Deep Koopman Autoencoders

• Deep Koopman autoencoders are models that learn a nonlinear mapping of system states into a latent space where dynamics are approximated as linear via the Koopman operator.
• They integrate encoder–decoder architectures with latent linearity, combining reconstruction, prediction, and operator estimation in both discrete and continuous time.
• Variants extend the framework for control-oriented applications, spectral regularization, and robust long-term forecasting in diverse, complex systems.

Searching arXiv for recent and foundational papers on deep Koopman autoencoders and closely related variants. Deep Koopman autoencoders are encoder–operator–decoder architectures that learn a nonlinear lifting of a dynamical system into latent coordinates in which evolution is approximately linear under a finite-dimensional Koopman operator, then map latent rollouts back to the original observation or state space. In representative formulations, the encoder maps $x$ to $y=g(x)$, the latent dynamics follow either $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$ or a continuous-time generator $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$, and the decoder implements $\mathbf{x}=g^{-1}(\mathbf{y})$ (Dey et al., 2022). Closely related formulations use learned embeddings together with EDMD in latent space, or replace a fixed discrete latent map with continuous-time latent integration and matrix-exponential rollout, but the common objective remains the same: to replace nonlinear state evolution by a simpler latent linear dynamics model without discarding reconstructive or predictive fidelity (Alford-Lago et al., 2021, Grozavescu et al., 2 Feb 2026).

1. Canonical architecture and operator-theoretic formulation

A standard deep Koopman autoencoder learns a nonlinear coordinate transformation into a latent space where the Koopman action is approximately closed. In DLKoopman, the encoder and decoder are MLPs, the latent dimension is user-specified by encoded_size, and the learned latent dynamics are either evolved by a Koopman matrix estimated from encoded data or by a linear neural layer, depending on the prediction mode (Dey et al., 2022). In DLDMD, the encoder $\mathcal{E}$ is explicitly interpreted as a learned observable map $\mathcal{E}:\mathbb{R}^{N_s}\to\mathbb{R}^{N_o}$, and the latent space can be higher-dimensional than the original state space because the goal is lifting rather than compression (Alford-Lago et al., 2021).

The same basic pattern appears in continuous-time variants. One formulation writes the latent dynamics as

$$\dot{\mathbf{z}}=\mathbf{K}_{\mathrm{cont}}(\phi)\,\mathbf{z},$$

so that the exact latent solution is

$\mathbf{z}_\tau=\exp(\mathbf{K}\tau)\mathbf{z}_0,$

and decoding yields forecasts at arbitrary physical times rather than only at the training step size (Grozavescu et al., 2 Feb 2026). A probabilistic continuous-time formulation similarly imposes

$\dot{\mathbf{\Phi}}=\mathbf{\Phi}\mathbf{K},$

but augments the latent operator with a stability-guaranteeing parameterization and Bayesian uncertainty quantification (Pan et al., 2019).

This standard form should not be conflated with a purely compressive autoencoder. Several formulations emphasize that the learned latent variables are intended to behave as Koopman observables or approximate Koopman eigenfunction coordinates, so latent dimension, spectrum, and rollout stability are structural design variables rather than incidental neural-network hyperparameters (Alford-Lago et al., 2021).

2. Learning objectives and computational realizations

Training objectives in deep Koopman autoencoders typically combine reconstruction, latent linearity, and prediction. DLKoopman states the aggregate loss as

$y=g(x)$0

with all three primary losses computed using MSE, and supports both a snapshot-based StatePred mode and a trajectory-based TrajPred mode (Dey et al., 2022). DLDMD adds an explicit EDMD residual term,

$y=g(x)$1

thereby coupling invertibility of the encoder–decoder pair with one-step Koopman consistency and multi-step state-space prediction (Alford-Lago et al., 2021).

The computational realization of the latent operator varies. DLKoopman’s StatePred computes

$y=g(x)$2

from encoded snapshots, followed by eigendecomposition-based evolution, while TrajPred represents the Koopman matrix directly as a linear neural network layer with equal input/output size, no bias, and no activation (Dey et al., 2022). DLDMD instead runs EDMD on latent snapshots,

$y=g(x)$3

then diagonalizes $y=g(x)$4 to recover spectra, modes, and forecasts (Alford-Lago et al., 2021).

Identification-oriented formulations shift emphasis from reconstruction toward simulation error. In deep state-space encoders for Koopman-form system identification, the latent model

$y=g(x)$5

is trained by minimizing multi-step prediction error on trajectory subsections, enabling efficient batch optimization without full-horizon backpropagation through entire datasets (Iacob et al., 2021). This suggests two persistent lines of KAE development: one prioritizes reusable latent forecasting modules, and the other treats the encoder and latent operator as identification components of a state-space model.

