Neural Koopman Embeddings

Updated 11 March 2026

Neural Koopman embeddings are a framework that leverages deep neural networks to lift and linearize nonlinear dynamical systems in a latent space.
The approach integrates methods like autoencoders, graph neural networks, and time-delay embeddings to achieve precise system identification, prediction, and control.
Applications span multimodal data fusion, reduced-order modeling, and nonlinear control, demonstrating significant performance and stability improvements.

Neural Koopman embeddings constitute a framework for learning finite-dimensional, often deep neural, coordinate transformations in which nonlinear dynamical systems evolve linearly (or nearly linearly) in latent space. This paradigm leverages Koopman operator theory—in which a nonlinear system is globally linearized by lifting the dynamics to an appropriate space of observables—while employing modern neural architectures to flexibly parameterize the requisite embeddings, operators, and their composition with additional domain constraints. The resulting pipeline endows nonlinear system identification, prediction, control, and scientific inference with the computational tractability and analysis tools native to linear systems, while retaining the powerful representation learning capabilities of neural networks.

1. Koopman Operator Theory and Neural Parameterization

Koopman operator theory asserts that nonlinear dynamical systems admit an (infinite-dimensional) linear representation: for a discrete-time system $x_{k+1} = f(x_k)$ , the Koopman operator $\mathcal{K}$ acts on observables $g$ by $(\mathcal{K}g)(x) = g(f(x))$ . If a finite dictionary $\{\phi_i(x)\}$ spans a Koopman-invariant subspace, then the evolution in the lifted space $z_k = \phi(x_k)$ obeys $z_{k+1} = K z_k$ for a finite matrix $K$ .

Neural Koopman embeddings replace the fixed dictionary with a neural encoder $\phi_\theta$ , possibly concatenated with the identity on $x$ , and jointly learn both the embedding and a finite $K$ . Autoencoders ( $x \leftrightarrow z$ ) are often employed for invertibility, and the linearity constraint is enforced by multi-step prediction losses, with additional regularization and inductive bias tailored to system class (e.g., stability, symplecticity, compositional structure).

Notably, recent architectures generalize $K$ to depend on conditioning variables, auxiliary tasks, or nonlinear maps (e.g., $U_\psi$ in operator-driven fusion), while maintaining the defining property that the latent evolution is, to varying approximation, governed by a Koopman-type operator (Mazumder et al., 22 Aug 2025, Frion et al., 2023, Fan et al., 2024).

2. Neural Koopman Embedding Architectures

2.1 Classical and Autoencoder Approaches

Early neural Koopman embeddings employ autoencoder architectures with encoders (MLPs or convolutional nets) $\phi_\theta : x \to z$ , decoders $\psi_\eta: z \to x$ , and a learned matrix or operator $K$ such that $z_{k+1} \approx K z_k$ and $x_{k+1} \approx \psi(K^t \phi(x_0))$ for long-horizon prediction (Frion et al., 2023, Uchida et al., 2022, Aswani et al., 2024). Reconstruction and latent linearity/prediction losses jointly tune these components.

2.2 Graph Neural, Attention, and Multimodal Extensions

Neural Koopman embeddings have been applied to compositional physical systems, graph-structured data, and multimodal biological settings. For example, "NeuroKoop" (Mazumder et al., 22 Aug 2025) fuses structural and functional connectome graphs via two GCN encoders, bi-directional cross-modal attention, and a neural Koopman operator $U_\psi$ , unrolling the latent state for several steps, with domain-specific modulation (working memory scores) and adversarial alignment regularizers. Message-passing architectures (KMPNN (Yeh et al., 2023)) lift node and edge features via deep GNNs, followed by a diagonal or block-structured linear Koopman operator and a GNN decoder, optimizing for multi-step prediction and invariant subspace learning.

2.3 Data-Driven Dictionary and Time-Delay Embeddings

Dynamic Mode Decomposition (DMD) and its neural extensions (e.g., DMD-residual autoencoders (Pan et al., 2019), hybrid SDP-initialized AEs (Estornell et al., 25 Apr 2025)) use explicit or learned dictionaries of observables, with options to augment via time-delayed (Hankel) coordinates to ensure approximate closure and improve the accuracy of finite-dimensional projections, especially for systems exhibiting memory (Aswani et al., 2024, Jayarathne et al., 2023).

