Learning Koopman Representations with Controllability Guarantees
Authors: Keyan Miao
Han Wang
Xuda Ding
Konstantinos Gatsis
Andreas Krause
Antonis Papachristodoulou
Venue: ICLR 2026 Poster (2026)
Keywords: Dynamical System
Koopman Operator
Control
Controllability
Nonlinear System
Primary area: learning on time series and dynamical systems
TL;DR: Learning koopman representations of nonlinear dynamical systems with controllability guarantees

Abstract: Learning nonlinear dynamical models from data is central to control. Two fundamental challenges exist: (1) how to learn accurate models from limited data, and (2) how to ensure the learned models are suitable for control design of the nominal system. We address both by enforcing a critical \emph{a priori} property of the nominal system during learning: \emph{controllability}. Controllability guarantees the existence of control policies that can drive the learned model from any initial state to any desired state. From a modeling perspective, it captures key structural features of the nominal system, thereby improving data efficiency. For downstream control, it enables the use of modern techniques such as model predictive control (MPC). Our approach is based on controllability-preserving Koopman representation learning. Rather than learning dynamics directly in the nominal state space, we learn in a latent space where the system admits a linear representation. We prove that controllability of the learned latent model implies controllability in the nominal state space. To enforce this property, we introduce a novel canonical parameterization of the latent dynamics matrices. We further incorporate Gramian-based regularization to shape the degree of controllability, yielding well-conditioned models for control. Implemented as an end-to-end Neural ODE framework, our method learns models that are both predictive and controllable from limited data. Experiments on nonlinear benchmarks demonstrate accurate long-horizon prediction, reliable MPC performance, and substantially improved data efficiency.

Published as a conference paper at ICLR 2026
LEARNING KOOPMAN REPRESENTATIONS WITH CON- TROLLABILITY GUARANTEES
Keyan Miao Han Wang?* Xuda Ding² Andreas Krause? Antonis Papachristodoulou"
Konstantinos Gatsis
'University of Oxford, UK
2ETH Zürich, Switzerland
Bosch Central Research
VVillanova University, USA
(keyan.mi miao, antonis 0eng.. ox. ac. uk konstantinos, gwtsistviiiii... wtt.taattiooosttssieviiiii aatesiiivvlllooaaa edu
hanwang@control.er ethz.ch krausea@ethz.chh
ABSTRACT
Learning nonlinear dynamical models from data is central to control. Two funda- mental challenges exist: (I) how learn accurate models from limited data, and (2) how to ensure the learned models are suitable for control design of the nominal system. We address both by enforcing critical a priori property of the nominal system during learning: controllability. Controllability guarantees the existence of control policies that can drive the learned model from any initial state to any de- sired state. From modeling perspective, it captures key structural features of the nominal system. thereby improving data efficiency. For downstream control, it en- ables the use of modern techniques such model predictive control (MPC). Our Koopman representation learning. Rather than learning dynamics directly in the nominal state space, learn in latent space where the system admits a linear representation. We prove that con- trollability of the learned latent model implies controllability in the nominal state space. To enforee this property, we introduce novel canonical parameterization of the latent dynamics matrices. We further incorporate Gramian-baas based regular- ization to shape the degree of controllability. yielding well-conditioned models for control. Implemented as an end-to-end Neural ODE framework, our method learns models that are both predictive and controllable from limited data. Exper- iments on nonlincar benchmarks demonstrate accurate long-horizon prediction, reliable MPC performance, and substantially improved data efficiency.
approach is based on patrrollabblityypeeevvvii
INTRODUCTION
Learning dynamical models from data is crucial for control design, analysis, and verification. For linear systems, methods such ARX/ARMAX and subspace identification are well
tablished, with strong theory and efficient algorithms (Ljugg 1998). Nonlinear extensions such as NARX (Billings) 2013). Volterra models, WineerrsteinWiener structures (Billings) 1980). Gaussian processes (Kocijan 2016), and grey-box approaches remain more challeng- ing, as they often impose restrictive assumptions or scale poorly with system dimension. Deep learning-based methods, including neural state-space models (Rangapuram al. (008)), recur- rent architectures (Hochreiter & Schmidhuber] 1997] Chung et al. 2014), and Neural ODEs (Chen et al. 2018 Rahman al. 2022), offer expressive param- eterizations capable of capturing complex nonlinear dynamics. While effective for trajectory prediction, the resulting models are often ill-suited for control: their nonlinear structure hinders the application of tools such as MPC, and they rarely provide guaran- tees on critical closed-loop properties such as safety and stability.
Search Space (All Parameterization)
Admissible Models
Search Space (Controllable)
Real Model
Admissible Controllable Models
Figure Encoding priors such as controlla- bility reduces the search space
'Han Wang contributed equally this work with Keyan Miao, serving corresponding author.

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Published as a conference paper at ICLR 2026
A broader limitation of both classical and learning-basee identification is the difficulty of incorpo- rating structural priors of the nominal system. For control, one of the most critical priors is con- trollability (Klamka (963). As illustrated in Figure [T] restricting the search to controllable models greatly reduces the parameter space, thereby improving data efficiency. However, encoding control- lability during training is challenging: even verifying it for nonlinear systems requires complex rank conditions over infinitely many Lie brackets (Isidori 1985). Consequently, most learning proce- dures focus solely on trajectory fitting, leaving structural properties be checked or enforced only after training. This disconnect often produces models that predict well but are unsuitable for con- trol. While such analysis routine for linear systems, it remains a major challenge in the nonlinear setting. A promising path to bridge this gap is through the Koopman operator.
Koopman- modeling has long been explored in control theory to render nonlinear systems more amenable to analysis and feedback. The key idea is to approximate nonlinear dynamics with linear surrogates in lifted space, enabling the use of linear systems theory (Koopman 193T). Early meth- ods such as dynamic mode decomposition (DMD) and its extensions relied on pre-specified basis functions (Williams et al. 2015 Kaiser et al. 2021 Brunton et al. 2022), while recent approaches integrate representation learning via autoencoders and neural networks. The common motivation is to obtain models that interface naturally with linear controllers, particularly MPC (Korda & Mezić, 2018). The appeal of Koopman representation learning lies in combining the expressiveness of nonlinear modeling with the tractability of linear control synthesis.
Related work and Gap. Learning- -based Koopman approaches approximate nonlinear dynamics by training neural networks to construct observables, thereby lifting the system into a space where linear dynamics can be identified (Lusch et al. 2018 Yeung et al. 2019). The main benefit of lifting is that in the lifted space, traditional optimization based control methods can be easily implemented
because of the linear dynamics (Korda& & Mezić 2018
Zinage & Bakolas 2023). Existing variants
differ in how operators are estimated (Han et al. 2020 Wang et al. [021 Shi & Meng 2022 Xiao
et al., 2022) and in whether the operators are fixed or time varying (Li et al. 2025), yet the overall pipeline largely targets multi-step prediction. Structural properties, however, are seldom addressed. Controllability has been considered only rarely, typically by adding the Kalman rank condition of the lifted system as a loss (Han et al., 2020), which neither guarantees nor reflects the property. A recent paper (Choi et al.) 2024) studied controllability preservation conditions, but it is limited to theoretical analysis under assumptions of exact representation. No computational methods were proposed in that work. Other priors have been explored even less. The first attempt incorporated stabilizability through LMI-based parameterizations (Fan al., 2024). and a more recent direction embeds control inputs nonlinearly via neural networks (Guo et al. 2025), which requires post hoc Lie bracket checks and nonlinear optimization. In short, existing methods emphasize reconstruction but neither guarantee controllability nor exploit it as structural prior to reduce the search space and guide training. This gap motivates our framework, which enforces controllability by construction, yielding models that are both predictive and reliable for control. Further discussion of related work is provided in Appendix A
Our contributions. The contributions of this work are as follows:
We propose a bbased framework for learning nonlinear dynamical models within a neu- ral ODE architecture. The approach yields linear surrogate models that are both accurate for pre- diction and efficient for control. The proposed framework accommodates irregular multi rate sampling data and allows the learned continuous time dynamics to be used at control frequencies that differ from those in the training data without requiring modification or retraining.
To ensure controllability of the learned model, we introduce a canonical parameterization of the Koopman operators. This guarantees controllability by construction in both single- and multi- input settings, while preserving expressiveness through learnable similarity transforms.
To enhance control performance, we incorporate controllability Gramians into the training ob- jective to increase the degree of controllability. By shaping their spectrum, we obtain better- conditioned models that reduce control effort in downstream tasks.
We validate the framework on several nonlinear benchmarks. Experiments demonstrate improved data efficiency, higher prediction accuracy, and superior MPC performance, with greater feasibil- ity and more reliable closed-loop behavior than unstructured baselines.
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2 PRELIMINARIES
2.1 PROBLEM SETUP
We study the learning of nonlinear dynamical models. Specifically, we consider system
*(t) f=((), u(t), zxt) R", u(t) Rm
where fxx u) is unknown but locally controllable as a priori.
