Deep Koopman Model

• Deep Koopman Models are data-driven dynamical systems that lift nonlinear state variables into a latent space where a linear operator governs the dynamics.
• They integrate encoder–linear propagator–decoder architectures to enable efficient prediction, estimation, and control across diverse applications such as robotics and manufacturing.
• Training involves reconstruction, linearity, and multi-step consistency losses, though challenges remain in achieving exact global linearization for complex nonlinear systems.

A deep Koopman model is a data-driven dynamical model that learns a nonlinear lifting of measured variables into a latent space in which the dynamics are represented by a finite-dimensional linear operator, often of the form $z_{k+1}=Az_k+Bu_k$, and then maps latent predictions back to physical variables for prediction, estimation, or control. In the literature covered here, this idea appears as encoder–linear-propagator–decoder architectures, state-augmented lifted models with direct linear readout, probabilistic and stochastic variants, and control-oriented formulations for MPC, LQR, safety filtering, and online adaptation (Morton et al., 2019, Iacob et al., 2021, Shi et al., 2022, Forootani et al., 4 Aug 2025).

1. Mathematical formulation

The common starting point is a discrete-time nonlinear system

$x_{k+1}=f(x_k,u_k),$

or, in autonomous settings, $x_{k+1}=T(x_k)$. Koopman theory shifts attention from the state itself to observables, so that a nonlinear transition in $x$ is represented by linear evolution of lifted coordinates. In practical deep Koopman formulations, the lifting is learned from data rather than fixed a priori, and the learned latent state is propagated by a matrix $A$, sometimes with an input matrix $B$, bilinear operators $H_i$, or time-varying operators $K_t$ (El-Hussieny, 19 Aug 2025, Abtahi et al., 4 Mar 2025, Shi et al., 2022, Chen et al., 2023).

A standard controlled formulation writes

$z_k=\phi_x(x_k), \qquad z_{k+1}=Az_k+Bu_k,$

where $\phi_x$ is a neural encoder. In autoencoder-style models, the predicted physical state is recovered by a decoder,

$x_{k+1}=f(x_k,u_k),$0

Other variants avoid a nonlinear decoder by augmenting the lifted state with the original state itself, for example

$x_{k+1}=f(x_k,u_k),$1

so that physical variables remain an explicit subspace of the latent representation (Abtahi et al., 4 Mar 2025, Abtahi et al., 16 Jul 2025, Gao et al., 16 Nov 2025).

The phrase “linearization” in this literature therefore refers to a learned coordinate transform, not to local Taylor linearization around a single operating point. This suggests a broader operating range than classical local models, but it does not imply exact finite-dimensional closure for arbitrary nonlinear systems. The approximation quality depends on the existence, or learnability, of a finite-dimensional approximately invariant observable subspace (Iacob et al., 2021, Slote et al., 7 May 2026).

2. Architectural forms

Deep Koopman models are not a single architecture but a family of related constructions. The main differences concern how observables are parameterized, how control enters latent dynamics, and whether state reconstruction is nonlinear, linear, or implicit.

Variant	Latent dynamics	Representative papers
Autoencoder Koopman	$x_{k+1}=f(x_k,u_k),$2, decoded by neural $x_{k+1}=f(x_k,u_k),$3	(El-Hussieny, 19 Aug 2025, Lv et al., 1 May 2025, Morton et al., 2019)
State-augmented Koopman	$x_{k+1}=f(x_k,u_k),$4, readout $x_{k+1}=f(x_k,u_k),$5	(Abtahi et al., 4 Mar 2025, Abtahi et al., 16 Jul 2025, Gao et al., 16 Nov 2025)
Nonlinear-control Koopman	$x_{k+1}=f(x_k,u_k),$6 or $x_{k+1}=f(x_k,u_k),$7	(Shi et al., 2022)
Bilinear Koopman	$x_{k+1}=f(x_k,u_k),$8	(Abtahi et al., 16 Jul 2025)
Probabilistic or stochastic Koopman	latent distributions or control-conditioned operators	(Morton et al., 2019, Garmaev et al., 2023, Chen et al., 2023)
Structured propagator Koopman	spectrally constrained $x_{k+1}=f(x_k,u_k),$9, Transformer or KAN encoder	(Forootani et al., 4 Aug 2025, Slote et al., 7 May 2026)

In concrete implementations, latent dimensions vary substantially with task and architecture. Reported choices include 5 for battery degradation, 8 for quadrotor DK-MPC, 12 for soft-robot control, 26 and 30 for pasteurization-unit Koopman models, 60 for Frenet-frame vehicle MDK-Net, and 66 for the deep bilinear vehicle model (Garmaev et al., 2023, El-Hussieny, 19 Aug 2025, Lv et al., 1 May 2025, Valábek et al., 6 Nov 2025, Abtahi et al., 4 Mar 2025, Abtahi et al., 16 Jul 2025).

