across all benchmarks demonstrates that the network achieves high reconstruction and multi-step prediction accuracy while avoiding overfitting.

  5. Knowl 5 — Parsimonious Global Linearization of the Nonlinear Pendulum

    empirical result

    For the undamped nonlinear pendulum governed by x¨=−sin⁡(x)\ddot{x} = -\sin(x), expressed as the first-order system x˙1=x2,x˙2=−sin⁡(x1)\dot{x}_1 = x_2, \dot{x}_2 = -\sin(x_1), the oscillation frequency decreases continuously as the Hamiltonian energy increases, resulting in a continuous Koopman spectrum.

    The deep Koopman framework with an auxiliary network discovers a 2D intrinsic coordinate representation y=(φ1(x),φ2(x))y = (\varphi_1(x), \varphi_2(x)) in which nonlinear trajectories are transformed into exact concentric circles undergoing pure rotation. The auxiliary network parameterizes the continuous frequency ω(R)\omega(R) as a function of the radius R2=y12+y22R^2 = y_1^2 + y_2^2, capturing the energy-dependent period lengthening without requiring infinite harmonic series expansion terms. In these learned coordinates, the radius traces the level sets of the Hamiltonian energy, matching the theoretical action-angle representation of Koopman eigenfunctions.

  6. Knowl 6 — Invariant Slow Manifold Identification in Discrete Spectrum Dynamics

    empirical result

    For the 2D dynamical system with a discrete eigenvalue spectrum governed by:

    x˙1=μx1\dot{x}_1 = \mu x_1

    x˙2=λ(x2−x12)\dot{x}_2 = \lambda (x_2 - x_1^2)

    with μ=−0.05\mu = -0.05 and λ=−1\lambda = -1, the phase space contains an attracting parabolic slow manifold x2=x12x_2 = x_1^2.

    The deep Koopman autoencoder identifies nonlinear coordinates y=φ(x)y = \varphi(x) that flatten this parabolic manifold into a linear subspace, globally linearizing the dynamics. Furthermore, when the auxiliary eigenvalue network is allowed the freedom to vary across state space, it autonomously converges to narrow intervals centered on the true discrete constants without requiring a priori knowledge that the spectrum is discrete.

  7. Knowl 7 — Koopman Embedding of Transient and Limit Cycle Fluid Dynamics

    empirical result

    For the 3D mean-field model of vortex shedding in fluid flow past a circular cylinder at Reynolds number 100:

    x˙1=μx1−ωx2+Ax1x3\dot{x}_1 = \mu x_1 - \omega x_2 + A x_1 x_3

    x˙2=ωx1+μx2+Ax2x3\dot{x}_2 = \omega x_1 + \mu x_2 + A x_2 x_3

    x˙3=−λ(x3−x12−x22)\dot{x}_3 = -\lambda (x_3 - x_1^2 - x_2^2)

    with μ=0.1,ω=1,A=−0.1,λ=10\mu = 0.1, \omega = 1, A = -0.1, \lambda = 10, the system contains an unstable origin, a slow manifold x3=x12+x22x_3 = x_1^2 + x_2^2, and a stable limit cycle at r=1r=1.

    The deep Koopman network captures both on-manifold limit cycle oscillations and off-manifold transient dynamics using a 3D latent representation. Two auxiliary networks parameterize the eigenvalues: one maps y12+y22y_1^2 + y_2^2 to the damping rate μ(R)\mu(R) and frequency ω(R)\omega(R), and the second maps y3y_3 to the fast attraction rate λ\lambda. The network correctly identifies that ω≈−1\omega \approx -1 is constant across phase space while μ(R)\mu(R) switches stability across the limit cycle radius, successfully predicting trajectories starting off the attractor from initial conditions alone.

  8. Knowl 8 — Deep Koopman Training Protocol and Architecture Specifications

    experimental setup

    Deep Koopman networks use fully connected feedforward layers with Rectified Linear Unit (ReLU) activations, f(x)=max⁡{0,x}f(x) = \max\{0, x\}, for all hidden layers and linear activation on output layers. Weight matrices are initialized from a uniform distribution U[−s,s]U[-s, s] where s=1/as = 1/\sqrt{a} with input dimension aa, and bias vectors are initialized to 0.

