---
title: Koopman Embeddings for Nonlinear Dynamics
url: https://www.emergentmind.com/topics/koopman-embeddings
type: topic
---

# Koopman Embeddings for Nonlinear Dynamics

Koopman embeddings are data-driven coordinate transformations that recast the evolution of nonlinear dynamical systems into an approximately linear regime through the action of Koopman operators—linear but generally infinite-dimensional objects that act on observables of the system state. This perspective allows nonlinear dynamics to be analyzed, predicted, and controlled using linear techniques, provided suitable finite-dimensional embeddings (often constructed as nonlinear functions or neural network outputs) can be identified such that evolution in those coordinates is well-approximated by a linear operator. Rigorous theory establishes the connection between operator-theoretic, geometric, and control-theoretic aspects of dynamical systems, while a rapidly developing suite of algorithmic tools—ranging from Dynamic Mode Decomposition (DMD) and Extended DMD (EDMD) to modern deep autoencoder architectures—enables practical construction of Koopman embeddings from finite trajectory data [2102.12086].

## 1. Mathematical Foundations of Koopman Embeddings

Given a dynamical system $x_{k+1}=F(x_k)$ (discrete time), the Koopman operator $\mathcal{K}$ acts linearly on observables $g:\mathbb{R}^n\rightarrow\mathbb{C}$ by composition: $\mathcal{K}g(x) = g(F(x))$. Koopman eigenfunctions $\varphi$ satisfy $\mathcal{K}\varphi(x) = \lambda\varphi(x)$, with $\lambda\in\mathbb{C}$. If a finite set of such eigenfunctions $\{\varphi_1,\ldots,\varphi_r\}$ is found, the embedded coordinate $z = [\varphi_1(x),\ldots,\varphi_r(x)]^\top$ evolves linearly: $z_{k+1} = K z_k$, with $K = \mathrm{diag}(\lambda_1, \ldots, \lambda_r)$. Observables $g$ that lie in the span of these eigenfunctions follow $g(x_k) = v^\top z_k = \sum_{j} v_j \lambda_j^k \varphi_j(x_0)$ [2102.12086].

The embedding challenge centers on constructing finite sets of functions that span approximately invariant subspaces for $\mathcal K$, yielding linear finite-dimensional evolution in $z$-space [2102.12086].

## 2. Algorithms for Constructing Koopman Embeddings

### 2.1 Linear and Kernel-Based Approaches

- **Dynamic Mode Decomposition (DMD):** Based on snapshot pairs and SVD reduction, DMD yields eigenvalues and modes approximating the spectrum and invariant features of the Koopman operator [2102.12086].
- **Extended DMD (EDMD):** Generalizes DMD by introducing a flexible, user-defined dictionary of basis functions, enabling richer nonlinear embeddings [2102.12086].
- **Kernel Methods:** Kernel-based algorithms approximate $\mathcal{K}$ in an RKHS, using the kernel trick to avoid explicit high-dimensional expansions. Eigenfunctions approximate Koopman modes as sums over kernel evaluations [2102.12086, 2501.16489].

### 2.2 Deep Learning Strategies

- **Autoencoder-Based Embeddings:** Deep encoders $\phi_\theta$ map raw state $x$ (or time-delay vectors) into a latent space $z$, where the operator $K$ models linear (or locally linear) dynamics: $z_{k+1} = K z_k$ [1712.09707, 2306.05224]. The decoder reconstructs $x$ from $z$.
- **“Lift and Regress”:** A feed-forward encoder produces high-dimensional hidden coordinates $z=\phi_\theta(x)$, and $K$ is fitted by minimizing $\|\phi(x_{k+1}) - K\phi(x_k)\|^2$, favoring linearization in latent space [2102.12086].
- **Auxiliary Networks for Continuous Spectra:** For systems with non-isolated frequencies (e.g., undamped pendulum), auxiliary networks parameterize frequency and growth rate as functions of latent state radius, enabling efficient, compact embeddings for continuous Koopman spectra [1712.09707].
- **Message-Passing Networks for Graph Dynamics:** Exploiting underlying geometric structure via graph-aware neural networks yields globally valid linearizations of large-scale network dynamics [2305.09060].