Several recent architectures depart from the original MLP autoencoder pattern. DeepKoopFormer uses a Transformer encoder, a spectrally constrained linear Koopman propagator, and a linear decoder, with an orthogonal–diagonal–orthogonal parameterization and Lyapunov-style regularization (Forootani et al., 4 Aug 2025). Deep-Koopman-KANDy replaces encoder and decoder with two-layer Kolmogorov–Arnold Networks and then performs post-hoc symbolic readout of learned observables (Slote et al., 7 May 2026). COLoKe uses an identity-augmented lifting $x_{k+1}=T(x_k)$0 and updates the embedding online only when a conformal-style multi-step consistency score exceeds a calibrated threshold (Gao et al., 16 Nov 2025).

3. Identification and training objectives

The training objective in most deep Koopman models combines latent linearity with state fidelity. A canonical example is the quadrotor DK-MPC model, which uses a reconstruction loss, a linear dynamics loss, a stability loss based on the spectral radius of $x_{k+1}=T(x_k)$1, and $x_{k+1}=T(x_k)$2 weight regularization: $x_{k+1}=T(x_k)$3 Its reported weights are $x_{k+1}=T(x_k)$4, $x_{k+1}=T(x_k)$5, $x_{k+1}=T(x_k)$6, and $x_{k+1}=T(x_k)$7, with Min–Max scaling to $x_{k+1}=T(x_k)$8, Adam, learning rate $x_{k+1}=T(x_k)$9, batch size 32, and 50 epochs (El-Hussieny, 19 Aug 2025).

A central development in later work is explicit multi-step consistency. MDK-Net adds a discounted multi-step loss over an 80-step training trajectory, together with a spectral penalty

$x$0

and uses $x$1 for single-step, multi-step, stability, and regularization terms (Abtahi et al., 4 Mar 2025). The deep bilinear Frenet-frame vehicle model adopts the same logic, with recursive multi-step rollout in lifted space and a separate regularizer on the bilinear matrices $x$2 (Abtahi et al., 16 Jul 2025). The nonlinear system identification framework in deepSI also uses multi-step simulation error on data subsections, explicitly to make long-horizon prediction errors computable in parallel during batch optimization (Iacob et al., 2021, Iacob et al., 13 Jul 2025).

Probabilistic variants modify the training criterion accordingly. The Deep Variational Koopman model learns distributions over Koopman observations and optimizes an ELBO with reconstruction and KL terms, yielding a distribution over linear latent dynamics models rather than a single $x$3 pair (Morton et al., 2019). The stochastic deep Koopman framework for multistage manufacturing uses VAEs to represent latent quality indicators as Gaussian random variables and propagates their means and log-variances linearly through stagewise Koopman matrices (Chen et al., 2023). Deep Probabilistic Koopman dispenses with latent time stepping entirely and learns time-varying distribution parameters from explicit sinusoidal Koopman coordinates, training by negative log-likelihood for Gaussian, Gamma, skew-normal, categorical, or Poisson outputs (Mallen et al., 2021).

This diversity of objectives indicates that “deep Koopman training” is best understood as a family of constraints enforcing latent linear evolution, rather than a single standardized loss. Reconstruction is optional, multi-step rollout is increasingly prominent, and stability regularization ranges from spectral-radius penalties to strict norm constraints and Lyapunov-style energy penalties (Forootani et al., 4 Aug 2025, Abtahi et al., 16 Jul 2025).

4. Control, estimation, and optimization

The principal control appeal of deep Koopman models is that nonlinear prediction is displaced into the encoder, while the optimizer sees a linear or linearized latent model. For MPC, the usual structure is

$x$4

with the measured state and the reference both encoded into the latent space, a quadratic stage cost, and box constraints on the physical inputs. This yields a convex QP whenever the latent dynamics are linear and the constraints remain affine (El-Hussieny, 19 Aug 2025, Lv et al., 1 May 2025, Abtahi et al., 4 Mar 2025).

