MetaKoopman: Bayesian Meta-Learning of Koopman Operators for Modeling Structured Dynamics under Distribution Shifts
Authors: Mahmoud Selim
Sriharsha Bhat
Karl Henrik Johansson
Venue: NeurIPS 2025 poster ()
Keywords: Meta Learning
Koopman Theory
Motion Planning
Primary area: reinforcement_learning
TL;DR: MetaKoopman is a Bayesian meta-learning method for the Koopman operator that enables fast, uncertainty-aware prediction and planning for nonlinear dynamics under distribution shifts.

Abstract: Modeling and forecasting nonlinear dynamics under distribution shifts is essential for robust decision-making in real-world systems. In this work, we propose **MetaKoopman**, a Bayesian meta-learning framework for modeling nonlinear dynamics through linear latent representations. MetaKoopman learns a Matrix Normal-Inverse Wishart (*MNIW*) prior over the Koopman operator, enabling closed-form Bayesian updates conditioned on recent trajectory segments. Moreover, it provides a closed-form posterior predictive distribution over future state trajectories, capturing both epistemic and aleatoric uncertainty in the learned dynamics. We evaluate MetaKoopman on a full-scale autonomous truck and trailer system across a wide range of adverse winter scenarios—including snow, ice, and mixed-friction conditions—as well as in simulated control tasks with diverse distribution shifts. MetaKoopman consistently outperforms prior approaches in multi-step prediction accuracy, uncertainty calibration and robustness to distributional shifts. Field experiments further demonstrate its effectiveness in dynamically feasible motion planning, particularly during evasive maneuvers and operation at the limits of traction. Project website: [https://mahmoud-selim.github.io/MetaKoopman/](https://mahmoud-selim.github.io/MetaKoopman/)

Motanoopman: Bayesian Meta-Learning of Koopman Operators for Modeling Structured Dynamics under Distribution Shifts
Mahmoud Selim
Sriharsha Bhat²
Karl H. Johansson
!TRATON, mahmoud. selim@se. traton. com
KTH Royal Institute of Technology. {mase2, svbhat, kallej)okth.se
Abstract
Modeling and forecasting nonlinear dynamics under distribution shifts is essen- tial for robust recision-making in real-world systems. In this work, we propose MetaKoopman, Bayesian meta-learning framework for modeling nonlinear dy- namics through linear latent representations. MetaKoopman learns a Matrix Normal-Inverse Wishart (MNIW) prior over the Koopman operator, enabling closed-fooe Bayesian updates conditioned on recent trajectory segments. Moreover, it provides a closed-for posterior predictive distribution over future state trajecto- ries, capturing both epistemic and aleatoric uncertainty in the learned dynamics. We evaluate MetaKoopman
full-scale autonomous truck and trailer system across a wide range adverse winter scenarios scenarios-including including snow, ice, and -friction conditions- well in simulated control tasks with diverse distribution shifts. MetaKoopman consistently outperforms prior approaches in multi-step prediction accuracy, uncertainty calibration and robustness distributional shifts. Field ex- periments further demonstrate its effectiveness in dynamically feasible motion planning, particularly during evasive maneuvers and operation the limits of trac- tion. Project website: https: hppp://mmann h/ta/mommmm--eeeeee/e selim. selim.githul github. io/ No/MetaKoopman//
Introduction
Modern autonomous systems must operate in complex, nonstationary environments, where the underlying system dynamics neither fixed nor fully known. Terrain properties shift from asphalt to ice; payload distributions may change with each mission; mechanical degradation or actuator latency can accumulate over time. Learning single predictive model that accounts for all such variability is difficult-if not infeasible. These sources of variation induce what known distributional shift: mismatch between the training conditions used learn model and the test-time conditions in which is deployed. In safety-criticn applications such as autonomous driving or robotic control, even small prediction errors can compound, leading unstable or unsafe behavior. These risks are further amplified when models are unable to assess or convey uncertainty in their design learning-basee systems that
predictions. This raises fundamental question: how remain both accurate and reliable when the dynamics themselves evolve?
From modeling perspective, this challenge exposes central limitation of standard approaches: the assumption globally valid dynamics model that remains accurate across diverse environments and latent conditions. In practice, such models exceptionally difficult-if not impossible- to obtain. Real-world dynamics are inherently time-varying due to hardware wear, variable payloads, and environmental factors that either unobserved partially observable best. Meta-learning [2] provides one path forward by training models to quickly adapt to new conditions using limited data. However, many meta-learning frameworks rely on gradient-based inner loops iterative optimization procedures [10, 11, 45), which can introduce latency and computational overhead that make them
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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unsuitable for real-time tial-time deployment. In contrast, Koopman operator theory [16] represents nonlinear dynamics through linear space operators, enabling highly efficient multi-step prediction and seamless integration with classical control and planning methods. Yet Koopman- models are typically trained offline and remain static at test time [43, 3, 1, 17], limiting their ability to respond to changing dynamics. This motivates our approach: a model that integrates the structured efficiency of Koopman operators with the adaptivity and data efficiency of meta-learning, while enabling online updates and principled uncertainty quantification.
In this work, we introduce MetaKoopman, a Bayesian meta-learning framework for modeling dynamics under distribution shift. At its core, MetaKoopman places a Matrix Normal-Invers Wishart (MNIW) prior over the Koopman operator, which is meta-learned across diverse environments. In doing so, MetaKoopman learns a distribution over the underlying dynamics, capturing uncertainty in system evolution. During deployment, this prior is adapted online through closed-form Bayesian updates using recent trajectory data to obtain an updated belief (the posterior) over the dynamics. This posterior is then used to generate probabilistic forecasts (posterior predictive distribution) over future states, allowing the model to quantify both epistemic and aleatoric uncertainty in its predictions. The resulting uncertainty-aware model is integrated into a sampling-based motion planner via a variational action encoder, enabling the efficient generation of dynamically feasible trajectories. MetaKoopman is evaluated across a suite of simulated benchmarks that expose models to varying terrain slopes, friction conditions, and payload dynamics. In all cases, it achieves superior multi-step prediction accuracy, calibrated uncertainty, and robustness to distributional shifts. Beyond simulation, deploy MetaKoopman on 37.5-ton autonomous truck-rrille system operating in harsh winter conditions, including snow, ice, and mixed-friction surfaces. In one safety-critical test, where a virtual obstacle appears react safely. In contrast, MetaKoopman rapidly infers changing surface dynamics and initiates a lane change, demonstrating how structured Bayesian adaptation enables safer and more reliable control in real-world autonomous systems.
icy road, non-adaptive models mispredict braking feasibility and fail to
Our contributions can be summarized as follows:
1. Bayesian meta-learning of Koopman operators First to combine Koopman structure with Bayesian meta-learning, giving analytic test-time posterior over the Koopman operator. In addi- tion, we derive closed-form posterior-predictii distribution, enabling principled uncertainty quantification in forecasts.
2. Meta-learned tempering for adaptation scalar 3, meta-learned during meta- training rescales the MNIW prior precision at test time. This simple yet effective mechanism increases responsiveness to online adaptations while maintaining calibrated uncertainty.
3. Variational action encoder- - acttnntaction sampling accelerates trajectory rollouts by an order of magnitude, removing the main model-based planning bottleneck.
4. Comprehensive evaluation We collect truck dynamics dataset in adverse weather (snow, ice, and mixed friction) and use with five simulated environments exhibiting distribution shifts to demonstrate state-of-the-art accuracy, calibrated uncertainty, and robustness.
5. Real-world deployment - Live 37.5-ton truck trials show adaptive braking on ice and safe evasive snow maneuvers where static models fail.
2 Related Work
Koopman operator theory [16] provides a linear, though infinit-diimensional, framework for analyzing nonlinear dynamical systems by lifting them into a space of observable functions, where their evolution can be described linearly. Early methods such as dynamic mode decomposition (DMD) [31. 32] and extended dynamic mode decomposition (EDMD) [42, 19] approximate the operator using manually selected observables. process that is often brittle and task-specific. To address this, recent advances leverage deep learning to automatically learn expressive embeddings [20, 26, 43, 37], leading to neural extensions such as deep-DMI -MD and modeling capacity and scalability of Koopman- -based methods. These developments have enabled accurate prediction and control across a range of domains, particularly where linearity allows the use of classical controllers such as LQR and MPC [27, 21, 17, 1J. Applications span soft robotics [3, 33), aerial robotics [34, 22), robotic manipulation [4], and vehicle dynamics [6, 40], demonstrating the versatility of Koopman- based modeling and control across diverse nonlinear systems.
which significantly enhance the