3. Control-oriented and task-relevant formulations

A major branch of deep Koopman autoencoder research is explicitly control-oriented. DeepKoCo learns a Koopman-based latent representation from images, but does so with a tailored lossy autoencoder that reconstructs and predicts observed scalar costs rather than pixels. Its image pipeline stacks the last $y=g(x)$6 frames, maps them through a convolutional encoder $y=g(x)$7, forms a latent state from the observation–action pair, and imposes linear latent evolution so that planning can be carried out with linear MPC rather than sample-based nonlinear MPC (Heijden et al., 2020). The defining design choice is that the representation is trained to predict costs, not to reproduce all observed dynamics, which the paper links to robustness against distractor dynamics.

For control-affine systems, several papers replace a purely linear latent propagator with bilinear or input-affine structure. tcBLRAN models

$y=g(x)$8

then discretizes this into a bilinearly recurrent autoencoder whose latent evolution is linear in $y=g(x)$9, affine in the control $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$0, and bilinear overall (Chakrabarti et al., 24 Mar 2025). A related Wiener-type Koopman formulation for MIMO input-affine systems instead keeps the latent dynamics linear time-invariant,

$\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$1

while delegating plant nonlinearity to a nonlinear decoder $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$2, yielding a control-oriented surrogate with a Wiener block structure (Schulze et al., 2022).

These control-centric variants preserve the core KAE principle but modify what is meant by a “useful” latent linearization. In autonomous forecasting, linearity is often judged by predictive accuracy alone. In DeepKoCo, linearity is valuable because it enables linear MPC with only a single evaluation of the neural networks per control step; in bilinear and Wiener-type formulations, linear or bilinear latent structure is valuable because it aligns with model reduction and control synthesis requirements rather than only one-step forecasting (Heijden et al., 2020, Chakrabarti et al., 24 Mar 2025).

4. Spectral structure, consistency, and robustness

A recurring theme in the literature is that latent linearity alone is insufficient for robust long-horizon behavior. Temporally consistent Koopman autoencoders introduce a latent consistency regularizer based on the statement that a latent function space is Koopman invariant if and only if there exists $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$3 such that

$\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$4

for all $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$5 and $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$6. The resulting temporal consistency loss compares predictions against one another across horizons rather than only against labels, and is reported to improve long-term forecasting especially under limited and noisy data (Nayak et al., 2024). The control-affine tcBLRAN extends the same principle to bilinear latent dynamics, enforcing agreement between predictions that arrive at the same future latent state from different rollout origins (Chakrabarti et al., 24 Mar 2025).

A distinct line of work addresses inference-time drift rather than training-time consistency. “Course Correcting Koopman Representations” argues that pure latent unrolling can produce invalid long-horizon behavior, including decoded phase-line crossings and failure to switch between basins of attraction. Its proposed Periodic Reencoding decodes and re-encodes every $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$7 steps, treating latent linearity as locally valid rather than globally reliable over arbitrarily long horizons (Fathi et al., 2023). A later variant augments the Koopman predictor with an attention-free latent memory block and dynamic re-encoding triggers such as EWMA, CUSUM, and sequential two-sample tests, with the explicit goal of keeping predictions on the learned manifold over horizons up to 1000 steps (Nagdi et al., 22 Jun 2026).

Robustness has also been pursued through spectral regularization and bias correction. “Eigenvalue initialisation and regularisation for Koopman autoencoders” proposes eigeninit, which samples initial Koopman operators from prescribed eigenvalue distributions, and eigenloss, which penalizes $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$8; on the reported datasets these schemes improve convergence rate by up to a factor of $\mathbf{y}_{i+1}=\mathbf{K}\mathbf{y}_i$9 and reduce cumulative long-term prediction error by up to a factor of $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$0 (Miller et al., 2022). “Deep Robust Koopman Learning from Noisy Data” identifies a different failure mode: noise-induced bias introduced by nonlinear lifting. It therefore learns both forward and backward latent dynamics,

$$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$1

and synthesizes a reduced-bias operator

$$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$2

for which the paper derives lower bias than the forward-only estimator under small-noise assumptions (Singh et al., 5 Jan 2026).

These developments directly address a common misconception: a successful one-step latent linearization does not by itself guarantee stable long-horizon rollouts, disentangled factors, or unbiased operator estimates. The literature instead treats spectrum design, temporal consistency, noise symmetry, and periodic projection back to the data manifold as separate, technically nontrivial problems.

5. Structured latent spaces and domain-specific extensions

Deep Koopman autoencoders have been extended beyond forecasting into representation-structuring objectives. Structured Koopman autoencoders for multifactor sequential disentanglement begin from a standard KAE with batchwise Koopman layer

$$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$3

but add a spectral regularizer that partitions the latent space into static and dynamic components. Static eigenvalues are pulled toward $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$4, while dynamic eigenvalues are pushed away from the static region, yielding a structured spectrum that supports multifactor disentanglement, factor swapping, and incremental swap operations in the Koopman eigenbasis (Berman et al., 2023). The paper is explicit that a vanilla KAE does not guarantee disentanglement even when it models dynamics well.