This pattern appears across multiple domains. The quadrotor DK-MPC framework embeds the current state and reference trajectory, solves a QP in latent coordinates, and applies only the first control input in receding horizon fashion (El-Hussieny, 19 Aug 2025). The soft-robot Chordata controller uses a learned latent model $x$5 with $x$6 and $x$7, enabling convex MPC for a multi-segment soft body whose true dynamics are described as high dimensional, nonlinear, time-varying, and of effectively infinite degree of freedom (Lv et al., 1 May 2025). The Frenet-frame vehicle models separate control inputs from exogenous road curvature and use Koopman-space tracking MPC with known curvature over the horizon (Abtahi et al., 4 Mar 2025, Abtahi et al., 16 Jul 2025).

Some control-oriented models go beyond control-affine latent dynamics. The deep nonlinear-control Koopman framework introduces an auxiliary control network $x$8 and considers three cases: $x$9, $A$0, and $A$1. LQR is then designed in the latent space, with the control recovered directly, through a state-dependent inverse, or—when no convenient inverse exists—not used for control at all in the fully nonlinear case (Shi et al., 2022). The deep bilinear Koopman vehicle model preserves convexity by freezing the lifted state in bilinear terms over each horizon and converting the MPC problem back into a QP (Abtahi et al., 16 Jul 2025). The deep Koopman safety command governor for autonomous vehicles combines a learned linear latent model with control barrier functions, so that state safety conditions become linear inequalities in the input and the online problem remains a QP (Chen et al., 2024).

Estimation has also become part of the deep Koopman toolkit. One recent estimator combines an EDMD-based Koopman core with a DDPG actor–critic that learns nonlinear correction terms in the lifted dynamics, and then transfers the estimator across diffeomorphic coordinate transformations via learned linear maps $A$2 and $A$3 (Sun et al., 2024). Another line of work adds innovation noise explicitly to the Koopman model,

$A$4

and uses deep state-space encoders plus multiple-shooting prediction error to obtain statistically consistent identification under general process and measurement noise (Iacob et al., 13 Jul 2025).

5. Applications and empirical record

Deep Koopman models have been reported in aerial robotics, ground vehicles, soft robotics, degradation modeling, multistage manufacturing, forecasting, thermal process control, and online estimation. The empirical pattern across these domains is consistent: relative to local linear models, hand-crafted Koopman dictionaries, or conventional nonlinear baselines, learned latent observables often improve long-horizon prediction and preserve optimization structure (El-Hussieny, 19 Aug 2025, Abtahi et al., 4 Mar 2025, Lv et al., 1 May 2025, Valábek et al., 6 Nov 2025).

In quadrotor control, DK-MPC achieved $A$5 for all horizons $A$6, with computation time consistently below 5 ms per step, while nonlinear MPC showed negative $A$7 for short horizons and nearly 60 ms at $A$8, which the paper notes can be problematic for 100 Hz control (El-Hussieny, 19 Aug 2025). In multi-segment soft-robot control, DK-MPC reached average tracking errors of 2.79 mm on an “O” trajectory and 3.11 mm averaged over “O”, “T”, “H”, and “U”, whereas the RBF-based Koopman baseline produced 22.03 mm on “O” and 22.49 mm on average (Lv et al., 1 May 2025).

In road-vehicle modeling, MDK-Net in the Frenet frame reduced 80-step open-loop MSE by orders of magnitude relative to an identified LTI model, for example $A$9 MSE 2.908 versus 6937.684 and $B$0 MSE 0.761 versus 11.242 on the reported test set (Abtahi et al., 4 Mar 2025). The deep bilinear Frenet-frame model further improved open-loop RMSE relative to both EDMDK and a linear deep Koopman model, and in hardware-in-the-loop closed-loop control the proposed CER-MDBK-MPC achieved 33% improvement over CER-MDK-MPC, 38% over EDMDK-MPC, and 68% over LTI-MPC, with task turnaround time about 0.58 ms at a 25 ms sampling period (Abtahi et al., 16 Jul 2025).

