Scaling Law of Neural Koopman Operators

Abulikemu Abuduweili; Yuyang Pang; Feihan Li; Changliu Liu

cs.ROFeb 23, 2026

Scaling Law of Neural Koopman Operators

Authors: Abulikemu Abuduweili, Yuyang Pang, Feihan Li, Changliu Liu

Organizations: Authors are with the Robotics Institute, Carnegie Mellon University, Pittsburgh, PA 15213, USA.

Abstract

Data-driven neural Koopman operator theory has emerged as a powerful tool for linearizing and controlling nonlinear robotic systems. However, the performance of these data-driven models fundamentally depends on the trade-off between sample size and model dimensions, a relationship for which the scaling laws have remained unclear. This paper establishes a rigorous framework to address this challenge by deriving and empirically validating scaling laws that connect sample size, latent space dimension, and downstream control quality. We derive a theoretical upper bound on the Koopman approximation error, explicitly decomposing it into sampling error and projection error. We show that these terms decay at specific rates relative to dataset size and latent dimension, providing a rigorous basis for the scaling law. Based on the theoretical results, we introduce two lightweight regularizers for the neural Koopman operator: a covariance loss to help stabilize the learned latent features and an inverse control loss to ensure the model aligns with physical actuation. The results from systematic experiments across six robotic environments confirm that model fitting error follows the derived scaling laws, and the regularizers improve dynamic model fitting fidelity, with enhanced closed-loop control performance. Together, our results provide a simple recipe for allocating effort between data collection and model capacity when learning Koopman dynamics for control.

Figures & tables

Panel of Figure 1 — Fig. 1 : Overview. (a) Mechanical systems (e.g., robot arm, quadruped) exhibiting complex nonlinear dynamics in physical state space.e. (b) Our proposed Koopman architecture uses an encoder network $\Psi_{\theta}$ to map the physical state $x_{t}$ into a latent space $z_{t}$ , governed by linear dynamics: $z_{t+1}=\mathbf{A}z_{t}+\mathbf{B}u_{t}$ .

Environment	State Dim.	Action Dim.
Damping Pendulum	2	1
Double Pendulum	4	2
Franka Panda	14	7
Kinova Gen3	14	7
Unitree Go2	35	12
Unitree G1	53	23

TABLE I : Summary of environment dimensions. Note: Unitree G1 and Go2 train on trimmed state-action spaces.

Fig. 2 : Scaling of prediction error with training sample size (

m

). Subplots display error curves across latent dimension multipliers (

n_{\text{multi}}

). Dots denote random seeds, and solid lines show fitted power-law trends. The legend reports the scaling exponent

\alpha

(error decay rate). Increasing dataset size consistently reduces error via a power law across environments, empirically validating theoretical sampling error convergence.

Fig. 3 : Scaling of prediction error with latent dimension

n

. Subplots display error curves across training sample sizes (

m

). Increasing latent dimension generally reduces error, empirically validating projection error convergence as Koopman subspace capacity expands.

Environment	$\alpha_{m}$ (Sample Size)	$\alpha_{n}$ (Latent Dimension)
Damping Pendulum	1.15	4.3
Double Pendulum	0.75	2.89
Franka	0.49	1.16
Kinova	1.44	2.58
Go2	0.65	0.47
G1	0.52	0.34

TABLE II : Fitted power-law exponents (

\alpha

) for prediction error decay. Sample efficiency

\alpha_{m}

is averaged across latent dimensions and seeds. Spectral decay

\alpha_{n}

is averaged across seeds in high-data regimes (

m\geq 10^{4}

), excluding smaller samples to isolate asymptotic projection error from finite-sample overfitting.

Fig. 4 : Scaling of prediction error with latent dimension (

n

) under the coupled scaling law

m=\text{coeff}\cdot n\ln n

. As the data sufficiency coefficient increases, the error curves converge toward the asymptotic full sample baseline.

Fig. 5 : Franka arm tracking error versus training samples (left) and latent dimension (right). Auxiliary losses (+Both) yield consistent performance boosts.

Fig. 6 : G1 performance scales positively with sample size (left) and latent capacity (right). Combined auxiliary objectives (+Both) achieve the highest survival rates.

Fig. 7 : Go2 control performance (survival steps). Auxiliary losses enable better scaling, especially in high-capacity regimes (right) where the baseline struggles.

Metrics	Environment	+ Cov loss	+ Ctrl loss	+ Both
Track Err ( $\downarrow$ )	Franka	6.18%	14.30%	25.09%
Survival Rate ( $\uparrow$ )	G1	2.09%	7.29%	9.60%
Survival Rate ( $\uparrow$ )	Go2	13.6%	59.62%	74.76%

TABLE III : Relative control performance improvement over baseline via Covariance (

\mathcal{L}_{\text{cov}}

) and Inverse Control (

\mathcal{L}_{\text{ctrl}}

) auxiliary losses.

Environemnt	EDMD	NNDM	Koopman	Koopman
Environemnt	(Poly)	(MLP)	(Base NN)	(+Aux)
Franka	0.441	0.268	0.122	0.121
G1	0.271	0.116	0.130	0.132
Go2	0.428	0.163	0.204	0.209

TABLE IV : Open-loop prediction error comparing classical EDMD (polynomial), unstructured Neural Dynamics (NNDM), and our Neural Koopman architecture.