Definition (Controllability (Kalman et al. 1960)). The control system ① is said to have Control-
lability (or 'be controllable' for any z(fo) Io Rn and ær R", there exists continuous control signal u(.): [to,t] fy]= - R. such that
xlt,) = S." ((=(t),u(t)dd
ZT
=(t6)=xo
The task is to construct, from data, a dynamical model ((((t), utt) that accurately captures the input-state evolution ( and remains controllable for control design. This entails not only perform- ing supervised learning but also mechanism that preserves controllability in the learned model (())))
Data of (1) is sampled as time series from the initial state zg, using a continuous control signal u(-): : [0, dxl + R":
= [=) x(d1) x(22) U (u( u(d1) u(d2) u(dx)]
x(dx)
2.2
NRURAL ODES WITH INPUT
Neural Ordinary Differential Equations (ODEs) (Chen et al. 2018) extends the idea of deep residual networks (He al. 2016). It has shown great success in learning continuous time dynamical models. Definition 2 (Neural ODE with input). With h R²= IR= hy Rh R representing the input network and output network respectively, Neural ODE with imput is system of the form
(to) = hn((x(t))
1 (t) = F (t.z (t) (t), teS yt) = hyy (z(t)
where S = : [a,tt (fo,ty E IR+) is the depth domain and F is a neural network referred to as ODENet with parameter 6; u(t) the input at time
The terminal state æ(ty), obtained by solving the initial value problem (IVP), represents the evolved system state. Neural ODEs, depth corresponds to continuous evolution over time, with ResNets interpretable as Fuler discretizations. To encode structural priors into the learned model, adopt the Koopman representation framework, which introduces a linear map F(.).
2.3 KOOPMAN OPERATOR
For the nonlinear control system I the goal of Koopman-based modeling is to obtain a linear representation of the dynamics in higher-dimension. nonlinear evolution of (æ, u) is described by linear operator acting on functions (z, u). Koopman operator theory establishes the existence of infinite-dine dimensioimin operator A. acting on observables ((r, u):
space of observables. In this setting, the
d (a((), (V)((()), w(()). di
In applications, usually uses tinite-dimensional approximation for the Koopman operator on finite-dimensions function space. A standard construction is to select observables of the form:
W u)
2 ((x) E N
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RV
Linear
Az Bu
Representation
Reconstruction 2 -
Lift R
Nonlinear
f(x, u)
Representation
Figure 2: Koopman- based dynamical model representation.
Encoder
Koopman Operator (Controllable)
z(t)
Current State x(t)
A
BG
Observable Encoder the
0
2017
Control Input u(t)
Invertible
Ag PorG
Transformation P6
B, = P
Forward Prediction
x(G,)
le(t))e=t
Initial x(tn)
ODE Solver
(2()))-
Atz(1) Bou(()
Input Sequence (u(t))
Figure 3: Illustration of the Koopman based Neural ODE learning pipeline.
where @ n N with N >> is set of nonlinear lifiting functions applied to the state and the input is retained its original coordinates. Since prediction and control only require the lifted state dynamics, it suffices to retain the first N rows of this operator. This leads the following Koopman representation as an ODE:
(()) A=(() + Bu(l), #()) M(((()), =(1) ddx(())
In many applications, f is chosen to be linear, i.c., h(z) = C'z with C' RTxN, The functionality of the Koopman operator illustrated Figure with further details provided in Appendix B.
KOOPMAN LEARNING FRAMEWORK
Building on the integration of the Koopman operator with Neural ODEs, propose a Koopman- based system learning framework, illustrated in Figure 3 This end-to-end pipeline couples neural encoder. a controllable Koopman operator, and differentiable ODE solver. The goal is learn Neural ODE of the form
(tt Atz(t)= Bau(t), x(t) = (o(((t)) = (x(()) vo()(())) a(t) v(t
v(t) = Cz((),
(3)
that guarantees controllability while modeling the unknown but controllable dynamics à = f(x, u).
The input network h specified as the observable encoder dn in the "Encoder" block. To pre- serve the nominal state explicitly, the first components of are included through identity lifting
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Koopman Framework (containing identity lift)
Koopman Lincar
2
Koopman Lincar
OOC (Po Vr)
SOC (Zo yr)
Nonlinear System
SOC Tools: Rank Condition
Controllability
(xo + XT)
Canonical Form
3
Figure 4: Overview of the relations among controllability notions. ! We show that the control- lability of the original nonlinear system equivalent OOC of Koopman linear representation (in main text); ② In the Koopman linear representation, we prove that OOC coincides with SOC (Lemma aand Theorem [ proved in Appendix D 3 Based on this equivalence, we for the first time introduce SOC canonical form which can be used for OOC canonical form for Koopman representation (Theorem
proved in Appendix [): 4 Finally, we leverage this canonical form to parameterize the neural Koopman operator, which enables constructing learning-base Koopman representations of the original nonlinear system.
(o(t) E RR), while the remaining N - n entries are nonlinear observables the (x: to learned. The "Koopman Operator (Controllable)) block specifies the structure of Ag and Bo that enforces controllability of the learned Neural ODE. The "Forward Prediction" block follows standard prac- tice: given initial state zo and input signal, the ODE solver propagates the dynamics obtain the terminal state.
Finally, recover the nominal state st), define the output network hy(2)) C2 with
a(t) v() Cz((), C [4, 0),
that s:() is obtained directly the output of the Neural ODE.
3.1 CONTROLLABILITY ANALYSIS
Given that the nominal nonlinear system is controllable and the learned Neural ODE B yields the output è y Cz, ask whether the system can be driven from any initial condition any desired target output y(ty). This consideration motivates the following definition.
Definition 3 (State-to- Output Controllability (SOC)). System (3) is said to be state-to-output con- trollable on (Z C IR y S R²) if. for any initial state condition 2(to) output yr E y, there exists a continuous input signal [Lo, []] + R. such that y(l,) = yr.
E Z and a target
Definition 4 (Output-to-Outpp)
Controllability (OOC)). System 33 is said output-to-output control- lable on y C IR" if, for any initial ouiput y(to) = yo E y and target output yr Y, there exists continuous inpul signal
[Lo + Rm such that y(t,) Vr.
We are interested in OOC and SOC for system 3 because its output y recovers the state æ of original nonlinear system. If (3) is OOC, then the corresponding nonlinear system is controllable in the sense of Definition T However, verifying OOC is generally more difficult than SOC, which can be checked using the Kalman rank condition, no comparably simple algebraic test exists for OOC (Danhane et al. 2023). Within our Koopman representation framework with identity lift, we show that SOC and OOC coincide in the following lemma.
Lemma 1. Consider system 3h. Define the set 2 C R.N to be such that Z == ((R"). Then, the system is O0C on R. and only if is SOC (2, RT).
While verifying OOC is generally difficult for linear systems, Lemma Commamesstalisthe establishes the equivalence between OOC on R and SOC (2, R.). This equivalence allows us verify OOC of system (3)
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through SOC. OOC then implies controllabilitt per Definition T for the corresponding learned nonlinear system. The verification criterion is stated the following theorem.
Theorem 1.
The system
00C on R if and only if the controllability matrix
C Bo Bo
Ag - Bo)
is full-rank.
The proofs are provided in Appendix D From training perspective, one could promote SOC of the Koopman linear system by adding loss term of the form min (C) However, as in many learning problems, this approach provides no guarantees. To address these issues, we propose direct parameterization strategy that explicitly constrains the matrices Ag and Bg, ensuring that the learned Neural ODE (3 is always OOC. An overview of the conceptual framework and our contributions is provided Figure in Appendix
3.2 CONTROLLABILITY CANONICAL FORM
The following theorem shows the parameterization for Ag and Bg that ensures OOC of (3) Theorem 2. Consider system Then, the system is OOC on IR if there exist matrices A$
R Bī E RB Xm and Po E R NXN, where
1
0
O
0
A5-
Pa = diag(PI, (5)
a1
02
ON bN
bi
b2
di
61
C2
CN
dv-n
such that
Ag Po Bg = BaB
(6)
Pi E RR and P3 E N- are both full rank matrices. The first rows of Ao only consists of and 1, and all are on the superdiagonal. All the other elements of Ao are free. The first n rows of Bg is O, the nih row is 1, and all the other rows are free.
The proof is provided in Appendix D Equipped with Theorems T and we can learn a Neural ODE B that models the original nonlinear system ( while preserving controllability
important to note that Theorem 2 applies to single-input systems (m = 1). The results can be extended to multi-input settings the Brunovský decomposition, discussed in Appendix
3.3 DEGREE OF CONTROLLABILITY VIA GRAMIANS
Beyond the binary notion of controllability, the degree of controllability determines the energy quired to steer the system (Killtth 1980). To quantify this, we use finite-horizon output Gramian
W [ (Ce (C) Bo) dr.
which measures how inputs excite the physical state over a finite time window. A small Amin (W?) indicates directions the state space that require disproportionately high input energy reach. A large condition number x( x(WF) implies unbalanced controllability and ill-conditioned optimization problems. To address this, we introduce the regularizer
R Rgram (Ag, Bg) = Amin (W?)