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Recent studies have further investigated adaptive and time-varying Koopman operators for modeling systems with evolving dynamics. For example, an adaptive Koopman embedding was introduced to augment nominal Koopman model with online neural module that updates operator matrices in real time, improving robustness to parametric variations and external disturbances in control tasks [35]. Similarly, the Koopman Neural Forecaster (KNF) [41] employs global and local operators to capture both stable and evolving dynamics in non-stationary systems, However, their approach is confined to time-series forecasting and does not account for control inputs. Other approaches explore explicitly time-varying Koopman operators for modeling changing systems [46, 14], but these typically require online optimization and lack principled uncertainty estimation. In contrast, our method enables closed- form Bayesian adaptation using a meta-leamed prior, providing both computational efficiency and principled uncertainty quantification.
Meta-learning provides framework for learning models that rapidly adapt to new tasks by leveraging experience across related tasks [10, 9, 25, 29]. In dynamical systems, meta-learning has been applied learn models that adapt online to changes in dynamics or environment [24, 30, 7, 8]. Approaches such Bayesian MAML [45], PLATIPUS [11], and PEARL [28] demonstrate that learning a prior over model parameters enables fast adaptation from limited recent experience, addressing the challenge of distributional shift in real-world domains. Critically, these methods avoid the need for globally accurate dynamics models by focusing on fast, local adaptation-enabaee enabling robustness to perturbations such as hardware failures, terrain variation, or sensor noise. However, many approaches either rely on optimization-ban based adaptation, which is computationally expensive at test time, or lack principled uncertainty quantification. In contrast, method performs closed- form Bayesian adaptation from short trajectory segments, combining computational efficiency with tractable uncertainty estimation.
3 Preliminaries
3.1 Modeling Nonlinear Dynamical Systems with the Koopman Operator
We consider a discrete-time, nonlinear dynamical system governed by the state evolution
F++1 [ uz),
(1)
where 2t E R" denotes the system state, Ut E R denotes the control input, and f is an unknown nonlinear function. The objective is to predict future states Æt+1, T242,... Et-h over a horizon h, given the current state and current and future control inputs.
The Koopman operator, denoted as K F F, is an infinite infinite-dimensie dimensional linear operator that describes the evolution of nonlinear dynamical system. Here, F refers the set of all measurement functions or observables, which form an infinite-dimensional Hilbert space. More specifically, given an observable function v/: the evolution of the system using the Koopman operator can expressed as:
(Z u,) = (ff(x u.), 44+1) = (1 u1+1)
(2)
In principle, the Koopman operator provides a complete linear description of the system's dynamics in the lifted space. However, operating directly in the computationally intractable. In the following section, we introduce a finiteddimensional approximation of the Koopman operator that can be learned from data.
Hilbert space F is
3.2 Meta-Lcarrning for Fast Adaptation Across Trajectories
Meta-leariig provides a framework for training models that can rapidly adapt new tasks using limited data. In our setting, a task T corresponds to a short trajectory segment collected from particular operating condition-suce such as a specific road surface, slope, or mass distribution- where the goal is to predict future states conditioned on recent state-action history. Let p(T) denote distribution over such tasks, each defined by a trajectory dataset Df = {(x,, u)} sampled from a distinct environment.
The meta-learning objective is find shared model parameters 9 and an adaptation procedure A on an adaptation dataset D'F, such that the adapted parameters 6' = A(D", 8) minimize a supervised loss on a held-out query set Dist drawn from the same task:
min EE~xCCT (D) 0')l s.t. 0' = A(D 0). 0.A
(3)
3

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Keys
belief Koopman
Jewu
Powd
Stwal
The Koopman openator
Dictrbbtton over the Koopman operator
Staten actions
Context Encoder
Action Encoder
Koopman slates
Integrates over all probable states
Pr4
Encoda actions
Concatenate operation
Figure I: An Overview of the MetaKoopman framework. (1) Past states and actions are encoded into a latent representation using the context encoder. (2) This latent representation is used to update the posterior belief over the Koopman operator. (3) The updated operator is used to predict future latent states, which are decoded back to the original state space for forecasting and planning.
This formulation enables the model to generalize across environments by learning a shared initializa- tion- prior-thar can be quickly adapted to new environments from a small number of observations. In our case, the parameters define a probabilistic prior over the Koopman operator, which up- dated online using recent trajectory data. This structure supports task-specific, uncertainty-aware predictions, and forms the foundation of the MetaKoopman framework described next.
The MetaKoopman Framework
We now present the MetaKoopman framework, which combines Koopman-based modeling with Bayesian meta-learing to enable adaptive and uncertainty-awar dynamics prediction. The method built around a latent linear dynamical system, where the Koopman operator is inferred via closed-form Bayesian updates from a meta-learned prior. We first describe the model structure (Section 4.1), then present Bayesian inference and predictive forecasting (Section 4.2), followed by meta- meta-learning of the prior (Section 4.3), and finally its integration with a real-time planner (Section 4.4).
4.1 Latent Koopman Dynamics Model
Building on the preliminaries, directly operating with the infinite-dimensionall Koopman operator is infeasible real-world systems. To make Koopman modeling tractable, we learn a finiteddimensional approximation to the embedding function IR" x m - Rd, where d > + which lifts state-action pairs into a latent space where the dynamics evolve approximately linearly.
We denote the resulting embedding as & g(I,, uz) E Rd and decompose it into two parts: state-related embedding 2n E RD and control-related embedding in E IRd-", such that ž = (#,, ūz). The evolution of the system is then modeled linearly in latent space as:
Te+
K
where K. Rixd is minte-aimmesiooal Koopman operator. Unlike prior methods that embed each state-action pair independently, we learn embeddings that capture temporal dependencies across multiple steps. Specifically, define sequence- based embedding delayed inputs:
l ād] Ga (Zt-qu Ut-qg Lt, Ut).
(5)
We refer Go as the context encoder, it maps window of past states and actions into compact embedding that summarizes local dynamies. This sequence- representation captures temporal dependencies that help the Koopman operator more accurately linearize the system evolution.