### 2.3 Algorithmic Pipeline (Standard Structure)

| Algorithm      | Encoder           | Operator   | Decoder | Embedding Type    |
|---------------|-------------------|------------|---------|-------------------|
| DMD           | Linear projection | Least-squares | Linear | Modal             |
| EDMD          | Dictionary basis  | Least-squares | Linear | Modal/nonlinear   |
| Kernel DMD    | Kernel functions  | Kernel regression | Linear | Nonparametric    |
| Deep Koopman AE | Deep NN (MLP)   | Matrix (learned) | Deep NN (MLP) | Nonlinear        |
| Koopman MPNN  | Graph MPNN        | Diagonal   | MPNN    | Graph-structured  |

## 3. Theoretical Guarantees, Limitations, and Spectral Structure

The convergence of EDMD-type algorithms is established for ergodic dynamics and invariant dictionaries: $K \rightarrow P_{\mathcal{D}} \mathcal{K} P_{\mathcal{D}}$ as $m \rightarrow \infty$; in $L^2$ the approximation converges strongly to $\mathcal{K}$ [2102.12086]. Koopman embedding theory is tightly connected to contraction analysis and Lyapunov stability: every discrete-time contracting nonlinear model admits a finite-dimensional stable Koopman embedding, proved via direct parameterization of stable $K$ [2110.06509].

Koopman operators may exhibit pure point, continuous, and residual spectra. Classical algorithms target point spectrum; continuous-spectrum systems (mixing or quasi-periodic) are more challenging, necessitating harmonic averaging or continuous auxiliary networks for accurate embedding [2102.12086, 1712.09707].

A significant limitation is the non-existence of one-to-one continuous finite-dimensional Koopman immersions for nonlinear systems with multiple omega-limit sets: any continuous embedding must collapse all limit sets in the image, and learned approximate immersions exhibit the same failure in the regime of dense data and small sampling interval [2312.17045]. This fundamental obstruction shapes the choice of embedding domain and the practical utility of the representations.

## 4. Extensions for Systems with Inputs, Control, and Hybrid Modeling

Koopman embeddings have been extended to nonlinear systems with inputs by lifting to linear parameter-varying (LPV) forms: under mild differentiability and convexity conditions, the lifted dynamics take $z_{k+1} = A z_k + B(z_k, u_k)u_k$, with $B$ state- or input-dependent [2207.12132]. For control-affine systems, bilinear or affine models emerge, and error bounds quantify the approximation quality of using a constant $B$ in linear time-invariant (LTI) Koopman models.

Delay-coordinate embeddings (Hankel maps) enable linear modeling of non-smooth periodic or hybrid systems by stacking current and history states, so that $X_{k+1} = A X_k + B U_k$, with $X_k$ the time-delay vector [2507.14455]. This approach remains valid as long as the system's modal sequence and event timing are periodic and consistent. It facilitates history-augmented control synthesis through LQR in the lifted space.

For block-oriented polynomial systems, exact finite-dimensional Koopman embeddings can be constructed by stacking Kronecker monomials, yielding polynomial-input time-invariant lifted models, which reduce to bilinear form under no-feedthrough conditions [2507.15093].

## 5. Structure-Preserving Embeddings, Stability, and Hamiltonian Systems

Recent advances in deep learning architectures incorporate additional geometric and stability constraints. For canonical nonlinear Hamiltonian systems, embeddings are sought that preserve symplectic structure via explicit Jacobian penalties in the loss function: $(D\varphi(x))^\top J_{2m} D\varphi(x) = J_{2n}$ [2308.13835]. Cubicization (embedding via higher-degree monomials) overcomes limitations for systems with continuous Koopman spectra (e.g., nonlinear pendulum), enabling long-term stable prediction with sum-of-squares parameterization of the learned Hamiltonian to guarantee boundedness.