+ (KWW))
with y > 0 a trade-off parameter. This term encourages a larger smallest eigenvalue while discour- aging large condition numbers, promoting balanced controllability across directions.

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Algorithm 1: Training Neural ODEs based Koopman representation with controllability guar-
antees
Input: Dataset D, horizon (4o, Lr], decay y, Gramian horizon weight Agram
Output: Parameters 0
Initialize { Ag. 0g. 0p):
for epoch = 1 to E do
Sample mini-batch B C D;
for trajectory (I, u) E B do
// Encode initial state // Koopman dynamics =(t) ODESolver Agz + Bou),Itotts] [to, /// Rollout in lifted space
z(1o) Vo (=(x((()))
(Ag, Bb) t Pa (Ag. Po)
#(t) - Cz(t);
// Recover state prediction // Trajectory loss
Sped - -
Ryram +- (?);
// Gramian shaping
t Sped + Araam Rgram:
// Total loss
0 - Optimizer(O,, ;
// Parameter update
3.4 Loss FUNCTIONS DESIGN
The state matrices Ag and Bg are parameterized as (6), where (4%, BB) denotes the canonical controllable form from (5), and Pa is a trainable invertible transformation. This construction pre- serves controllability by design while maintaining flexibility: without Ps, the input would have only limited influence the identity coordinates.
Given an input trajectory u [4. > Rm and initial condition x=(4), the Neural ODE dynamics 3) with initial state =(to) 1 (o(((())) generate predicted state trajectory a(t) Cz(t). We
define the prediction loss as
1, to '" w(() (|a(() xx())l dt.
Spod (0)
where will) nonnegative weight function. This loss evaluates accuracy over entire rollout rather than only at the next step, thereby encouraging stable long- monghhorzzon predictions. We choose w(t) with a decay rate that emphasizes carly prediction errors while still accounting for the full trajectory. The integral is computed in discretized form, with the discretization determined by the dataset's 's sampling scheme.
To complement predictive accuracy with controllability, augment this loss with the Gramian regularizer from Section 3.3 yielding the full training objective
min 9
Sprea (f) + Agram Rgram (Ag. Bo).
Gradients are computed through the ODE solver, enabling end-to-end formulation that yields models accurate over long horizons, controllable by design, and numerically well conditioned for downstream control.
3.5 MODEL PREDICTIVE CONTROL ON LEARNED MODELS
We deploy MPC on the learned Neural ODE by discretizing with sampling time At under zero-order hold:
At Aor Bo dr,
A0 = Aodi,
By
Czk-

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At time k solve the following finite-horizon MPC on the discrete- time surrogate:
k+H-1
Σ
[1C2z]k - x2eff12 112 + -lugj U -1-11x1122) + 1[ - 2/0ff
min [ #1
z=
s.t. *|1||k = Agas1k + BigugR: j == k -1-1.
Aajk = (o(()))
Czjjk E X, Ujik E U.
Here rer is a reference the physical coordinates, and Q, R, P = 0 are standard MPC weights; the smpot-smo term ||uيjæ 1 l can be replaced by || Il if desired. Since the opti- mization posed entirely the linear surrogate (Ag. By C). 7 is convex quadratic program. At each sampling instant we apply the first control "kjk. measure 21+1, reset 2 I|++1 = V (z(+)))), and repeat in the beeding-horizon fashion.
EMPIRICAL EVALUATION
The experimental evaluation designed to assess both model learning accuracy and control perfor- mance. We first study widely used nonlinear control benchmarks: pendulum swing-up, mountain car, and cartpole stabilization.
MLP
DKD
State
-0.30
Ours
0.35
To evaluate data efficiency, train each model with different fractions of the available dataset (1%, 5%, 10%, 50%, 100%) and measure both short- and long-horizon prediction errors. The first baseline is the Deep Koopman Operator (DKO) (Han et al., 20200 Wang et al. 2021; Xiao et al. 2022
11
13
15
Shi & Meng 2022), which employs deep neu-
State
-0.5
ral networks to learn Koopman representations. Al-
-1.0
True
though existing variants differ in technical details, their overall pipeline is similar; in our experiments we adopt a continuous-time realization. The sec- ond baseline is multilayer perceptron (MLP) (Chua) jet al. 2018) trained directly on the nonlinear dynam- ics, representing purely data-driven approach with- out structural priors.
-1.3
DKO
Ours
2.0
Step
Figure 5: 15 Steps Prediction Results on test case of Pendulum
We compare these against method on (i) trajectory prediction error, (ii) cumulative input energy for control. These metrics reflect both the model fitting ability and the downstream control utility of the learned dynamics. In the end, we extend our framework to the multi-input setting and validate on a six-dimensional gene regulatory network (GRN) (Elowitz & Leibler 2000) system. All exper- imental settings, including dataset generation, training details, and hyperparameters, are described in Appendix
Prediction performance Table D reports model learning errors across varying training dataset sizes. Our framework consistently outperforms both DKO and MLP when less than 30% of the data available. With abundant data, our framework and DKO achieve comparable accuracy, indicating that both are sufficiently expressive to capture the underlying dynamics. The key distinction is that our method guarantees controllability by construction, whereas DKO can only assess a posteriori without guarantees during training. Figure [ trained with only 30% of the pendulum dataset, shows that our model closely tracks the true trajectory, while the baselines deviate despite using the same data. This advantage stems from enforcing controllability parameter search space and guides optimization toward meaningful solutions. Consequently, our model attains lower errors with fewer samples and converges faster during training. The training curves in Figure 6 support this view. Our losses decrease rapidly and stabilize at a lower level, whereas DKO starts with large errors and converges slowly. Although nonlinear, the discrete MI.P
structural prior, which reduces the
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6.0290
DKO
DKO
Ours
0.44
6.0150
0.39
6:1125
0.831
0.32
0.492
6.0075
6.0090
0.911
0.88
0.00
6.0000
stop
stop
stop
(a) Mountain Car
(b) Pendulum
(c) Cartpole
Figure 6: Loss versus training step on Mountain Car, Pendulum, and Cartpole. Our method con- verges faster than DKO.
Table I: Comparison of prediction (MSE) across environments with varying fractions of train- ing data. Our method shows clear advantages under limited training data, highlighting its data efficiency.
Environment
Method
1%
5%
10%
30%
50%
100%
MLP
0.0758
0.0571
0.0178
0.00510
0.00418
0.0002
Mountain Car
DKO
0.0200
0.00022
0.00016
S 1 x 10
Ours
0.0032
0.00019
0.00011
MLP
1.1778
0.5998
0.2610
0.0470
0.0218
0.0091
Pendulum
DKO
1.5347
0.1079
0.0390
0.0086
S 6 x 10
Ours
0.3747
0.0318
0.0114
0.0061
0.0465
0.0296
0.0202
0.00727
0.0039
0.0021
Cartpole
DKO
0.1306
0.01452
0.007585
0.00064
S 10
Ours
0.0095
0.0024
0.001153
0.000571
baseline struggles to capture continuous dynamics, leading to rapid error accumulation over rollouts. In summary, controllability provides not only a downstream guarantee but also powerful inductive bias that improves data efficiency, accelerates convergence, and stabilizes prediction.
Control Performance We next evaluate control performance by embedding the learned models into MPC. As shown in Table 2 differences in prediction accuracy translate directly into control outcomes. With limited training data, DKO often fails to complete the task due to inaccurate or uncontrollable models that render MPC infeasible. In contrast, by encoding controllability structural prior, our method learns more accurate models from scarce data and produces MPC solu- tions that require substantially less input energy.
Figure 7 illustrates this effect in the mountain car task. With only 1% of the dataset, DKO fails to stabilize the car under MPC, whereas our model successfully drives the system the target. Even with 5% of the data, our approach requires substantially less control effort than DKO, highlighting the advantages of accurate and controllable dynamics. Similar trends are observed in pendulum swing-up up and cartpole stabilization.
Another line of work introduces vare-narying Koopman operators (L 2025 where the state matrix A is generated at each step from past trajectories rather than fixed. This increases flexibility and can improve prediction, but the cost of higher computational burden for MPC, since A must be recomputed at each step. Moreover, since trajectory history is required (e.g., 30 prior steps), real system must first be driven with preliminary inputs before control actions can be computed, com- plicating deployment. Controllability analysis is also more challenging for time-varying Koopman models. General nonlinear models such as Neural ODEs and LSTMs can achieve strong predictive accuracy by directly parameterizing the dynamics. Their limitation lies in downstream use: embed- ding them in MPC typically leads to nonconvex optimization problems that are much slower to solve and lack convergence guarantees, unlike linear formulations.

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Table MPC input cost comparison selected dataset fractions. "Fail" indicates that the controller was unable to complete the task.
Environment
Task
Data
DKO Cost
Our Cost
Rel. (%)
1%
Fail
186.59
Mountain Car
Hilltop
5%
297.51
165.21
44.5
5%
Fail
100.25
Pendulum
Swing-up & hold
10%
239.2
43.50
81.8
Cartpole
Balance
30%
Fail
19.61
Mountain Car
Control Actions
0.05
0XD (5%)
010 (15)
DED (15)
DK0 (5%)
0.04
Ours [1%
0sm (3%)
Oura [5%)
0sm (2%)
Velocity
0.03
Start
Control
Goal
0.80
*
-0.31
-2
-0.22
Position
Time
(a) Phase trajectories
(b) Control inputs
Figure 7: Comparison of Mountain Car experiments on control task.