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These temporally enriched embeddings provide more informative representation of local dynamics than single-step inputs, supporting more accurate modeling and inference. The use of delayed coordinates is further supported by Takens's ' theorem [38], which motivates the use of time-delay embeddings for reconstructing system dynamics. We analyze the effect of history length in the appendix, and find that moderate windows provide the best tradeoff between adaptability and predictive accuracy.
To perform multi-step prediction, we additionally encode future control inputs ue+ly.... Ug+h using a dedicated action encoder. This ensures that the action embeddings are consistent with the latent dynamics inferred from the recent trajectory context. Finally, the predicted latent states are mapped back the original state space using learned linear observation matrix C E R=*x"
i = C..-
An overview of the high-level model architecture is provided in Figure I; additional implementation details are deferred to the Appendix.
4.2 Closed-Form Bayesian Inference for Koopman Dynamics
To support data-efficient and uncertainty-awar adaptation, we formulate Koopman operator learning Bayesian inference problem. Rather than estimating fixed operator, place distribution over plausible operators and update using recent trajectory segments. This enables principled uncertainty quantification in both the model's parameters and its predictions, and facilitates rapid adaptation to new environmental conditions.
Probabilistic Formulation. We extend the classical Koopman model to account for process noise, measurement error, and embedding imperfections. Specifically, we model latent dynamics as:
कt+1 K2² +,
(0 E),
Where K E Raxd is the stochastic Koopman operator, and Σ RTX* is the process noise covariance. Under the assumption of i..dd noise, the likelihood of latent transitions & - T+41 is:
N p(X K, - II N E) Σ),
(8)
where X E RIXN 7 E R and N is the number of data points. The conjugate prior for this likelihood is the matrix normal inverse Wishart distribution MNIW:
K. MNIW(M, V , i),
where KT| ~ MN(M, V,V,, and Σ IW $).
Lemma 4.1. Given the likelihood (Eq.8), the posterior over K, E remains in the MNIW family:
K.E ~ MNIW(M, V. D. 0.)),
(10)
with the posterior parameters given by:
M =S18811
=
0= N 8
V = $ Szje
(11)
and
= = 8 + MV, S., 277 +V, S.n = X + MVMT,
Sel= Sun - S22 1-1 TT
(12)
22
Proof. The proof follows directly from [23].
Lemma 4.2. Given the posterior in Eq. (10), the one-step posterior predictive distribution can be well-approximated by a multivariate Gaussian:
3 12, X, 2 ~ N (Mar, & (1+
(13)
5

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and the multi-step predictions for future latent states can be approximated by a moment- matched Gaussian, obtained by recursively propagating the first two moments and re-approxmmating by Gaussian al each step:
Me+1 = M Uzzt:
(14a)
E441 = MY M + it (1+ (I+ Pz, + ur(V (r(VE Ez,))-
(14b)
where (u,EE,) denote the latent state, Et mean and covariance and (Mz,tr Dz,) denote the mean and covariance of the regressor 2.
Proof. See the appendix for the derivation and discussion of the Gaussian approximation.
4.3
MetaKoopman: Bayesian Meta-Learning of the Koopman Operator
The Bayesian formulation in Section 4.2 enables posterior inference over the Koopman operator. However, this relies on the quality of the prior. Rather than specifying it manually, MetaKoopman meta-learns the prior parameters across a distribution tasks. This yields prior that supports fast, uncertainty-awarr adaptation via closed-form Bayesian updates. This section describes the meta- learning procedure (Section 4.3.1) and the test-time adaptation strategy with tempering (Section 4.3.2).
4.3.1 Meta-Learning the Prior
We parameterize the prior over the Koopman operator and noise covariance MNIW distribution with parameters 0 [M, V, i. J... The goal is to meta-learn them across distribution of tasks trajectory segment from a distinct operating condition.
~ p(T), where each task corresponds
For each task T, we split its trajectory data into an adaptation set Di and held-out evaluation set Di Using Eq. 10, we compute posterior parameters from D4, then evaluate the posterior predictive (Eq. 14) on Disu. The meta-objective is minimize the negative log likelihood of the ground truth under the posterior predictive distribution:
min ETTP(T) [Cys (0'(D4)) min EyHp(T) [- log (Diss) (((D)))),
(15)
where 8' (D4) denotes the posterior parameters obtained by updating the prior 0 with D This formulation encourages learning prior that is both expressive and adaptable: it captures structure shared across tasks, while allowing efficient posterior inference from limited data. Unlike gradient- based meta-learning methods, adaptation here is fully analytical, requiring no inner-loop optimization.
4.3.2 Online Adaptation with Tempering
To modulate the strength of prior information during adaptation, we introduce a scalar tempering factor 3 E (0, 1] that scales the prior precision. Specifically, we temper the prior as:
3 -ーー
(16)
Smaller values of 3 reduce the influence of the prior, allowing the model adapt more aggressively to new data. Importantly, we meta-learn 3 jointly with the MNIW prior parameters, treating it as learnable scalar optimized via the meta-objective in Eq. 15. This allows MetaKoopman discover an optimal tradeoff between prior knowledge and new evidence, tailored to the task distribution. Algorithm details the procedure for meta-learning the Koopman prior.
4.4 Uncertainty-Awarr Planning with Variational Action Encoding
To integrate MetaKoopman into a sampling cameling-based motion planner, we must generate a large number of dynamically feasible trajectories in real time. This typically requires sampling actions and encoding them into latent space using the action encoder, but doing so for each sample is computationally expensive, which can bottleneck the planning process.
To mitigate this, we introduce variational action encoder that enables efficient sampling of future action embeddings directly from standard Gaussian distribution. Instead of deterministically