In manufacturing, stochastic deep Koopman models reinterpret the latent state as stagewise quality information rather than generic system state. The SDK formulation defines per-stage encoded latent quality $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$5, propagates cumulative quality by

$$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$6

and predicts stage outputs with $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$7. Its stochastic extension replaces the deterministic autoencoder with a VAE, propagating means and log-variances separately and adding stagewise KL terms to the training objective (Chen et al., 2023). This yields a stochastic Koopman autoencoder with supervised quality prediction and explicit traceability across production stages.

The framework has also been transferred to nonphysical “dynamics.” “Koopman Autoencoders Learn Neural Representation Dynamics” treats layer depth in a trained network as the time coordinate of a discrete dynamical system, learns a Koopman autoencoder surrogate for layer-to-layer representation evolution, and augments the objective with an encoder isometry term to preserve representation topology (Aswani et al., 19 May 2025). A more lightweight operator-learning extension studies adversarial regularization of Koopman autoencoders by adding a discriminator on the latent space and reports improvements on small-data ODE benchmarks such as pendulum, Lorenz, and a fluid attractor equation (Enyeart et al., 2024).

A plausible implication is that “deep Koopman autoencoder” denotes not a single architecture class but a family of latent-linearization models whose inductive bias can be repurposed for disentanglement, quality propagation, operator learning, or neural representation analysis whenever a linear latent evolution is considered structurally meaningful.

6. Empirical behavior, benchmark patterns, and limitations

Empirical results across the literature are heterogeneous but reveal recurrent patterns. DeepKoCo reports similar final performance to model-free baselines on clean pendulum swing-up, while being considerably more robust to distractor dynamics; on the manipulator tracking task, the baselines fail when the target moves, whereas DeepKoCo learns to track arbitrary circular trajectories, including from image observations (Heijden et al., 2020). DLKoopman reports a test prediction ANAE of $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$8 on the NACA0012 airfoil pressure example and supports interpolation at $$\frac{d\mathbf{y}}{di}=\mathbf{\mathcal K}\mathbf{y}$$9 and extrapolation at $\mathbf{x}=g^{-1}(\mathbf{y})$0 in StatePred mode (Dey et al., 2022). In multistage manufacturing, SDK achieves the lowest reported total test MSE, $\mathbf{x}=g^{-1}(\mathbf{y})$1, among ANN, RF, PGAT, DMMTL, S-AEK, E-AEK, and SDK (Chen et al., 2023).

Long-horizon forecasting is a particularly prominent evaluation regime. tcKAE reports, on clean pendulum data with $\mathbf{x}=g^{-1}(\mathbf{y})$2, average errors of $\mathbf{x}=g^{-1}(\mathbf{y})$3 for DAE, $\mathbf{x}=g^{-1}(\mathbf{y})$4 for cKAE, and $\mathbf{x}=g^{-1}(\mathbf{y})$5 for tcKAE, while also showing lower variance and better robustness under 40 dB and 30 dB noise (Nayak et al., 2024). An SVD-regularized consistent KAE for long-term prediction reports average prediction error over 1000 steps of $\mathbf{x}=g^{-1}(\mathbf{y})$6 for USVD-CKAE, compared with $\mathbf{x}=g^{-1}(\mathbf{y})$7 for Vanilla KAE and $\mathbf{x}=g^{-1}(\mathbf{y})$8 for CKAE, while converging around epoch 400 and using 168M FLOPS rather than the 362M of ISVD-CKAE (Choi et al., 2024). A continuous-time latent-dynamics KAE for CFD reports that a 240-step rollout takes about $\mathbf{x}=g^{-1}(\mathbf{y})$9 ms, versus about $\mathcal{E}$0 ms for ACDM, and that the same latent ODE trained at $\mathcal{E}$1 s can be queried at $\mathcal{E}$2, $\mathcal{E}$3, or $\mathcal{E}$4 s with aligned rollouts at common physical times (Grozavescu et al., 2 Feb 2026).

The literature is equally explicit about limitations. DLDMD notes that latent dimension must be tuned manually, large well-sampled datasets are often required, and prediction quality degrades for chaotic or noisy systems such as Lorenz-63 (Alford-Lago et al., 2021). tcKAE reports a smaller advantage on sea surface temperature because the method assumes strong time invariance, whereas SST is influenced by seasonal and nonstationary effects (Nayak et al., 2024). Continuous-time CFD KAEs identify spectral bias and smoothing from $\mathcal{E}$5-style reconstruction, sensitivity of training stabilization, and the strength of the global linearity assumption for shocks and highly nonlinear phenomena (Grozavescu et al., 2 Feb 2026). Noise-robust and memory-corrected variants likewise imply that latent Markovian closure is generally approximate rather than exact (Singh et al., 5 Jan 2026, Gupta et al., 2023).

Taken together, these results indicate that deep Koopman autoencoders are most effective when the learned latent space is not only approximately linear but also spectrally well-conditioned, temporally coherent, and matched to the structural demands of the application. The central research problem has therefore shifted from merely fitting a latent linear operator to managing invariance error, spectral geometry, control relevance, and out-of-manifold drift over long horizons.