Multi-input extension. We extend our framework to the multi-input setting and validate it other more complex robotics environments including Reacher (Mujoco) and 7-DoF Franka manipulator environment used in Shi & Meng (2022). As shown in Appendix G the results demonstrate that our method can scale to larger and -nplt systems and still show higher data efficiency. We also ex- six-dimensional gene regulatory network (GRN) (Elowitz & Leibler 2000)
tend our approach with three control inputs. Under the same protocol as the single-input tasks, our controllability. preserving Koopman model attains test MSE of 1.0 10-1, compared to 3.0 10 for DKO. On the associated control task, where one state must be regulated at a setpoint of 6, our method achieves an input cost of 80.96, lower than the 89.98 required by DKO. These results demonstrate that the proposed canonical parameterization and training procedure scale beyond single-input sys- tems, while also highlighting the added challenges of multi-input settings, where optimization harder to stabilize and performance degrades more quickly with limited data.
CONCLUSION
We proposed an end-to-end framework for learning nonlinear dynamical models with controllability guarantees, built on Koopman representations learned through Neural ODEs. To ensure controlla- bility, we introduced novel OOC controllability canonical forms that parameterize the state matrices of the Koopman surrogate, providing guarantees by construction. Simulations on nonlinear bench- marks demonstrate superior data efficiency and control performance compared to state-of-the-an open challenges: multi-input controllability introduces
baselines. At the same time, there additional complexity in initialization and parameterization, and in practice it is often unclear how assign control inputs to state variables. Addressing these issues is important direction for fu- ture work, and progress here could significantly broaden the applicability of controllability-awaar system identification.
ACKNOWLEDGMENTS
This work was supported by NCCR Automation, grant agreement 51NF40.225155 from the Swiss National Science Foundation. This support funded Keyan Miao's research visit at the ETH AI Center, during which part of this work was carried out. Keyan Miao was also supported by the Engineering and Physical Sciences Research Council (EPSRC) through grant EP/T517811/1, and
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Antonis Papachristodoulou through the EEBio Programme Grant EP/Y014073/1. The authors would also like to thank Dr. Armin Lederer for helpful discussions and insightful feedback.
Reproducibility
Statement. Source
code can be found at for theoretical results, clear explanations
https://gttuu..com/ MiaooConttolllleeeeeee----eee of any assumptions and a complete proof of the claims can be found the Appendix [D] for any datasets used in the experiments, complete description of the data processing steps can be found in Appendix G
Ethics Statement
This work involves only simulated environments and raises no ethical concerns.
Use of LLMs We used a large language model (LLM) solely aid and polish the writing and improve the clarity of exposition. All research ideas, technical contributions, experiments, and analyses are our own.
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A DETAILED RELATED WORK
Learning-based Koopman approaches (Lusch et al., 2018: Yeung et al. 2019 Ricardo Constante- Amores al., 2024) approximate nonlinear dynamies by training neural networks construct ob- servables, thereby lifting the system into a space where linear dynamics are identified. Despite differences in implementation, the pipeline is largely consistent: an encoder maps the state into latent space, and operators (A, B) are then estimated either through direct regression (e.g.. least squares) or as trainable parameters within an end-to-end framework (Han et al., 2020] Wang al. 2021 Shi & Meng, 2022 Xiao et al. 2022). Variants further differ in whether the operators are kept fixed or allowed to vary with time (Li al. 2025), but the overall objective remains multi- step trajectory reconstruction. Crucially, structural properties of the underlying system are rarely incorporated.
Very few works have even attempted to consider controllability in this setting, and those that do typically include the Kalman rank condition as a loss term (Han et al, 2020). This suffers from two major issues. First. placing the condition in the loss provides guarantee of preserving trollability. Second, and more importantly, requiring controllability of the entire lifted system is unrealistic and misguided: the lift is often very high dimensional while the input dimension remains small, making global controllability virtually impossible; moreover, in control tasks the relevant quantity is the reachability of the original physical state, not arbitrary auxiliary observables. As result, such lifted-space conditions fail to capture the property that actually matters.
Research on incorporating structural priors into Koopman learning remains limited. The first such attempt considered stabilizability, introducing LMI-based parameterizations of the Koopman op- erator (Fan al. 2024). This was the carliest demonstration that parameterization can embed tystem-theoretic properties into Koopman models, but stabilizability is weaker requirement than controllability and cannot guarantee capabilities such as trajectory tracking. It is also worth noting that, more broadly, some recent works have introduced properties such as stability, stabilizability, or dissipativity into dynamics or control learning, for example via Neural System Level Synthesis (Fu- rieri et al. 2022). or SDP/QP formulations Yin et al. 2021 Tang et al. 2024 Okamoto & Kojima 2025). While these demonstrate that inductive biases can be beneficial, such approaches either im- pose heavy computational burdens through projection-basee
optimization, rely on the availability of base stabilizing controllers, and in any case highlight that structural priors are gaining attention while simple and efficient mechanisms remain highly needed.
very recent effort, parametric Koopman models, embeds the control input nonlinearly via neural networks (Guo et al. 2025). Here controllability can only be verified post hoc using Lie bracket conditions, and because the input enters nonlinearly, the resulting control problems remain nonlinear optimizations, limiting the practical benefit of the linear surrogate.
In summary, existing Koopman learning methods focus primarily on reconstruction accuracy, with occasional efforts to enforce stabilizability or to post hoc analyze controllability. None provide guarantees of controllability during training, nor do they shape its degree in way that affects the feasibility and efficiency of MPC. Moreover, by neglecting controllability as structural prior, they miss the opportunity to reduce the effective parameter search space and to guide the training process itself. This gap motivates our framework, which enforces controllability by construction and explic- itly regularizes the degree of reachability, yielding models that are both predictive and reliable for control.
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B DETAILED PRELIMINARIES
B.I KOOPMAN OPERATOR
To handle inputs explicitly, the system be reformulated augmented state space,
(t)) x(1)= = [t))
x()== -(())))(()))))))))))))))))))))) (ft((t, u(t)) u(t)
The dynamics are autonomous in (t),, so one can define Koopman operator acting on observables Ru+m RR This operator is linear but infinite-dimensionall and satisfies
d di (((())) (C)()))))
In other words, the nonlinear flow of (x, u) induces linear dynamics in the lifted coordinates (+))..
Since K. is infiiitedimeesionnll, Anite-dimensional approximation is required for practical use. A standard construction is to select observables of the form
(xx
z == ((x) R.Y
where á R" (N > n) is set of nonlinear lifting functions applied to the state, and the input u is retained in its original coordinates. With this choice, the operator reduces to block-matrix structure,
d x1) dt u(t))
((()) et))
Kv
K
Here A E NNXN governs the autonomous dynamics of the lifted state, E Nx describes how control inputs affect the lifted state, and the lower blocks (denoted by *) capture the evolution of the input coordinates themselves. Since prediction and control only require the lifted state dynam- ics, it suffices retain the first N rows of this operator. This leads the following Koopman representation as an ODE:
a(t) = Az(t) + Bu(t), a(t) h(z(t)), =(t)= o(x(t))
A, B, h) constitutes finite-dimensionss approximation of the infinite-dimensional The tuple (, controlled Koopman operator. In many applications h is taken to be linear, i.c. h(z) = Cz with C'e RaxN
B.2 CONTROLLABILITY OF LINEAR SYSTEMS
Consider the linear control system defined globally
a(t) Az(t) + Bu(t),
2(t) R. u(t) e R"
with A RR NxN and B Nxm.
Dislike the challenging nonlinear controllability verification problem, the problem is tractable for linear systems.
Proposition 1 (Kalman Rank Condition). The linear system with matrices (A, B3 is controllable if and only the controllabiliry matrix
C C=
[
AB
A²B
AN
has full row runk, i.e. rank(C) N. In this case, the columns of C span the entire stute space, so any state can be reached from any initial condition by a suitable input.
Equivalently, controllability can also be characterized through the controllability Gramian
J.1 1. 10
WT
ABBB
T dT.
which is positive definite for some sufficiently large time interval if and only if (A, B) is controllable. The eigenvalues Wr further quantify the directions that are more or less easily influenced by the inputs.
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Proposition 2 (Controllable Canonical Form). the linear system with matrices (A, B) is control- lable, there exists an invertible matrix P such that the transformed pair
Ag = P-1 AP,
B PP
lakes block companion form. In the single-inpul case (m 1), Ag is the companion matrix
0
0
0
A
B3
0
011
0/
aN
where (a1.... (01....,000) ..x) are real coefficients. In the multi-inpul case (m > 1), Ag can be wrillen in Brunovský block form with each block corresponding inpul channel. These canonical represen- tations show that every controllable pair is equivalent, to similarity transform, to structured form where controllability is explicit.