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Algorithm 1 Meta-Learning of the Koopman Prior
Require:
Distribution over tasks p(T); initial prior parameters {AM, V, v. $ 33
1: while not converged do
2: Sample batch { B
P(T)
3:
for each task T: do
4:
Split data into Di, T.
5:
Temper the prior: V = 3-- = 3- using Eq. (16).
Compute posterior parameters using Eq. (10) on Di; 7: Evaluate log-likeliiee ligeliio on Di using Eq. (14).
6:
8: end for
9: Update using gradients of total NLL across tasks (Eg. (15)).
end while
mapping each future action to a latent embedding, the encoder models a distribution qs uz) over latent actions üj parameterized by mean and diagonal covariance matrix, where denotes the parameters of the variational action encoder. Following the standard VAE [15], We regularize this distribution during training to match a standard normal prior p(ū:) = (0, T) by minimizing the KI. divergence:
1+ Σ
Littll
LNLL +
DEL (4o u.) || I)).
(17)
This encourages the latent future action space to follow standard Gaussian, enabling direct sampling U ~ (0, 1) at test time without evaluating the action encoder. This allows the planner to efficiently generate thousands of trajectories by simulating them only through the linear Koopman dynamics.
We refer to this mechanism as Cost- Guided Latent Action Sampling (CLAS), which enables cost- based scoring of sampled trajectories while leveraging uncertainty- predictions. Implementation details and ablations are provided in the appendix.
Cost-Guided
Experimental Results
5.1 Experimental Setup
Our experiments aim evaluate the capabilities of MetaKoopman in both simulated and real- world control settings. We investigate the following key questions: (i) How accurately does the model predict future dynamics under distribution shift? (ii) Does adaptation from short trajectory segments meaningfully alter the model's predictions? (iii) How does MetaKoopman compare to existing adaptive and probabilistic basclines? (iv) Can the model support safer closed-loop control in real-world deployment scenarios?
To explore these questions, we evaluate MetaKoopman across suite of simulation benchmarks based on MuloCo [39], as well as a full-scale real-works autonomous truck. Additional details on the truck dataset collection, along with ablations and detailed comparisons -incluuunnnnnnnuunnnnn the effect of tempering. uncertainty calibration, computational complexity, the variational action encoder, and history length sensitivity-are provided in the appendix.
Evaluation environments. We evaluate MetaKoopman on a diverse set of simulation and real- world environments, each designed perturbations:
generalization under distribution shift adaptation structural
Real-World Autonomous Truck in adverse winter conditions: A 37.5-ton autonomous truck is evaluated under winter conditions including snow, ice, and mixed-friction friction roads, with shifts in tire-surface interaction and braking dynamics.
HalfCheetah-Slope: The agent is trained on mild inclines ([-10?, 10%) and tested steeper, unseen slopes ([-15%, 15%), evaluating generalization across varying terrain.
Hopper-Gravity; Gravity is scaled during training (00.9, 1.1)) and further perturbed at test time (0.85, 1.15),, simulating changes in payload environmental conditions.
7

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(a) HalfCheetah-Slope
(b) Hopper
(c) Truck Dataset
Figure 2: Validation MSE time across selected environments. MetaKoopman consistently
achieves lower prediction error and faster convergence compared to baselines.
Table I: Final validation loss (# std) across methods and environments. Best performance per environment is bolded.
Enviromment
MetaKKoopman
MAML
BayesianNN
NeuralODE
Hopper
0.0116
0.807
Truck-Datasel
0.099 8.0021
0.0888
0.1000
0.0917
0.0023
0.2042
0.1674
0,2923
0.0120
0,2589
0.0053
0.0926
0.0022
HallCChettahSSlpp
0.7490 0.0117
0.8808 00104
1,2800 + 0.0088
1,2927 0.0196
1,7008 0.0374
1 0.0210
1.998 + 0.2123
Walker-Friction: Surface friction is varied from μ/3 3/ during training, and extended to (u/5, 5p] at test time, exposing models to extreme differences in contact dynamics (e.g., dry vs. slippery terrain).
Ant-DisabledJointe The agent is trained with of the first six joints disabled and tested with arbitrary joint failures among all eight, modeling structural perturbations such as hardware damage actuator failure.
Panda-Damping" A 7-DoF Panda Lift manipulation task from robosuite, where joint damping coefficients are varied during training and extended beyond the seen range at test time, evaluating adaptation to shifts in contact and actuation dynamics typical of manipulation domains.
HalfCheetah -Stitaa--taiinay::: A standard unperturbed HalfCheetah environment is included to evaluate the robustness and the behavior in the absence of distribution shift, ensuring adaptive methods do not degrade under stationary conditions.
Baselines. We evaluate MetaKoopman by comparing against a a range of strong and widely used baselines that span deterministic, probabilistic, and meta-learned dynamics models: (1) Deep Koopman Operator (DKO) [33], employs neural networks to jointly learn observable functions and the Koopman operator, embedding single state-action pair rather than entire trajectories. However, lacks adaptation and uncertainty estimation. (2) Gradient- Based Adaptive Learner (GrBAL) [24] model-based meta-learner that performs online adaptation via gradient updates, but offers no calibrated uncertainty. (3) Bayesian Model-Agnostic Meta-Learner (Bayesian MAML) [45] meta- learns prior over dynamics models, combining adaptation with uncertainty estimation, but relies optimization-basee based updates at test time. (4) Bayesian Neural Networks (BNNs) model predictive uncertainty through approximate posterior sampling, but do not adapt to changing dynamies. (5) Ensemble models (EMLP) [18] train an ensemble of five neural networks, estimating uncertainty through inter-model variance. They provide robustness but lack explicit mechanisms for adaptation. (6) Finally, Neural Ordinary Differential Equations (Neural ODEs) [5] model continuous- -imme dy- namies but lack both uncertainty estimation and test-time adaptability. These baselines contextualize MetaKoopman's s contributions in terms of structure, adaptability, and principled uncertainty.
5.2 Predictive Dynamics under Distribution Shift
We evaluate MetaKoopman's 's predictive accuracy HalfCheetah-Slone Sloope,, Hopper-Gravity, and real-world truck dataset. As shown in Figure 2, MetaKoopman consistently achieves lower MSE across all environments. Table further summarizes the final validation losses, where MetaKoopman outperforms other approaches in both simulated and real-world settings. The results demonstrate that the combination of structured Koopman modeling and Bayesian adaptation leads to accurate, data-efficient predictions that remain reliable under distributional shifts, Extended evaluations are provided in the appendix.