In degradation and reliability applications, the Deep Koopman Operator and Koopman-Inspired Degradation Model yielded strong RUL performance under varying operating conditions. For varying-current batteries, KIDM+LR achieved MSE $B$1, MAE $B$2, and MAPE $B$3, versus AE+LR at MSE $B$4, MAE $B$5, and MAPE $B$6 (Garmaev et al., 2023). In multistage manufacturing, the stochastic deep Koopman model reported total MSE $B$7, outperforming E-AEK at $B$8, PGAT at $B$9, and RF at $H_i$0 on the MCMP dataset (Chen et al., 2023).

In forecasting and process optimization, DeepKoopFormer is reported to consistently outperform standard LSTM and baseline Transformer models in accuracy, robustness to noise, and long-term forecasting stability across synthetic systems, climate data, cryptocurrency, and electricity generation (Forootani et al., 4 Aug 2025). Deep Probabilistic Koopman achieved a mean relative improvement over the vanilla benchmark of $H_i$1 in GEFCom 2017 electricity demand forecasting and outperformed all 177 competitors listed there (Mallen et al., 2021). In a laboratory pasteurization unit, a deep Koopman model with linear projection achieved a 45% improvement in open-loop prediction accuracy over N4SID, and the resulting Koopman EMPC reduced total economic cost by 32% and steady-state electrical energy by 10.2% relative to the N4SID EMPC baseline (Valábek et al., 6 Nov 2025).

6. Limitations, interpretability, and future directions

A persistent misconception is that deep Koopman models provide exact global linearizations of arbitrary nonlinear systems. The literature reviewed here does not support that claim. It supports learned finite-dimensional approximations whose quality depends on data coverage, architectural choice, and whether a useful approximately invariant subspace exists at all (Iacob et al., 2021, Iacob et al., 13 Jul 2025). Several papers state this directly: quadrotor and soft-robot performance may degrade outside the training distribution; robustness to strong disturbances or unseen maneuvers remains future work; and a single global time-invariant $H_i$2 may be inadequate for regime switches or pronounced nonstationarity (El-Hussieny, 19 Aug 2025, Lv et al., 1 May 2025, Forootani et al., 4 Aug 2025).

Another limitation concerns control inputs. Many Koopman models assume $H_i$3, but this is often insufficient when the original dynamics are nonlinear in control. Recent work therefore introduces state-dependent or fully nonlinear control encoders, bilinear lifted dynamics, or control-conditioned operator sequences. This suggests that the distinction between “linear in latent state” and “linear in physical input” is central, and that control design may require invertibility or horizon-wise linearization that is absent from simpler formulations (Shi et al., 2022, Abtahi et al., 16 Jul 2025, Garmaev et al., 2023).

Interpretability is also mixed. State-augmented lifted coordinates with $H_i$4 keep physical variables explicit and eliminate the need for a nonlinear decoder (Abtahi et al., 4 Mar 2025, Abtahi et al., 16 Jul 2025). Spectral penalties, eigenvalue inspection, and Lyapunov regularization give some operator-level interpretability (Forootani et al., 4 Aug 2025, Garmaev et al., 2023). Deep-Koopman-KANDy goes further by performing post-hoc symbolic dictionary readout; on Lorenz it recovered $H_i$5 with perfect recall and Jaccard score $H_i$6, whereas on the Arnold cat map it failed to find sparse closure, which the paper treats as the expected outcome because finite-dimensional Koopman closure is provably impossible there (Slote et al., 7 May 2026). This suggests that deep Koopman models can also serve as diagnostic tools: failure to find a stable sparse latent closure may reflect a property of the dynamics rather than merely a training defect.

Current research directions in the supplied literature include online conformal updating of Koopman embeddings from streaming data (Gao et al., 16 Nov 2025), disturbance-aware and innovation-form identification under general noise (Iacob et al., 13 Jul 2025), richer structured propagators and stability constraints (Forootani et al., 4 Aug 2025), explicit disturbance robustness and agile-flight extensions for robotics (El-Hussieny, 19 Aug 2025), and broader transfer across operating conditions, coordinate systems, and control regimes (Sun et al., 2024). Across these developments, the core theme remains unchanged: a deep Koopman model is not a single algorithm, but a modeling principle that seeks linear evolution in a learned observable space while preserving enough structure for prediction, estimation, or optimization.