Definition 5 (State-to-Outpuc Controllability (SOC)). Consider the linear system é(t) = Az(t) Bu(t) with output y(t = Cz(t), y(t) E RP. The triple (A, B, C) is state-to-output controllable if. for any initial state z0) z(oo) and target output yT R2, there exists finite horizon and input sequence u(t) 1mm & E [o, ty), such that y(tz) = yT.
Output controllability can be characterized by the output controllability matrix
Cy = [CB CAB ... CAN-IB] E PM
The system (A, B, C') is output controllable if and only if rank(Cy)) = p. i.. the columns of Cy span the entire output space. When C I, this reduces to the Kalman rank condition for state control- lability. Output controllability generalizes state controllability by focusing only on the directions visible through C. When C the two notions coincide.
If (A, B) is not controllable, there exist directions in the state space that cannot be influenced by any admissible input, which limits the effectiveness of feedback and may render stabilization or trajec- tory tracking tasks impossible. Even when system is controllable in the binary sense, the degree of controllability encoded by the Gramian eigenvalues strongly affects input energy requirements and numerical conditioning in control algorithms. For this reason, ensuring promoting controllability in identified models is essential when the ultimate goal is to use them for control.
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C
CONCEPTUAL OVERVIEW AND CONTRIBUTIONS
Koopman Framework (containing identity lift)
Koopman Linear
2
Koopman Linear
OOC
SOC (zo - YT)
(Yo - YT)
4
Nonlinear System
SOC Tools: Rank Condition
Controllability (xo - XT)
Canonical Form 3
Figure 8: Overview of the relations among controllability notions.
Fig illustrates how different notions of controllability are connected in our framework and our contributions:
We show that the controllability of the original nonlinear system is equivalent to OOC of Koopman linear representation (in main text).
In the Koopman linear representation, proved in Appendix D).
prove that 00C coincides with SOC
(Lemma[Tand Theorem |
Based on this equivalence, we for the first time introduce a SOC canonical form which can be used for OOC canonical form for Koopman representation (Theorem proved in Appendix D
Finally, we leverage this canonical form to parameterize the neural Koopman operator, which enables constructing learning-based Koopman representations of the original non- linear system.
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D PROOF
Proof of Lemma] 7 We begin by proving that the map 2 + RR with y = T= is bijective. R", if y1 y/2 then 21 = o(y1) = 22 Φ This indicates that is injective. On the other hand, any y1 Z corresponds one (31). This shows that T is surjective. We conclude that the map T is bijective.
The map is well-defined from the definition of set 2. For any 21, 22
We then prove sufficiency of the claimant. From the definition, for any Yo R", the system reach any output yr E IR". Consider 2n E 2 and 2r E 2, let 1/o = C2g and yr = Czr. Given that 7 is bijective, it holds that 2o <(yo) and ž7 ((r)) From OOC, the system starts from output Yo can reach yr, which indicates that the system starts from state can reach 27- Given that states 20 and 2T arbitrarily chosen, SOC holds. Necessity of the claimant holds with similar arguments.
Proof of Theorem We begin by proving that the system C is full-rank. Consider an arbitrary E 2 and yr E R if there exists an input sequence u(-): : [to ty] + I that drives the system from zo to yr, then we have
SOC on (2, RT) and only
(CAnltt)-in 1o) 20 + f
YT =
Aolly 1)) Bou(F)dd
(8)
Rearranging the terms and applying Taylor expansion for Adllyy -o- -m= have
Ao
00
( " (1y lg T) u(T)(lT)
yr Ao to)
C
(9)
yr - Coee 20 spans R" because yr and 2g independent, and yr spans R" Under this, we conclude that
deduce that the existence of u(T) is equivalent to C is full rank. Using Lemma the system is OOC on I2 if and only if the controllability matrix C is full-rank.
In the proof of Theorem 1 we use the fact that yr Ae 20 spans IR because yr and 2o independent, and yr spans R" The subspace 2 does not influence the result. This indicates that for any subspace H C TR the Koopman linear system is SOC on (H, R²) if and only if it is SOC (R R*).
Proof of Theorem are given by
The proof based on some algebraic calculation. From 5 Ag and Ab
0
0
Ag P
ABBBo Po
(10)
(n 1)th row
(n 2)th row
Using induction, Ag Bo for < k: & n are given by
0
A Ba = Po
(11)
(n-kith row
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The OOC controllability matrix has the following structure
column
0
0
(n 1)th column
C = CP
(n
1)th row
(n)tth
(12)
0
0
nith column
0
0
(n
1)th column
P1
0
(n 1)th row
[1x)tth row
Clearly, C has full row rank as P1 has full rank and the matrix that multiplies Pi also has full rank. From Theorem system (3) under canonical form is OOC on R".
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E EXTENSIONS
E.1 MULTI-INPUT PARAMETERIZATION
We extend Theorem to multi-input systems with m 2 2. The idea is to consider the Brunovsky canonical form (Luenberger 2003).
Theorem 3. Consider system with 2 Then, the system OOC on R. and only if there exist matrices Ağ E IR N B R Nxm and P E RN NxN, where
1
B
fi
A2
0
0
0
91
B2
Ag
[2
f3
Am
By
92
93
Bm
P = diag(P1 (Pl P2).
b1
b2
bn
d1
d2
dm
C2
CN
61
62
cm
(13)
such that
Ao P
Bo PoBS
(14)
P R" P2 R-N- are both full rank matrices. The blocks As N.NN and B, R. N:x1 are in the forms of
0
0
0
1
0
0
0
A;
B; =
(15)
0
ONN.
where Em N, the elements in rows from 1 to N of both Ag and Bğ are free.
Proof. To show the structure of the OOC matrix C, consider using mathematical induction. For
2 < k < N, suppose that
ABB
0
0
A
B2
0
A Bo = Ps
g'
g/
A
Bm
di
dg
:
el
e/2
e
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Given that focus the structure of the matrix, use g ......... em to represent variables matrices, with a slight abuse of notation. From this, we obtain
Ag
Be
As
A
Bg
A
A&B
B1
f1
A2
0
0
gí
1
B2
Po
J2
J3
A m
g/2
93
A Bm m
b1
bg
bn
d'
d2
d'
C1
C2
CN
eq
6/
(16)
A
91
A
k+1
B2
Po
g2
A k++1
Bm
m
di
d2
d'm
éi
which satisfies the induction for ++ 1. be verified that AZ also fulfills the structural assump- tion. Using the result, the OOC matrix C is given by
B
B1
A A
B1
0
9
B2
0
g2
A2 B2
93
A
N-1
B2
=CP
94
Bm
Am Bm
96
A
Bm
97
9'
B
0
0
A B
0
AN B
0
gi
B2
g
A2
0
g's
A
Bg
0
=P
.94 Bm
gb
Am B.n
96
A By
(17)
From the companion structure of each A, and B,, it is known that for each i:
[ A,B, full rank. Inserting zero blocks into the matrix will not change the rank, we then immediately obtain that C is full rank as well. Using Theorem T we conclude that system is OOC.
B,
(18)
E.2 SIMPLIFIED PARAMETERIZATION
In the following, we propose another simplified canonical form for single input case. Proposition 3. Consider syslem with (single-input). Then, an OOC canonical form is given by
À
Opx(N-n) A'
B
Ag O(N x
Bo
(19)
B'
where À E Ruxn and ß gax1 are given
0
0
0
1
0
Ā
0
(20)
0
.00
On
an
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A' (N)-)))(N-N) and B' E R(N-n)xl are free matrices.
Proof. From 19 the state controllability matrix is given by
B
AB
A A
AN-1B
C,
(21)
B'
A'B'
A'N
From Theorem I the OOC controllability matrix given by
C CC. = [
AB
A'
AN-1
(22)
From 20
the matrix
[
AB
A
A
(23)
full rank. From the Hamilton Theorem, C is also full rank.
This canonical form requires Ag be block diagonal, unlike the one in Theorem 2 This form can be conservative in practice, particularly for the Koopman linear system. This is because the states and ((x) in 2 have no correlation.
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F DISCUSSIONS OF LIMITATIONS
F.1 THE STRUCTURE OF Pg
The canonical forms in Theorems and reply on an invertible matrix Ps. This matrix is critical as it provides extra degree of freedom in linearly transforming the coordinates. In these theorems, the matrix Po is constructed as block diagonal Pp diag(Pr (1g. P2) because of the OOC structure: is the first n rows of [Bg Ag
(4g) (4g)4=1 BBo that needs to be full rank. Too see why the
block diagonal structure important, we re-evaluate the OOC matrix C for m First, is clear that
= CPPPB
ABBi
(45) N-1 BB]
:
Let matrices Cf Ruxx, C E Bax(N-n), Cj E RIN ((N-,)xn, C2 E ((N--))(N-m) be such that
Ci
C
C3
The companion forms of Ag and Bğ in Equations
and
13
ensure that
rank(C)) n rank (BB)
A
(Ap) N B₂l) 2 n
Assuming that Py is full rank but not block diagonal:
Pi
P2
P
Ps
Pa
Then we have
C- CPC
= [PIC + P2Cs PIC + P2C
Although Pt, P2 and ej are ensured be full rank, + is not ensured be full rank. One sufficient condition for C to be full rank is
P2 02 Pi is full rank.