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Approach ice patch
on
designated point
Figure 4: As the truck approaches a previously unobserved ice patch on slippery road, MetaKoopman enables the planner adapt, allowing the vehicle to brake accurately low friction surface.
5.3 Does adaptation meaningfully change the model?
To evaluate whether adaptation meaningfully al- ters the underlying dynamics model, we assess MetaKoopman's capability refine its predictions after incorporating recent trajectory history. Using the wallwwrldd truck dataset, compare the distribu- tion of normalized mean squared error (MSE) across forecasted trajectories before and after performing a Bayesian meta-update of the Koopman operator.
somalee MAL LAMO
As shown in Fig. 3, prior adaptation the model ex- hibits higher prediction error, reflecting a mismatch between the meta-learned prior and the current op- erating conditions-
Figure MSE distribution before and after Koopman operator adaptation on the truck dataset. The post-update distribution (blue) shows shift toward lower errors compared to the pre-update (orange)
as changes in surface fric- tion or tire-road interaction. After performing the Bayesian meta-update using recent trajectory data, the error distribution shifts noticeably toward lower values. This shift indicates improved alignment with the evolving system dynamics, refining its predictive model to remain consistent with the observed environment.
5.4
Closed-Loop Impact: Safer Planning Through Adaptive Uncertainty-Awarr Dynamics
While learning-based dynamics models have shown promise in simulation, their application to real- world autonomous systems remains limited. A key challenge is how to evaluate not just predictive accuracy, but also how such models influence downstream decision decision-making eespeiallyyy in conditions that involve distribution shifts, environmental uncertainty, and high-stakes planning. Do these models meaningfully adapt in time? Can they influence control choices that improve safety or robustness?
To answer these questions, we conduct a full-scale deployment on a 37.5-ton autonomous truck and trailer system during series of tests in a northern climate during winter 2025. These tests were carried out on a closed-course proving ground designed to simulate extreme low-traction environments, including snow-covered roads, polished ice, and mixed-friction (mu-split) surfaces.'
Precise stops on polished iee. In the first scenario, the vehicle is required stop at a designated marker after entering a zone of dramatically reduced surface friction. The road transitions from high-grip snow to polished ice, creating distribution shift that significantly alters braking dynamics. Models that do not adapt to this shift underestimate stopping distance and consistently overshoot the target. As shown in Figure 4. MetaKoopman adapts online to the changing traction conditions. adjusting its predictive dynamics model using recent trajectory data.
Evasive lane change snow. In this scenario, virtual obstacle appears unexpectedly covered road. The system must immediately decide whether to brake or initiate a lane change. Due to low traction, braking is physically infeasible, but non-adaptive models fail to account for this and attempt to stop, resulting in overshoot or collision. As shown in Figure 5, MetaKoopman detects the distribution shift through increased prediction error and adapts its dynamics model using recent trajectory data. This allows the planner to accurately assess braking feasibility and execute safe
snow-
All tests conducted with the assistance of a professional safety driver.

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heavy vahidle drives road
An obstacle appears suddenly
Too little friction to brake sa fely to standstill
Lane change executed
(a)
(b)
(d)
(e)
Figure 5: Evasive lane change on snow. (a-d): virtual obstacle appears mid-lane on a low-
traction road. Non-adaptive models underestimate stopping distance and attempt braking, resulting in failure. MetaKoopman adapts online and executes safe lane change to avoid collision. (e): Sensor visualization showing obstacle detection and the feasible trajectory selected by the planner.
evasive maneuver. The result is controlled lane change that avoids the obstacle while respecting vehicle dynamics and control limits. More the real-world setup and experiments can be found in the appendix.
6 Discussion
MetaKoopman bridges structured modeling and meta-learning through a tractable, uncertainty- aware framework for online adaptation under distributional shift. In formulation. uncertainty modeled explicitly over the Koopman operator, while the encoder remains deterministic. This design choice is deliberate: in many practical settings caatings-particularii those involving distributional shifts-the dominant source of variability arises from changes in system dynamics rather than in the representation itself. By keeping the encoder fixed and meta-learning only prior over the Koopman operator, MetaKoopman focuses uncertainty modeling where it matters most-in the evolving dynamics. This assumption is justified in structured environments where the encoder is trained across diverse dynamics, yielding stable latent representations while uncertainty in the learned dynamics governs predictive quality.
Several works have explored representation-leve uncertainty by placing distributions over latent embeddings, such as variational stochastic encoders in dynamical systems [12, 44. 13). Integrating such approaches could further capture uncertainty in sparse or ambiguous regions, complementing the current operator-level levell formulation. Overall, capturing epistemic uncertainty operator via the conjugate MNIW prior enables data-effic lata-efficient, closed-form Bayesian updates suitable for real-time planning and control. While extending uncertainty the representation space is promising future direction, our results show that the present formulation already achieves strong predictive accuracy and reliable uncertainty calibration under distributional shift.
the Koopman
More broadly, MetaKoopman's 's structure suggests several directions for future work. These include integration with reinforcement learning for uncertainty or partially observable environments, and application to decision-critical planning pipelines. Our results show that closed Bayesian meta-learning is not only tractable but viable in real-world autonomous systems, offering a practical path toward principled uncertainty in safety-critical control.
exploration, extension multi-agent
7 Conclusion
We introduced MetaKoopman, a Bayesian meta-learning framework for structured dynamics modeling under distribution shift. By meta-learning a conjugate prior over Koopman operators, MetaKoop- man enables closed-form posterior updates from short trajectory segments and produces calibrated posterior predictive distributions for planning. Our experiments demonstrate that MetaKoopman consistently improves multi-step prediction accuracy, adapts effectively to new dynamics, and enables safer closed-loop control in real-world deployment on heavy-duty autonomous truck. These results show that tractable Bayesian meta-learning can seale to safety-critical systems, and open the door to future extensions in uncertainty-awee control, reinforcement learning, and real-time robotics.
10