    Training configurations across systems:

    • Discrete Spectrum: Encoder/decoder have 2 hidden layers of width 30; auxiliary network has 3 hidden layers of width 10. Training uses 5,000 trajectories of length 51, batch size 256, α1=0.1,α2=10−7,α3=10−15,Sp=30\alpha_1 = 0.1, \alpha_2 = 10^{-7}, \alpha_3 = 10^{-15}, S_p = 30.
    • Pendulum: Encoder/decoder have 2 hidden layers of width 80; auxiliary network has 1 hidden layer of width 170. Training uses 15,000 trajectories of length 51, batch size 128, α1=0.001,α2=10−9,α3=10−14,Sp=30\alpha_1 = 0.001, \alpha_2 = 10^{-9}, \alpha_3 = 10^{-14}, S_p = 30.
    • Fluid Flow (on slow manifold): Encoder/decoder have 1 hidden layer of width 105; auxiliary network has 1 hidden layer of width 300. Training uses 15,000 trajectories of length 121, batch size 256, α1=0.1,α2=10−7,α3=10−13,Sp=30\alpha_1 = 0.1, \alpha_2 = 10^{-7}, \alpha_3 = 10^{-13}, S_p = 30.
    • Fluid Flow (off attractor): Encoder/decoder have 1 hidden layer of width 130; auxiliary network has 2 hidden layers of width 20. Training uses 20,000 trajectories of length 101, batch size 128, α1=0.1,α2=10−9,α3=10−13,Sp=30\alpha_1 = 0.1, \alpha_2 = 10^{-9}, \alpha_3 = 10^{-13}, S_p = 30.

    All models are optimized using Adam with a learning rate of 0.001. On the pendulum and fluid flow datasets, networks are pre-trained for 5 minutes as a standard autoencoder (minimizing only Lrecon\mathcal{L}_{\text{recon}}) before adding multi-step prediction and linearity losses. Early stopping is applied using validation error.

  9. Knowl 9 — Limitations of the Deep Koopman Embedding Framework

    limitation

    The deep Koopman framework has several key operational limitations:

    1. Manual Latent Dimensionality: The dimension pp of the intrinsic coordinate space is a hyperparameter that must be manually specified based on prior knowledge of the underlying dynamics rather than being discovered automatically.
    2. Manual Spectral Block Partitioning: The network requires pre-specifying the allocation and structure of real versus complex conjugate eigenvalue blocks, as well as separate auxiliary network branches for distinct eigenvalue groups.
    3. Limited Extrapolation: As with general deep neural networks, the models act as interpolators within the training distribution and exhibit degraded prediction accuracy when extrapolating to phase space regimes not covered by the training trajectories.
    4. Transient Data Dependency: Capturing transient phenomena (such as convergence to a slow manifold or limit cycle) requires sufficient sampling and density of transient trajectories in the training dataset.

Coverage note — None was omitted; all key contributions—the architecture, auxiliary eigenvalue network, composite loss, empirical benchmarks (discrete spectrum, nonlinear pendulum, fluid wake), training setup, and stated limitations—are fully covered.

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Citation

MLA
Lusch, B., et al. “Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics”. Nature Communications, vol. 9, no. 1, 2018, https://doi.org/10.1038/s41467-018-07210-0.
APA
Lusch, B., Kutz, J. N., & Brunton, S. L. (2018). Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications, 9(1). https://doi.org/10.1038/s41467-018-07210-0
Chicago
Lusch, B., J. N. Kutz, and S. L. Brunton. 2018. “Deep Learning for Universal Linear Embeddings of Nonlinear Dynamics”. Nature Communications 9 (1). https://doi.org/10.1038/s41467-018-07210-0.
Harvard
Lusch, B., Kutz, J.N. and Brunton, S.L. (2018) “Deep learning for universal linear embeddings of nonlinear dynamics”, Nature Communications, 9(1). Available at: https://doi.org/10.1038/s41467-018-07210-0.
Vancouver
1. Lusch B, Kutz JN, Brunton SL (2018) Deep learning for universal linear embeddings of nonlinear dynamics. Nature Communications. https://doi.org/10.1038/s41467-018-07210-0

BibTeX

@article{Lusch_2018, title={Deep learning for universal linear embeddings of nonlinear dynamics}, volume={9}, ISSN={2041-1723}, url={http://dx.doi.org/10.1038/s41467-018-07210-0}, DOI={10.1038/s41467-018-07210-0}, number={1}, journal={Nature Communications}, publisher={Springer Science and Business Media LLC}, author={Lusch, Bethany and Kutz, J. Nathan and Brunton, Steven L.}, year={2018}, month=Nov }
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