2.4 Structure-Preserving and Conditioning Variants

Certain formulations explicitly enforce physical structure (symplecticity in Hamiltonian systems (Goyal et al., 2023)) and stability (Schur-stable or contraction-based parameterizations (Fan et al., 2024), Lyapunov-certified bilinear systems (Zinage et al., 2022)). Extensions to control incorporate bilinear or state-dependent control terms, often via auxiliary networks (Deep Koopman with nonlinear control (Shi et al., 2022)), and can yield theoretical stabilizability guarantees.

Representative Architectures Table

Architecture Example	Encoder Type	Latent Dynamics	Key Constraints
NeuroKoop (Mazumder et al., 22 Aug 2025)	GCN (structural/functional)	MLP Koopman $U_\psi$ + modulation	Cross-modal attention, adversarial, cognitive covariates
KMPNN (Yeh et al., 2023)	MPNN (graph message passing)	Diagonal/complex $K$	Reconstruction, linearity, prediction loss
Hybrid SDP-AE (Estornell et al., 25 Apr 2025)	MLP, Hankel input (SDP order)	Linear $K$ (HAVOK)	SDP-based selection, eigenvalue nets
Symplectic AE (Goyal et al., 2023)	MLP (with symplectic penalty)	Linear or cubic Hamiltonian	Symplecticity, bounded stability

3. Training Objectives and Loss Functions

Neural Koopman embedding training objectives combine:

Reconstruction loss: $\mathcal{L}_{rec} = \|x - \psi(\phi(x))\|^2$ , ensuring invertibility and manifold preservation (Frion et al., 2023, Estornell et al., 25 Apr 2025).
Latent linearity and multi-step prediction loss: $\mathcal{L}_{lin} = \| \phi(x_{t+1}) - K \phi(x_t) \|^2$ and its multi-horizon or forward-decoding variants.
Adversarial regularization: e.g., to align fused and real embeddings in multimodal settings (Mazumder et al., 22 Aug 2025).
Physical structure and stability constraints: e.g., symplectic loss for Hamiltonian systems (Goyal et al., 2023), contraction/stability via parametric $K$ (Fan et al., 2024), Lyapunov constraints (Zinage et al., 2022).
Task losses: classification (e.g., prenatal exposure (Mazumder et al., 22 Aug 2025)), control-tracking (Uchida et al., 2022), or variational data assimilation (Frion et al., 2023).

Optimization employs Adam-type solvers, sometimes in staged protocols (SDP-initialized, pre-training, or alternating projections), with hyperparameters determined by validation (embedding dimension, memory/window size, regularization weights).

4. Empirical Performance, Guarantees, and Ablations

Neural Koopman embeddings exhibit state-of-the-art accuracy for nonlinear prediction and control across physical, biological, and engineering domains:

Classification: NeuroKoop achieves $82.33 \pm 0.02 \%$ accuracy in fusing adolescent connectomes, a 5% absolute gain over the best baseline. Koopman layer ablation shows a dramatic accuracy drop (Mazumder et al., 22 Aug 2025).
Prediction: Hybrid SDP-AE methods outperform pure neural autoencoders by order-of-magnitude improvements in 1-step MSE and accelerate convergence (Estornell et al., 25 Apr 2025). Message-passing implementations achieve $10^2$ – $10^4\times$ lower network-dynamics error than classical or generic autoencoder baselines (Yeh et al., 2023).
Control: Model predictive and LQR-based policies trained on Koopman embeddings provide robust and sample-efficient tracking even under strong parametric uncertainty and disturbances, with performance gains of $>87\%$ RMSE reduction demonstrated in coupled-oscillator, manipulation, and robotic benchmarks (Singh et al., 2024, Zinage et al., 2022, Uchida et al., 2022).
Stability: Explicitly stable or stabilizable parameterizations guarantee boundedness and/or contraction in the latent space, overcoming fragility of unconstrained deep Koopman approaches (Fan et al., 2024, Goyal et al., 2023, Zinage et al., 2022).