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Acknowledgments
This work was supported by Vinnova, Sweden, through the AllDrive project. We also want to express our gratitude to the AllDrive team for their valuable contributions, collaboration, and support throughout the winter tests.
References
[1] Ian Abraham, Gerardo De La Torre, and Todd D Murphey, Model-based control using koopman operators. arXiv preprint arXiv:1709.01566 2017.
[2] Daniel A Braun, Carsten Mehring. and Daniel M Wolpert. Structure learning in action. Be- havioural brain research, 206(2):157-165, 2010.
[3] Daniel Bruder, Xun Fu, R Brent Gillespie, David Remy, and Ram Vasudevan. Data-driven control of soft robots using koopman operator theory. IEEE Transactions Robotics, 37(3):948-
961, 2020.
[4] Hongyi Chen, Abulikemu Abuduweili, Aviral Agrawal, Yunhai Han, Harish Ravichandar, Changliu Liu, and Jeffrey Ichnowski. Korol: Learning visualizable object feature with koopman operator rollout for manipulation. arXiv preprint arXiv:2407. .05580054 2024.
[5] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
[6] Vít Cibulka, Tomáš Haniš, Milan Korda, and Martin Hromčík. Model predictive control of a vehicle using koopman operator. PTAC-PapersOnLLine, 53(2):4228-4233 2020.
[7] Ignasi Clavera, Jonas Rothfuss, John Schulman, Yasuhiro Fujita, Tamim Asfour, and Pieter Abbeel. Model-based reinforcement learning via meta-policy optimization. In Conference on Robot Learning, pages 617 629. PMLR, 2018.
[8] Ron Dorfiaan Idan Shenfeld, and Aviv Tamar. Offline meta reinforcement learning- identifiability challenges and effective data collection strategies. Advances in Neural Information Processing Systems, 34:4607-4618, 2021.
[9] Yan Duan, John Schulman, Xi Chen, Peter I. Bartlett, Ilya Sutskever, and Pieter Abbeel. R122 Fast reinforcement learning via slow reinforcement learning. arXiv preprint arXiv:1611.02777 2016.
[10] Chelsea Finn, Pieter Abbeel, and Sergey Levine. Model-agnostic meta-learning for fast adap- tation of deep networks. International conference on machine learning, pages 1126-1135. PMLR, 2017.
[11] Chelsea Finn, Kelvin Xu, and Sergey Levine. Probabilistic model-agnostic meta-leamiin. Advances neural information processing systems, 31, 2018
[12] Marco Fraccaro, Simon Kamronn, Ulrich Paquet, and Ole Winther. A disentangled recognition and nonlinear dynamics model for unsupervised learning. Advances in neural information processing systems, 30, 2017.
[13] Minghao Han, Jacob Euler-Rolle, and Robert K Katzschmann. Desko: Stability-assured robust control with deep stochastic koopman operator. International conference on learning representations (ICLR), 2022.
[14] Wenjian Hao, Bowen Huang. Wei Pan, Di Wu, and Shaoshuai Mou. Deep koopman learning of nonlinear vare-na systems. arXiv preprint arXiv:2210.06272, 2022.
[15] Diederik P Kingma and Max Welling. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114 2013.
[16] Bernard Koopman. Hamiltonian systems and transformation hilbert space. Proceedings of the National Academy of Sciences, 11553315-318, 1931.

---

[17] Milan Korda and Igor Mezić. Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control. Automatica, 93:149-160, 2018.
[18] Balaji Lakshminarayanan, Alexander Pritzel, and Charles Blundell. Simple and scalable predictive uncertainty estimation using deep ensembles. Advances in neural information processing systems, 30, 2017.
[19] Qianxiao Li, Felix Dietrich, Erik M Bollt, and Ioannis G Kevrekidis. Extended dynamic mode decomposition with dictionary learning: A data-driven adaptive spectral decomposition of the koopman operator. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(10), 2017.
[20] Bethany Lusch, J Nathan Kutz, and Steven L Brunton. Deep learning for universal linear embeddings of nonlinear dynamics. Nature communications, 9(1):4950, 2018.
[21] Giorgos Mamakoukas, Maria I. Castano, Xiaobo Tan, and Todd D Murphey. Derivative-base based koopman operators for real-time control of robotic systems. IEEE Transactions on Robotics, 3766)2213- 2021.
[22] Zeyad M Manaa, Ayman M Abdallah, Mohammad A Abido, and Syed S Azhar Ali. Koopman- lqr controller for quadrotor uavs from data. In 2024 IEEE International Conference on Smart Mobility (SM), pages 153-158. IEEE, 2024.
[23] Kevin P Murphy. Probabilistic machine learning: Advanced topics. MIT press, 2023.
[24] Anusha Nagabandi, Ignasi Clavera, Simin Liu, Ronald S Fearing, Pieter Abbeel, Sergey Levine, and Chelsea Finn. Learning to adapt in dynamic. real-world environments through meta- reinforcement learning. arXiv preprint arXiv:1803. 2018.
[25] Alex Nichol and John Schulman. Reptile: a scalable metalearning algorithm. arXiv preprint arXiv:1803.02999, 2(3):4, 2018.
[26] Samuel E Otto and Clarence W Rowley. Linearly recurrent autoencoder networks for learning dynamics. SIAM Journal on Applied Dynamical Systems, 18(1):558-55) 553,, 2019.
[27] Joshua L Proctor, Steven L Brunton, and J Nathan Kutz. Dynamic mode decomposition with control. SIAM Journal on Applied Dynamical Systems, 151) 2016.
[28] Kate Rakelly, Aurick Zhou, Chelsea Finn, Sergey Levine, and Deirdre Quillen. Efficient off-policy meta-reinforeemmnt lcarning via probabilistic context variables. In International conference on machine learning, pages 5331-5340. PMLR, 2019.
[29] Andrei A Rusu, Dushyant Rao, Jakub Sygnowski, Oriol Vinyals, Razvan Pascanu, Simon Osindero, and Raia Hadsell. Meta-learning with latent embedding optimization. arXiv preprint arXiv:1807.059660 2018.
[30] Steindór Sæmundsson, Katja Hofmann, and Mare Peter Deisenroth. Meta reinforcement learning with latent variable gaussian processes. arXiv preprint urXiv:1803.0755 2018
[31] Peter Schmid. Dynamic mode decomposition of numerical and experimental data. Journal of fluid mechanics, 656:5-28, 2010.
[32] Peter Schmid, Larry Li, Matthew P Juniper, and Oliver Pust. Applications of the dynamic mode decomposition. Theoretical and computational fluid dynamics, 25:249 2599, 2011.
[33] Haojie Shi and Max Q-H Meng. Deep koopman operator with control for nonlinear systems. IEEE Robotics and Automation Letters, 703):770-7707,, 2022.
[34] Lu Shi, Masih Haseli, Giorgos Mamakoukas, Daniel Bruder, Ian Abraham, Todd Murphey, Jorge Cortés, and Konstantinos Karydis. Koopman operators in robot learning. arXiv preprint arXiv:2408.042000 2024.
[35] Rajpal Singh, Chandan Kumar Sah, and Jishnu Keshavan. Adaptive koopman embedding for robust control of complex nonlinear dynamical systems. arXiv preprint arXiv:2405.09110 2024.
12