Metric	Env	EDMD	NNDM	Koopman	Koopman
Metric	Env	(Poly)	(MLP)	(Base NN)	(+Aux)
Track Err( $\downarrow$ )	Franka	0.442	0.514	0.437	0.256
Survival( $\uparrow$ )	G1	34.0	27.4	58.0	59.7
Survival( $\uparrow$ )	Go2	19.5	4.7	62.0	103.0

TABLE V : Closed-loop control performance comparing tracking error (Franka) and survival steps (G1, Go2). Neural Koopman combines deep learning expressivity with Linear MPC.

Panel of Figure 13 — Fig. 8 : Gram matrix condition number $\kappa(\mathbf{G})$ versus prediction error. Marker size and opacity denote sample size $m$ (smaller/darker indicates larger $m$ ). Lines show fitted trends with (orange) and without (blue) covariance regularization. Grey arrows connect identical sample sizes. The proposed covariance loss significantly reduces $\kappa(\mathbf{G})$ (leftward shift) and consistently correlates with lower error.

Panel of Figure 14 — Fig. 9 : Control matrix condition number $\kappa(\mathbf{B}^{\top}\mathbf{B})$ versus prediction error. Marker size/opacity denote sample size $m$ (smaller/darker indicates larger $m$ ), and grey arrows connect identical sample sizes. Lines show fitted trends with (orange) and without (blue) inverse control regularization. Unlike the Gram matrix, $\kappa(\mathbf{B}^{\top}\mathbf{B})$ correlates negatively with error; better models naturally evolve higher condition numbers.

Environment	State Dim.	Action Dim.
Polynomial	3	0
Damping Pendulum	2	1
Double Pendulum	4	2
Franka Panda	14	7
Kinova Gen3	14	7
Unitree Go2	35	12

TABLE VI : Summary of environments and original dimensions. Note that the Unitree G1 and Go2 models are trained on trimmed state–action spaces.

$n_{\text{poly}}$	Sample Size $(\alpha_{m},C_{m},R^{2})$	Latent Dim $(\alpha_{n},C_{n},R^{2})$
3	$(2.64,\;1.75{\times}10^{-8},\;1.00)$	$(0.28,\;1.18{\times}10^{-21},\;0.25)$
5	$(2.57,\;8.79{\times}10^{-8},\;1.00)$	$(0.20,\;2.72{\times}10^{-21},\;0.51)$
10	$(2.52,\;4.74{\times}10^{-7},\;1.00)$	$(0.14,\;4.51{\times}10^{-20},\;0.74)$
20	$(1.91,\;2.98{\times}10^{-6},\;1.00)$	$(0.08,\;4.54{\times}10^{-20},\;0.57)$
50	$(1.64,\;2.63{\times}10^{-5},\;0.99)$	$(0.02,\;4.17{\times}10^{-19},\;0.53)$

TABLE VII : Impact of Nonlinearity on Scaling. Higher

n_{\text{poly}}

leads to slower exponent decay term (lower

\alpha

) and higher error floors (

C

R^{2}

indicates fit reliability.

Degree ( $n_{\text{poly}}$ )	Baseline	+ Cov loss	Error Reduction
3	$9.51\times 10^{-6}$	$8.89\times 10^{-6}$	$6.5\%$
5	$2.12\times 10^{-5}$	$2.04\times 10^{-5}$	$4\%$
10	$6.22\times 10^{-4}$	$5.47\times 10^{-4}$	$12.1\%$
20	$1.34\times 10^{-2}$	$1.17\times 10^{-2}$	$12.7\%$
50	$1.98\times 10^{-1}$	$1.62\times 10^{-1}$	$18.1\%$

TABLE VIII : Impact of covariance regularization loss (

\mathcal{L}_{\text{cov}}

) on prediction error. Evaluated across varying polynomial degrees (

n_{\text{poly}}

) on largest model capacity (

n_{\text{multi}}=16

) and dataset size (

m=140\text{k}

Fig. 10 : Impact of Covariance Regularization on Latent Feature Correlation. Heatmaps display the correlation matrices of the learned embeddings $\phi(x)$ for varying polynomial degrees $n_{\text{poly}}\in\{3,10,50\}$ . Top Row (Baseline): Without regularization, features exhibit high cross-correlation (dense off-diagonal blocks). Bottom Row (w/ $\mathcal{L}_{\text{cov}}$ ): With regularization, the matrices become significantly sparser, demonstrating effective feature decorrelation. The mean off-diagonal correlation $|\overline{corr}|$ is reported for each case.

Koopman Operator Learning

Koopman operator learning builds finite-dimensional linear models of nonlinear dynamics by finding useful observable representations from data.

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This paper presents a model-based reinforcement learning (RL) framework for optimal closed-loop control of nonlinear robotic systems. The proposed approach learns linear lifted dynamics through Koopman operator theory and integrates the resulting model into an actor-critic architecture for policy optimization, where the policy represents a parameterized closed-loop controller. To reduce computational cost and mitigate model rollout errors, policy gradients are estimated using one-step predictions of the learned dynamics rather than multi-step propagation. This leads to an online mini-batch policy gradient framework that enables policy improvement from streamed interaction data. The proposed framework is evaluated on several simulated nonlinear control benchmarks and two real-world hardware platforms, including a Kinova Gen3 robotic arm and a Unitree Go1 quadruped. Experimental results demonstrate improved sample efficiency over model-free RL baselines, superior control performance relative to model-based RL baselines, and control performance comparable to classical model-based methods that rely on exact system dynamics.

Abstract

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