Ablation studies consistently show cross-modal attention, dynamic latent evolution, and physically congruent regularization as critical design elements (Mazumder et al., 22 Aug 2025, Estornell et al., 25 Apr 2025).

5. Applications: Multimodal Fusion, Scientific Inference, and Control

Neural Koopman embeddings have enabled a range of advanced scientific and engineering tasks:

Multimodal data fusion: Integration of structural and functional brain connectivity via cross-modal attention and Koopman latent flow, enabling classification and mechanistic insight into neurodevelopmental effects (Mazumder et al., 22 Aug 2025).
Reduced-order modeling: Hybrid frameworks produce compact, data-driven surrogates for high-dimensional physical and biological systems, with precise modal decomposition and uncertainty quantification via variational inference (Estornell et al., 25 Apr 2025, Pan et al., 2019).
Nonlinear control and robotics: Bilinear or adaptive neural Koopman controllers extend efficient linear control synthesis (MPC, LQR) to complex nonlinear plants, tracking targets under model mismatch and online disturbances (Singh et al., 2024, Shi et al., 2022, Zinage et al., 2022).
Scientific discovery and interpretability: Structure-preserving embeddings (e.g., symplectic, Hamiltonian) yield interpretable latent models with guaranteed stability, supporting inverse design and principled prediction in domains such as fluid dynamics or neural development (Goyal et al., 2023, Frion et al., 2023).

6. Limitations, Extensions, and Open Challenges

While neural Koopman embeddings advance the data-driven identification and control of nonlinear systems, several theoretical and practical challenges remain:

Finite-dimensional closure: Many dynamical systems (multi-attractor, chaotic) lack exact invariant finite-dimensional Koopman subspaces, requiring approximate or time-delay-augmented embeddings (Aswani et al., 2024).
Curse of dimensionality: Choice of embedding dimension and memory/order must balance fidelity and computational tractability; overparameterization may yield intractable SVDs or spurious modes (Estornell et al., 25 Apr 2025).
Stability and generalization: Without explicit structural constraints, neural embeddings may fail outside training distribution or destabilize under feedback (Fan et al., 2024, Uchida et al., 2022). Control-aware training and regularization mitigate but do not eliminate this issue.
Interpretability: While latent flows may align with Koopman modes, true diagonalization and eigenfunction extraction is limited except in special cases; direct physical meaning of neural coordinates is often elusive (Mazumder et al., 22 Aug 2025, Jayarathne et al., 2023).
Extension to complex architectures: Generalization to convolutional, recurrent, or spatio-temporal graph networks is an active area, with block-Toeplitz or delay-embedding techniques proposed but not universally validated (Aswani et al., 2024).
Online and adaptive learning: Recent conformal and adaptive schemes selectively update embeddings and operators in response to model error or system drift, balancing tracking accuracy with avoidance of catastrophic forgetting (Gao et al., 16 Nov 2025, Singh et al., 2024).

7. Outlook

Neural Koopman embeddings synthesize the analytic tractability of operator-theoretic methods with the representational power of deep learning, forming a foundation for interpretable, robust, and generalizable nonlinear system identification, prediction, data assimilation, and control. Principal directions for ongoing research include principled basis selection, scalable training for high-dimensional and multi-modal datasets, rigorous uncertainty quantification, and integration of strong priors and structure—physical, causal, or cognitive—directly into the neural-encoded Koopman framework.

References: (Mazumder et al., 22 Aug 2025, Uchida et al., 2022, Aswani et al., 2024, Estornell et al., 25 Apr 2025, Pan et al., 2019, Singh et al., 2024, Jayarathne et al., 2023, Frion et al., 2023, Gao et al., 16 Nov 2025, Gallos et al., 2023, Goyal et al., 2023, Li et al., 2019, Uchida et al., 2023, Fan et al., 2024, Yeh et al., 2023, Zinage et al., 2022, Shi et al., 2022).