(24)
For (24] holds and Ps is full rank, adopt block diagonal structure where both P2 and Pa are set to zero, and P1, P4 > 0. This will unavoidably introduce conservativeness in training. As we admit that making PaC + P2C full rank by construction is hard, can still consider constructing a full rank matrix Po with only P₂ = O to reduce conservativeness. This is left for future investigation.
We also provide some practical insight: any full rank matrix result a full rank C almost surely. If one fix full rank matrix Ci and not necessarily full rank matrix C₁, and randomly construct a matrix P from a continuous distribution. Then, PIC + P2C5 is almost surely full rank. This is because the determinant of is nontrivial polynomial in the entries of P.C + + P&Cs. The zero set of any nontrivial polynomial has Lebesgue measure zero in R= Since both Pı and P2 are constructed from a continuous joint distribution, it assigns probability zero to Lebesgue-null sets. If P.C.++PPPC almost surely full rank, then so is C.
F.2 THE MULTI-INPUT CANONICAL FORM
In Theorem 3 we provides multi-inpu canonical form (3). This is obtained from the single-input canonical form within Brunovsky structure. The whole matrix Aš consists of m small matrix A; E N.N. In linear system theory, the dimensions N,, also termed as controllability indices, can be determined via rank checks. More specifically, the index N; 2 3 for input is the smallest integer such that
rank([B,
Ag
A B)) rank([Bg
Ag
- = 1.
(25)
This condition can be readily verified a posteriori for given Ag and Bg. However, the indices are hard to be determined by construction. the correspondence of state and input the lifted space is unknown. In the experiments, we consider a conservative method that unifornly assigned indices. This is, clearly, not the optimal assignment in general.
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G EXPERIMENTS
G.1 DATA COLLECTION
For each environment we simulate continuous-time nonlinear dynamics under random control in- puts. At the beginning of each rollout an initial state 2o is drawn uniformly from prescribed range. The system then evolves according to its ground-truie truth ODE å f(x, u) while, at each step, the control input u(t) is sampled uniformly from the admissible bounds. Trajectories are recorded at fixed sampling intervals, yielding sequences length 250 steps.
Sampling details. The integration step size differs across environments: for mountain and GRN we use At 1.0s, s, while for pendulum and cartpole we use At 0.02s. Although our experiments regularly sampled data, modeling the system in continuous time allows the same framework to handle irregular sampling required.
Dataset size.
Mountain Car: 5000 training trajectories, 1000 test trajectories.
Pendulum: 5000 training trajectories, 1000 test trajectories.
Cartpole: 10000 training trajectories, 1000 trajectories.
GRN: 10000 training trajectories, 1000 test trajectories.
Reacher: 5000 training trajectories, 1000 test trajectories.
Franka: 5000 training trajectories, 1000 test trajectories.
Training usage. During training, we do use the entire 250-step rollouts directly. Instead, sliding window of length 15 is randomly sampled from a trajectory. This increases sample diversity and ensures that the learned models are trained on local temporal contexts.
G.2 EXPERIMENTS SETTINGS
Mountain Car dynamics.
The continuous-time Mountain Car dynamics are:
p p= v, 1 P 0.0025 c3s)
where p is the position, v the velocity, a the input action, and P the action scaling coefficient.
Table 3: Hyperparameters for Mountain Car.
Hyperparameter
Value
Batch size
Learning rate
10-3
Prediction horizon
15
Dimension of observables N
16
Number of layers of &
Hidden dimension of s)
64
Activation function
ReLU
Pendulum dynamics.
The continuous time Pendulum dynamics are:
i w
3g
à 21
sin 0
m/2 T,
where 0 is the angle, w the angular velocity, m the mass, the length, 7 the input torque, and g the gravity constant.
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Table 4: Hyperparameters for Pendulum.
Hyperparameter
Value
Batch size
Learning rate
1 x 10
Prediction horizon
Dimension of observables N
16
Number of layers of t
Hidden dimension of v
64
Activation function
ReLU
CartPole dynamics.
The continuous-time CartPole dynamics are:
2,
f + 102 sin 8
10 cos 0 M
M
0,
[f+1d" sin
g sin - cos 0
M
(
cos?
M
where x is cart position, 0 the pole angle, M the total mass, m the pole mass, the pole length, f the input force, and g the gravity constant.
Table 5: Hyperparameters for CartPole.
Hyperparameter
Value
Batch size
64
Learning rate
10-3
Prediction horizon
15
Dimension of observables N
25
Number of layers of &
Hidden dimension of sb
64
Activation function
ReLU
GRN dynamics.
The continuous-time GRN dynamics with 6 states and 3 inputs are:
d - +
+ U1
K
+
x6
1
42 -222
K
x
7
+ U2
43 = - +
+ U3.
K + x
4. = CC4 + 3x1
45 = CT5 + A02
is = cas + Ba3;
where Zi, Tg are the states, U1, U2 Ug are the control inputs, and 7, a, K, c, 3 are system param- eters.
Reacher (Mujoco). Reacher is a two-jointed robot arm. The goal is to move the robot's end effector (called fingertip) close to target as F shown. Note that we should exclude redundant or inn-inddeendent states when constructing the OOC canonical form; only the independent degrees of freedom should enter the canonical structure.
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Table 6: Hyperparameters for GRN.
Hyperparameter
Value
Batch size
Learning rate
1 x 10
Prediction horizon
Dimension of observables N
30
Number of layers of t
Hidden dimension of v)
64
Activation function
ReLU
Figure 9: Reacher Environment.
Franka (7 DoF Manipulator). Franka is 7 DoF robotic manipulator. The goal is to move the robot's end effector (called fingertip) close to target. The environment setting our experiments is the same as provided in Shi & Meng (2022). Note that we should exclude redundant or non- independent states when constructing the OOC canonical form; only the independent degrees of freedom should enter the canonical structure.
G.3 ADDITIONAL. RESULTS
Mountain Car
The comparison of prediction performance on Mountain Car Environment with 1% training dataset are shown in Figure I0
Pendulum
The comparison of control performance on Pendulum environment is shown is Fig-
ure [I
Cartpole
The comparison of prediction and control performance on Cartpole environment is 12 and Figure
shown in Figure
GRN
The results of prediction on GRN environment by our method shown Figure 16
Reacher The results of prediction on Reacher environment over different amount of training by DKO and our methos are shown in in Table 9 Also, the training loss versus step is shown in Figure 15a Our losses decrease rapidly and stabilize at lower level, whereas DKO starts with large errors and converges slowly.
Franka The results of prediction on Reacher environment over different amount of training by DKO and our methos are shown in Table 9 Also, the training loss versus step is shown in Figure 15b The results of prediction GRN environment by method is shown in Figure [6] It can be found that our method learns faster with fewer data needed to achieve satisfying prediction performance.
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Although may depend on the state, the surrogate system can still be regarded as linear in its effective input û. Consequently, the controllability analysis remains unchanged: long the encoder does not map all admissible inputs to zero, the pair (A, B) governs reachability in exactly Published as a conference paper at ICLR 2026
Table Hyperparameters for Reacher.
Hyperparameter
Value
Batch size
Learning rate
1 x 10
Prediction horizon
Dimension of observables N
16
Number of layers of t
Hidden dimension of v)
64 ReLU
Activation function
Table 8: Hyperparameters for Franka.
Hyperparameter
Value
Batch size
64
Learning rate
10-3
Prediction horizon
15
Dimension of observables N
16
Number of layers of &
Hidden dimension of g)
64
Activation funetion
ReLU
G.4 ROLE OF CONTROLLABILITY LOSS
Suitable penalty on controllability loss will also contribute to accuracy. However, too large penalty makes the model must trade off matching the true dynamics versus enlarging controllability, then may damage prediction accuracy. Although MPC can compensate for prediction inaccuracies to some extent, poor prediction quality results in higher control effort in the end. The experiments on Pendulum with different weights on gramian loss (0.005 and 0.05) are shown in Figure 17
G.5 ADDING ENCODER FOR INPUT
Adding an input encoder as shown Figure T [effectivel introduces a nonlinear dependence the control signal on the state, since the input the lifted dynamics becomes à go rather than the raw control signal. Formally, this representation takes the form:
(t) Ao=(t) + Boú(t), =(t) ()(x)))), a(t) = v(t) = Cz((), a(t) = (&(()) @ u(t)
the same way in the standard formulation without encoding. In this sense, the encoder enriches the expressiveness of the Koopman representation without undermining the controllability guarantees.
The prediction performance of our framework with input encoder is shown in Table IO It can be found the prediction is even more accurate. However, the reason why we still choose the framework without input encoder is: this case the effective input becomes u = g(t) O u, leading a bilinear system & Az + (C)( C u.). This breaks the standard linear structure used in MPC (which could be our future research direction); one could optimize over û, but then the original control is no longer explicit, making constraints and interpretation of the optimization objective difficult. We therefore focus on the direct-input setting, which integrates more naturally with control design.