A key theoretical result is that convex parameterizations directly enforce stability; all Schur-stable linear operators $K$ can be parameterized without explicit inequality constraints [2110.06509].

## 6. Learning Strategies, Robustness, and Online Adaptation

Hybrid frameworks combine semidefinite programming (SDP) to determine the minimal latent dimension and memory depth of an approximate Koopman invariant subspace, followed by deep autoencoder training that refines the mapping and enables explicit coordinate reconstruction [2504.18676]. Conformal online learning (COLoKe) uses a prediction-conformity mechanism to trigger adaptive updates only when model residual exceeds a dynamically tuned threshold, thereby limiting unnecessary retraining and avoiding overfitting in streaming and nonstationary contexts [2511.12760].

Nonparametric kernel approaches (operator stochastic approximation + online compression) enable memory-efficient, consistent learning of the Koopman operator directly in RKHS, with finite-sample error bounds and asymptotic guarantees [2501.16489].

## 7. Applications and Empirical Performance

A broad range of practical applications demonstrate the utility of Koopman embeddings:

- **Modal analysis and coherent structure identification** in fluid flows via DMD/EDMD [2102.12086].
- **Prediction and control of nonlinear oscillators and hybrid systems**, including Duffing, Van der Pol, Lotka–Volterra, and robotic locomotion [2102.12086, 2507.14455].
- **Networked dynamical systems and neural networks** via graph-message-passing Koopman encoders for global linearization [2305.09060].
- **Trajectory planning for robotic manipulators in dynamic environments**: deep Koopman embeddings enable real-time, multi-step collision-aware rollout in motion planners [2507.03878].
- **Structure-preserving embeddings for Hamiltonian PDEs**: cubicized autoencoders outperform linear/quadratic embeddings in long-term simulation accuracy [2308.13835].
- **Quantum algorithms for differential equations**: Koopman–von Neumann embedding maps suitable linear ODEs into Schrödinger-type equations for efficient quantum simulation, subject to strict spectral conditions [2311.15628].
- **Generalized multi-task representations**: Koopman embeddings capture reusable dynamical structure that transfers across prediction and control tasks, as demonstrated in Lorenz system experiments [2508.18954].

Across these domains, empirical studies show that modern Koopman embedding methods yield substantial gains in prediction accuracy, data efficiency, and robustness compared to classical linearization and baseline neural methods [2504.18676, 2511.12760, 2305.09060, 2508.18954].

## 8. Open Challenges, Limitations, and Future Directions

While progress is rapid, central challenges remain. These include:

- **Existence and scope:** Finite-dimensional, globally one-to-one continuous embeddings are generically impossible for nonlinear systems with multiple basins of attraction; the collapse of omega-limit sets in learned embeddings is now rigorously established [2312.17045]. This motivates chart-based or discontinuous (piecewise) embeddings and greater use of infinite-dimensional dictionaries (kernel, Hilbert-space methods).
- **Embedding selection:** The quality of dictionaries/basis functions critically affects the fidelity and generalizability of EDMD/Koopman AEs—overfitting and regularization remain active areas of study [2102.12086, 2306.05224].
- **Spectral and control-theoretic extension:** The design of embeddings for input–affine, hybrid, or high-dimensional systems requires further advances in parameter-varying and bilinear formulations, as well as integration with robust control synthesis [2207.12132, 2507.15093].
- **Noise resilience and scalability:** Deep methods for denoising and high-dimensional PDEs, automated hyperparameter tuning, and robust online updating are under development [2308.13835, 2511.12760, 2501.16489].
- **Quantum and computational complexity:** Understanding the spectral conditions and complexity theory underlying Koopman–von Neumann embeddings for classical-to-quantum mapping opens new domains for computational dynamical systems [2311.15628].

The field continues to advance rapidly on the interplay of operator theory, machine learning, nonlinear dynamics, and computational geometry, with Koopman embeddings at the interface of theoretical insight and practical modeling power.

Source: https://www.emergentmind.com/topics/koopman-embeddings