---

[36] Jianlin Su, Murtadha Ahmed, Yu Lu, Shengfeng Pan, Wen Bo, and Yunfeng Liu. Roformer: Enhanced transformer with rotary position embedding, Neurocomputiing. 568:127063, 2024.
[37] Naoya Takeishi, Yoshinobu Kawahara, and Takehisa Yairi. Learning koopman invariant sub- spaces for dynamic mode decomposition. Advances in neural information processing systems, 30, 2017.
[38] Floris Takens. Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Warwick 1980: proceedings of symposium held at the University of Warwick 1979/80, pages 366-381. Springer, 2006.
[39] Emanuel Todorov, Tom Erez, and Yuval Tassa. Mujoco: A physics engine for model-base control. In 2012 JEEE/RS/ international conference on intelligent robots and systems, pages 5026-5033. IEFE, 2012.
[40] Rongyao Wang, Yiqiang Han, and Umesh Vaidya. Deep koopman data-driven control framework for autonomous racing. In Proc. Int. Conf. Robot. Autom.(ICRA) Workshop Opportunities Challenges Auton. Racing, pages
[41] Rui Wang, Yihe Dong, Sercan Ô Arik, and Rose Yu. Koopman neural forecaster for time series with temporal distribution shifts. arXiv preprint arXiv:2210.036675 2022.
[42] Matthew Williams, Ioannis Kevrekidis, and Clarence W Rowley. A data driven approxima- tion of the koopman operator: Extending dynamic mode decomposition. Journal of Nonlinear Science, 25:1 307-333,, 2015.
[43] Enoch Yeung, Soumya Kundu, and Nathan Hodas. Learning deep neural network representations for koopman operators of nonlinear dynamical systems. In 2019 American Control Conference (ACC), pages 4832 4832-4839. IEEE, 2019.
[44] Cagatay Yildiz, Markus Heinonen, and Harri Lahdesmaki. Ode2vae: Deep generative second order odes with bayesian neural networks. Advances in Neural Information Processing Systems, 32, 2019.
[45] Jaesik Yoon, Taesup Kim, Ousmane Dia, Sungwoong Kim, Yoshua Bengio, and Sungjin Ahn. Bayesian model-agnostic meta-leaming. Advances in neural information processing systems, 31, 2018.
[46] Hao Zhang. Clarence W Rowley Eric A Deem, and Louis N Cattafesta. Online dynamic mode decomposition for time-varying systems. SIAM Journal on Applied Dynamical Systems, 1883):158-1169, 2019.
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A Posterior Predictive Distribution
Given embedding & in time the goal is to predict the probability over the state embedding for the next time-step p(@++1 [2, D), where D denotes the training dataset.
p(11+1|2,, D) /I 2,
dKGd
(18)
First, we look at f /(ff(+++1/// D, 2c D)p(K We know that:
p(KE)= MN E)
((x- M) ;)'>" E- (K- M) P-1)
exp
2 Ir
(19)
(2x) $ Σ |
and,
1
--1
( E
exp
(i++1 -xx) T (2141 - KZ+)
(20)
Then:
/ p(21411K, E, D)p(K(E)dd (
= =/
exp
2 tr
Σ-1
(81+1- - KCE1) - K&)) + (K-si)) -
p-1 (rc - 1id ))
(2T))
ntnk
Σ
l+n
1/3
=
1ー
K: ( + x--) KT -2 (5eti + (M-1) N-T
(2TT)
Rtnk
Σ
exp
+ -MM--1 + Fee (Zet1))
-1 tr
Σ-1
(K.SuakT - -25ac + Sab
(2TT)
B4nk
Σ
[V13
exp
Σ (K- Sall Sau (K-- SabSaa)) + Salp )
(2t)
ΣΣ
1=
14 _
exp
tr
(2TT)
ku (2Tr)
ntnk
[ 5 |Saaa
|
-
exp
2
1
tr
(Σ (X-1Sa) Sa(b)
(2T)) 3 |Σ| $
Saa -
exp tr -I-'Sa)) Saip)
(21)
where:
S 2, V-1
Sab = " T/T + -VV-1 -1
Sub = -/-1 -M f141 (8+41)
S.jo = Sab S-1
21

---

Substituting Eq. 21 into Eq. 18 yields:
SS ocl
D+n+1
[-1 tr (2-1 [Sa + ( x)
p(ūr+1(2,, =
exp
(2m) 1 M
1-
2 1
-- tr (E-1 (Saps + 4 $1)]
Σ
exp
dE
(2T) 8 11 22 - r (2)
|Saal 1 /11
2 T
D+1 2
(2x) 21
32 T.
Saab + |0 94n
T
D++ 2
Saal
- - 31
11 + v Sajol 4
(22)
(2 " (2) 1
Now, let's break Salb first. We know that:
Salo = San SabiS Sib
-MD-IN (ox+t)) - (āet1 + MV-1)
( V-2)
( M T
=MD-'M + ( - 2 Cār(T+1) T + ē CV-1MT (( + -
= -tt -I C2.) (it+1)² - 22 + -MV-M + +MV-'CV-IMM"
813
8
(ă+1 Sty T Sy (ōr41 11441-SA4 S.y.
Si
(23)
Now, we have reached a quadratic formula. We begin by using the Woodbury formula on S
S = (I 2 Cā.) -1
+ & (C-1 - =527 2c
1 + के (āæī) + -V-1 - [,5 =t
1 + 2े V.
(24)
Then,
S.S MY C (1 + VT
(4 -V (1 + ス/ Væ:
-1 =T
MV-1
1 (1 27 VE,)
de (1 + 2 Via)) a v)
ب (1+ E Vā.
x (1+ 2 V 1 (1 E V&)) -1 &T (1 + T V&.)
=M 2 (1+ x v2.) - - V& (1 के. V2.) -1 (1+ (1+
= -ME
(25)
22

---

Substituting 25 and 23 into 22 yields:
D+00 N 2
Saal
$ -
11 v-1 (izt1 - Mā) ( (1+ - ( (iz+1 - -M 1-
p( [2,, D)
(2x)aT,
V
Using the matrix determinant lemma, this equals to:
P+ten 2
V
p(8241|(,,
T
(i241 -Ma)
(it41 Māa)
(2TT) # IT 1)
(
1 2, V
1 2 V2
To M v (1+ 2
(26)
Which is multivariate Student t-distribution with mean M2, and scale matrix V (1+
Under large number of degrees of freedom (large number of training examples in our case), this distributions converges multivariate gaussian distribution with mean M and covariance
V/ 1 + 2 V..)
On the Gaussian Approximation. The posterior predictive distribution Eq. is multivariate Student-/ with degrees of freedom v' Me + N, where Mo is the prior's degrees of freedom and N is the number of test-time time adaptation points. Gaussian when y' >> 30. In our framework, this condition is reliably satisfied: vo is meta-learned and chosen to exceed the latent dimension 17, which is typically at least 32, and N adds further support. As result, /' is well above the threshold in all cases consider, justifying the Gaussian approximation used in predictive forecasting.
well known that the Student-t closely approximates a
Multi-step predictive via moment matching. We now extend the single-step posterior predictive in Eq. (26) to a multi-step forecasting rule. As shown in Eq. (26), the conditional one-step predictive a multivariate Student-t with mean MM 2 and stte-dependent scale T 10(1 + 2, 22). For large
degrees of freedom 0 adopt the standard Gaussian surrogate
i441 & s N(M 2,, [11+2TV4 VV.).
(27)
which preserves the exact first two moments of the Student-t. Although this marginal is generally non-Gaussian, its mean and covariance admit closed forms. We therefore follow a principled moment matching strategy: each horizon step we compute the exact mean and covariance of the marginal induced by (27), and we re-approximate that marginal by the Gaussian with those moments.
Regressor and state moments. Let ul,e B[ Σ,, Cov(Ez denote the regressor moments at time i, and μu := E[ū,], Σ, = Cov(T,) the state moments. We use the quadratic-form identity
E 2e] p V Pa. + E..))
(28)
Recursive multi-step moment matching. To forecast h steps ahead, we iterate the one-step calculation while recomputing the regressor moments from the current state moments at each step. Under the Gaussian surrogate (27), the marginal recursive noment-matcher mean and covariance can be given as follows:
Mt+1 E[ā++1] M pe,ts
(29)
Σ+1 Cov(21+1) E.E MT + & (1+ (1+HI, UI, + tr tr(VE..))
(30)
Proof. By the tower property, Me+1 = [ E)] = E[M M Mp2.1
For the covariance, apply the law of total variance:
Σ+11 Cov(E &)) + +15 Cov
[ M ME L 27 VVj.
23