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Traj dim o (Normali
Traj dim (kormal)
-07
Nurdino inddnu
-02
fuedimo ind.dig
-0.4
-0.4
-0.0
-00
-0.0
-10
Traj 1- dim (Nommal
Trjj 1 1-dim (Nommal)
0.0
The din Fred gim 0
0.0
Tuee din
Frud dims
088
Trjj 2 2dim (Normal)
2-d22mmm (Kormal)
0.0
Bueddin amdaad
Fue-din
-0.3
0.-
-03
-0.4
-04
-05
(a) Trajectorics 3 cases predicted by DKO.
(b) Trajectories 3 cases predicted by Ours.
Figure 10: Comparison of Prediction Performance on Mountain Car Environment with 1% training dataset
Pandulum
Control Actions
DKD (5%)
DKD (5%)
DKD (18%)
DKD (10%)
Oura 15%)
Ours (5%)
Anole Velocity
Ours (10%
Ours (10%)
Start
Control
-35
Phase trajectories
(b) Control inputs
Figure 11: Comparison of Pendulum experiments. (a) trajectories in angle-velocity space; (b) cor-
responding control sequences.
Table Comparison prediction emror (MSE) across environments with varying fractions of train- ing data for Reacher and Franka.
Environment
Method
1%
5%
10%
30%
50%
100%
DKO
3.47
1.55
0.5613
0.2572
0.0194
Reacher
5
Ours
0.0199
0.0135
0.0039
< 1 x 10
DKO Ours
0.0453
0.022
0.0068
10-4
1 x 10
Franka
0.039
0.00064
<
x 10
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Traj 00000-dim (Normal)
0.0
040
0.1
-0.2
-04
Time
(Normal)
Traj dim 1 (Normel)
Too den
rue din
0.5
C2
Time
(Normal)
T 2 2dim (Normal)
0.0
FU2:0m
-0.1
Fmod.01 mm
Time
(a) Trajectories cases predicted by DKO.
(b) Trajectories 3 cases predicted by Ours.
Figure 12: Comparison of Prediction Performance on Cartpole Environment with 10% training dataset
Tajertayy
Trajectiry
0.0
-0.1
5000 drocddrr
0.05
-02
0.50
02886 dimeentton
dmssscce
(a) Control performance of DKO (fail)
(b) Control performance of Ours
Figure 13: Comparison of Control performance on Cartpole with 30% training dataset
Table 10: Comparison of prediction error (MSE) across environments with varying fractions of training data. Our method shows clear advantages under limited training data, highlighting its data efficiency.
Environment
Method
1%
5%
10%
w/o Input Encoder
0.0032
0.00019
0.00011
Mountain Car
w/ Input Encoder 0.00087
0.00018
0.00011
w/o Input Encoder 0.3747
0.0318
0.0114
Pendulum
w/ Input Encoder
0.1338
0.0067
0.0065
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(Normal)
Time 14]
Time
(Normali
dim (Nermal)
0-
13
14
Tine h)
Timn
(Normal)
(Nommal)
0.0
3.
Time 151
10ne
Figure 14: Prediction Performance on GRN Environment
DKO
3.5
DKO
Ours
3.0
Ours
2.5
loss
2.0
1.5
0.4
1.0
0.2
0.5
0.0
0.0
2500 5000
7500
10600
12500 15000 17500 20000
500
1000
1500
2000
step
step
(a) Reacher
(b) Franka
Figure 15: Loss versus training step Reacher and Franka. Our method converges faster than DKO.
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Tra 2-dim (Normal)
Traj dim (Nermal
Tiae
Th.m
1000
1.033
Poat de
2,22
0.155
0,055
0.50
0.25
0.30
0.25
2:30
0.20
0.50
0.25
0.30
0.30
2.0
0.0
Time [X]
Traj 2-dim (Normall
Traj 2-dim (Normall
1.34
Prrd
1.34
1.52
131
8.2%
0.50
8.90
Time
Tha dim (Nermal)
Traj 2 dim 9 (Normal)
3.700
0.730
Tie
3.700
0.15
0.330
2.905
0.20
0.55
0.30
0.15
0.25
0.00
0.55
0.30
2.20
Time (3)
(a) Trajectories of end effector predicted by DKO.
(b) Trajectories of end effector predicted by Ours.
Figure 16: Prediction Performance (end effector position) on Franka Environment with 10% training dataset
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Control Actions
30-
25-
20
15
10
-5-
9000 desnnniinn
(a) Small weight: MPC control
(h) Small weight: MPC trajectory
Control Actions
State Trajectory
40
25
300
700
-3.0
-2.
-2.6
-1.5
-3.0
0.5
(c) Larger weight: control
(d) Larger weight: MPC trajectory
Figure 17: Comparison of MPC performance under small and larger weights settings. The larger weight setting shows larger eigenvalues but poorer predictive control (larger energy use and over- shoot in trajectory).
Encoder
z(t)
Current State x(t)
Observable Encoder the
Control Input u(t)
Input Encoder
a(t)
Figure 18: With Input Encoder
32


Reviews:
[{"review_id": "0gNRK5IDbt", "reviewer": "Reviewer_ybHH", "summary": "The paper proposes a Koopman-based representation learning framework, implemented as an end-to-end Neural ODE, that learns nonlinear dynamical models from limited data while ensuring the learned models remain suitable for control.  It enforces controllability by construction via a new canonical parameterization of the latent linear dynamics (A, B), and shows that controllability of the learned latent model implies controllability of the original (nominal) system.  The method also shapes the degree of controllability by adding a finite-horizon output-Gramian regularizer that enlarges the smallest eigenvalue and controls the condition number to promote well-conditioned models for control. For downstream control, the learned linear surrogate enables a convex quadratic-program MPC in the lifted space with receding-horizon execution.  Empirically, on standard nonlinear benchmarks (pendulum, mountain car, cartpole), the approach achieves more accurate long-horizon prediction and better MPC performance with improved data efficiency compared to Deep Koopman Operator and MLP baselines.", "questions": "1. Aren't Theorems 1 and 2 about any linear systems? Any specific connection to Koopman operators?\n2. Will the proposed approach work even in slightly highly dimensional non-linear systems, such as simple MuJoCo environments?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "1. Learning the Koopman operator is an interesting and important classical control system problem.", "weaknesses": "1.\tThis paper has marginal algorithmic or theoretical contributions. See the details below.\n2.\tThe main contribution is a method to learn the Koopman operator. However, the proposed methods are quite standard, typical in any Koopman operator learning approach. While the paper prominently states the connection with Neural ODE, it is not really clear why that is relevant or interesting. Lifting the state x to a high-dimensional variable z and modeling the non-linear system as a linear system at this lifted space is indeed the standard approach of Koopman theory. So, the reason for presenting neural ODE as new approach is not clear. \n3.\tThe theoretical results presented are standard results from linear systems theory. In particular, Theorem 1 about controllability and Theorem 2 about reparameterization to a suitable form are standard results. So, it is not clear why these are presented as novel results, and how they contribute to the Koopman operator theory. \n4.\tOnce the Koopman operator is learned, LQR or MPC are the standard ways to design a control policy. There is no novelty in that part.\n5.\tThe experiments are done on three very simple tasks: mountain car, pendulum, cartpole. It is not clear if the proposed methods can scale to even a slightly more difficult settings, say, mujoco environments.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "mwHB1VZqlt", "reviewer": "Reviewer_x559", "summary": "The paper consider the problem of modeling and control of nonlinear systems via Koopman operator theory (KOT). Particularly, the paper ensures controllability by using a certain canonical parameterization and also are able to tune the degree of controllability to ensure better control performance. In addition they use Neural ODE to learn a KOT based model Simulation results present the efficacy of their approach", "questions": "I have the following questions:\n\n1) Since the Koopman model is just a surrogate model, there is an approximation error between actual and approximated model. This is not considered in the paper. In addition, there exists works that consider this approximation rigorously (see [1]).\n2) There are also papers that consider the controllability of the KOT surrogate model. It is not clear to me on how your work is significantly different from [2].\n3) It is not always possible to make a nonlinear system controllable by ensuring controllability in the KOT model.\n4) There has been lots of work that convert unknown dynamics into Koopman based models and use the surrogate model for control design purposes (see [3,4,5] ). It is not clear how your work is different from them. Yes, you mention that you also ensure controllability of the KOT model, but [2] does this as well\n5) In simulation results, although you have compared your approach with some prior methods, the nonlinear systems considered are too simple. I would like examples of more complex robotic systems such as quadruped and humanoid where efficiency of KOT models is yet to shown. \n6) Showing that the proposed approach works on more complex nonlinear dynamics such as quadruped or humanoid needs to shown. In addition, they must be well motivated to show their efficacy compared to RL based control policies\n7) Ensuring controllability as one of your main contributions would not add significant contribution to the paper given the fact that there exists some work that do it\n\n\n\n[1] Mamakoukas, G., Di Cairano, S. and Vinod, A.P., 2022, June. Robust model predictive control with data-driven Koopman operators. In 2022 American Control Conference (ACC) (pp. 3885-3892). IEEE.\n\n[2] Choi, Joonwon, Minhyun Cho, Hyunsang Park, Vishnu Vijay, and Inseok Hwang. \"On The Controllability Preservation of Koopman Bilinear Surrogate Model.\" In 2024 IEEE 63rd Conference on Decision and Control (CDC), pp. 3457-3462. IEEE, 2024.\n\n[3] Korda, M. and Mezić, I., 2018. Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica, 93, pp.149-160.\n\n[4] Zinage, V. and Bakolas, E., 2023. Neural koopman lyapunov control. Neurocomputing, 527, pp.174-183.\n\n[5] Salzmann, T., Kaufmann, E., Arrizabalaga, J., Pavone, M., Scaramuzza, D. and Ryll, M., 2023. Real-time neural mpc: Deep learning model predictive control for quadrotors and agile robotic platforms. IEEE Robotics and Automation Letters, 8(4), pp.2397-2404.", "limitations": "", "rating": 4, "confidence": 5, "soundness": 3, "presentation": 3, "contribution": 1, "strengths": "The main strengths are as follows:\n1) The efficacy of their approach over other methods is clearly show.\n2) The literature review in the introduction is extensive.\n3) The use of Neural ODE is a smart choice over MLPs; one main reason being free from what discretization issues while converting the continuous to discrete systems.", "weaknesses": "I have worked on Koopman operators for a while now. I believe these are the following weakness:\n\n1) The KOT for control literature is quite rich today. For instance works that consider controllability for KOT based models, use Neural ODE or similar techniques exist in literature. \n2) The motivation for this work is not clear to me.\n3) The approximation error between the KOT and real system is not considered. This is particularly important because Koopman operator is an infinite dimensional linear operator and to make it of practical value, a finite approximation of the Koopman operator via EDMD is usually made.\n4) Missing real world examples of more sophisticated nonlinear systems such as quadruped or humanoid is missing. There are already hundreds of paper that consider standard and simple nonlinear systems.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "FhDtVI4fVx", "reviewer": "Reviewer_vF1q", "summary": "This paper presents a dynamics learning method that enforces controllability as a structural prior in order to improve the quality and usability of the learned dynamics. In particular, this paper:\n* Introduces a Koopman-based framework for learning linear surrogate models using Neural ODEs.\n* Enforces controllability of the learned model by design, by proposing a particular parameterization of the to-be-learned Koopman operator. This parameterization is based on the observation that enforcing state-to-output controllability in the lifted space (easy) is equivalent to enforcing output-to-output controllability in the original space (hard).\n* Provides experiments on four control settings (mountain car, pendulum, cartpole, gene regulatory network), showing improved quality of the learned dynamics and improved controllability of those dynamics compared to SOTA baselines.", "questions": "* What do results look like on larger experimental systems?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 4, "contribution": 3, "strengths": "* The idea of the paper is clear, simple, and well-grounded: Encode controllability as a structural prior enforced by design within neural ODE learning. The method and motivation are very clearly explained, and the clear notation facilitates understanding of how the Koopman operator is integrated into the neural ODE framework.\n* The proposed parameterization is backed by theoretical proofs, proving controllability.\n* The proposed loss regularization is a nice idea to ensure not just the binary notion of \"controllable,\" but also improve the degree of controllability.\n* The experimental results on mountain car, pendulum, and cartpole compare against SOTA baselines, and clearly show improved quality of dynamics learning in low-data regimes (comparable quality in higher-data regimes), as well as improve controllability of the learned dynamics when used within MPC.", "weaknesses": "* The experimental settings are all quite small, with the largest setting (GRN = gene regulatory network) including 6 dimensions and 3 control inputs, and with the best results on the smaller single-input settings (GRN is the only multi-input setting, and results are more more marginal). It would be nice to see additional and more convincing validations on larger systems.\n* For large systems, the eigenvalue computation needed for the Gramian regularization term might be expensive to compute. In general, it would be nice to see relative training cost between the different methods reported.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "0WTNyK1498", "reviewer": "Reviewer_dLx9", "summary": "This paper proposes a framework for learning Koopman representations of nonlinear control systems that are guaranteed to be controllable. The authors derive conditions linking output-to-output controllability (OOC) and standard state-output controllability (SOC), and introduce a canonical parameterization of the latent linear dynamics matrices A_\\theta, B_\\theta that ensures controllability by construction. Additionally, a controllability Gramian regularizer is used to improve the conditioning (“well-controllable” property). The learned model can be used directly in model predictive control (MPC). Experiments on benchmark systems demonstrate improved sample efficiency and control performance compared to standard Deep Koopman (DKO) and MLP-based models.", "questions": "Q1. The paper constrains A_\\theta, B_\\theta through a controllable canonical form plus similarity transform P_\\theta. Does this restriction reduce the expressive power of the learned Koopman operator? Please clarify whether your parameterization still provides a universal approximation property for arbitrary nonlinear dynamics in the lifted space, or discuss the trade-off between controllability enforcement and representational flexibility.\n\nQ2. The current method enforces controllability but not stability. Have you observed instability in the learned latent dynamics (e.g., exploding eigenvalues of A_\\theta)? It would be useful to regularize eigenvalues of A_\\theta or provide evidence that the MPC formulation mitigates instability during rollout.\n\t\nQ3. The Gramian term involves computing W_T^y. How is this computed efficiently during training, and how does it scale with the latent dimension N? It would be helpful to include complexity analysis or ablation to show that the regularizer does not dominate training time.\n\t\nQ4. The framework is called “Koopman,” but it remains unclear whether the learned observables correspond to Koopman-invariant subspaces or simply linear latent embeddings. Can you empirically or theoretically justify that \\varphi_\\theta(x) approximates Koopman eigenfunctions or invariant coordinates? Spectral or mode analysis (e.g., eigenvalue comparison with known systems) would strengthen the Koopman interpretation.\n\t\nQ5. Can your framework scale to higher-dimensional or partially observed systems (e.g., PDEs, power network, or soft-robot system)? Discuss computational limits and whether encoder-decoder architectures can handle such cases without losing controllability guarantees.\n\t\nQ6. The paper does not compare against recent Koopman or geometric control approaches (e.g., KEEC) that emphasize structural invariance. What are the pros and cons of controllability-based representations and equivariance-based representations?\n\nQ7. Parameters like the Gramian regularization weight and time horizon T may strongly affect results. How sensitive is performance to these settings? Suggest including an ablation or at least a qualitative discussion on how these hyperparameters influence both controllability conditioning and prediction accuracy.\n\nQ8. The paper claims the learned model can be “directly integrated” with MPC. Was this tested in a closed-loop simulation with constraints? Suggest presenting control performance metrics (tracking error, energy use, robustness) under realistic MPC settings to demonstrate deployability.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "S1. Embedding controllability directly into the Koopman parameterization is novel. Prior deep Koopman and neural ODE models treat controllability as an afterthought or soft constraint. The proposed canonical form and similarity transform provide an elegant, principled way to enforce this property.\n\nS2. The proofs linking OOC↔SOC and the use of the finite-time output controllability Gramian are mathematically well-grounded and consistent with classical control theory. This makes the paper attractive to both ML and control communities.\n\nS3. The ability to plug the learned model directly into linear MPC is a significant engineering advantage. The demonstration that the approach maintains performance under limited data conditions is also persuasive.", "weaknesses": "W1. There is limited novelty relative to the existing Koopman literature. While embedding controllability guarantees is new, the core architecture—learning a linear operator in latent space via encoder + linear dynamics + decoder—is largely inherited from standard deep Koopman frameworks. The contribution thus appears incremental in form but conceptual in framing. The authors should clarify in what sense their controllability parameterization extends beyond “structured Koopman learning” or prior work on “control-aware embeddings.”\n\nW2. Controllability parameterization may constrain expressive power. The use of a canonical controllable form plus similarity transform P_\\theta guarantees controllability but may significantly restrict the learned dynamics manifold. It is unclear whether the proposed structure can approximate arbitrary Koopman operators as the dimension of z increases, or whether expressiveness is traded off for structural rigidity. \n\nW3. There seems to be no stability guarantees. The method ensures controllability but says little about the stability of the learned system -  an equally critical property for control deployment. Without constraints on the eigenvalues of A_\\theta, the learned linear system may be unstable, making MPC optimization difficult or ill-posed.\n\nW4. All benchmarks are relatively small and well-known toy systems (Pendulum, CartPole, etc.). While these are standard, they do not convincingly demonstrate the method’s scalability or robustness to real-world noise and unmodelled dynamics. \n\nW5. Computational cost and training stability are unclear. The paper lacks analysis of training complexity, numerical conditioning, or the computational overhead introduced by the Gramian regularizer (which involves integrating matrix exponentials).\n\nW6. The connection to Koopman theory is not fully justified. Although the paper uses the Koopman terminology, the approach essentially learns a linear latent-space model via a neural encoder. 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Author rebuttal:

Decision: Accept (Poster)