---

using (28), this becomes:
Ext++1 E..4 M 1 (1+ Metn tr(VE
A.1
On Modeling and Interpreting Uncertainty
The posterior predictive distribution derived is approximated with multivariate gaussian distribu- tion, with scale matrix
M Σ M
T
(1+12,
M2, V P2,t - tr(V Eg,))
x
propagation of state uncertainty
alcatoric
epistomic modulation via
which naturally decomposes predictive uncertainty into two components. The matrix v represents aleatoric uncertainty, capturing irreducible system noise-such as sensor inaccuracies and actuation imprecision that persists across test-time conditions. The multiplicative term M7,+. V Ma,tti
(r(V),,,++) reflects epistemic uncertainty, expressing the model's confidence in the latent Koopman dynamics based on limited recent data. This formulation ensures that predictive variance increases in lass-lamiliar regions of the latent space, and contracts where the model has strong task specifcc evidence.
In our framework, uncertainty is modeled explicitly the Koopman operator, while the encoder remains deterministic. This modeling choice is guided by the observation that, in many practical particulaaly those involving distribution shift-mode uncertainty tends dominate. When test-time dynamics deviate from training-tine time conditions (c.g., due to changing surface friction or payload configurations),
primarily the uncertainty in the evolving system behavior-not the representation-tine that governs prediction quality. This is a reasonable assumption in structured systems where the encoder has been trained across a broad and diverse set of environments. In such settings, latent representations tend to be stable and informative, and the uncertainty in the learned dynamics the principal source of variability affecting performance.
By capturing epistemic uncertainty over and leveraging the conjugate structure of the Matrix Normal-Invers, Wishart prior, enable closed-form Bayesian updates that are both data-efficient and computationally lightweight. This design supports fast, adaptive inference suitable for real-time applications such motion planning and control, where latency is critical. While incorporating representation-level uncertainty is promising extension, our empirical results suggest that the current formulation provides strong predictive accuracy and reliable uncertainty estimates under challenging distribution shifts.
24

---

B Truck and Trailer Data Collection
One of the key contributions of this work is the application of motion planning techniques for au- tonomous truck and trailer systems. Autonomous driving datasets and maintain, with access often restricted to select (OEMs) and their suppliers. This study involved ex- tensive data collection from real autonomous truck and trailer platform to develop test datasets that accu- rately reflect the complex dynamics of these systems under diverse and challenging conditions, including harsh winter environments. Data was gathered over 12 months (Jan 2024 Jan 2025) using 37.5 37,n,tnn, 17-meter-long Scania autonomous tractor-semitrailer in various driving and weather scenarios, covering all seasonal variations (see Fig. 6). Each session was supervised by a safety driver and test engineer to ensure safety and system reliability.
typically expensive acquire
Figure 6: Truck and semi-trailer autonomous vehicle used for testing.
The dataset includes comprehensive uutonomy-relatee logs from various high-fidelity sensors. Special emphasis was placed on capturing edge cases and distribution shifts, particularly under challenging winter conditions. Routine tests included maneuvers on high-speed test track and test drives on public highways in different weather conditions including sun, snow and rain. Driving scenarios included straight line driving. negotiating curves and slopes, lane changes, cut-ins, stopping, following other actors and highway driving. Data collection was conducted either through fully autonomous operation or, for higher-risk maneuvers, via manual driving with onboard sensors and logging systems enabled.
To evaluate autonomous driving performance under extreme winter conditions, a series of targeted field tests were conducted during February and March 2025 on specialized proving grounds located in a northern climate. These controlled environments in- cluded variety of challenging low-traction surfaces such as wet asphalt, compacted snow, and polished ice. The autonomous system was evaluated across a diverse suite of scenarios specifically designed to challenge its planning and responsiveness under ad- verse conditions. These included high-speed corner- ing snow-covered roads, dynamic adaptation to sudden ice patches, sharp emergency braking, and evasive maneuvers to avoid virtual obstacles. These trajectories were incorporated into the test dataset for evaluating dynamic modeling performance.
particularly critical set of experiments was per- formed mu-split track (see Fig. 7), where one side of the vehicle experienced high-friction dry as- phalt while the opposite side traversed low-friction ice. This asymmetric traction environment is known to induce significant yaw moments, increasing the risk vehicle instability, trailer swing, and potential jackknifing. These mu-split tests constituted critical stress evaluation of the model's ability to generalize under distributional shift, revealing key aspects of its stability and robustness. They also played a central role in validating the planner's strategies for maintaining safe and dynamically feasible behavior across varying surface conditions and friction transitions.
Figure 7: Mu-split test scenario captured from within the vehicle cab, illustrating asymmetric surface conditions traveling on high-friction asphalt while the other traverses low-friction polished ice.
side of the vehicle
For data processing, raw trajectory data were preprocessed to enhance signal fidelity and facilitate effective model learning. A fourth-order Butterworth low-pass filter with 5 Hz cutoff frequency was
25

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applied to suppress high-freeug segh-freueency noise in the state signals. To ensure balanced feature scaling across the neural network, all input features were normalized the range [-1, 1]. Each system state was encoded as 7-dimensional vector comprising longitudinal and lateral velocities and accelerations, the tractor's yaw angle, the articulation angle between the tractor and trailer, and the front-wheel slip angle. The control input vector included throttle, brake, and steering commands. The resulting dataset contains over hours of curated trajectory data. To promote diversity in dynamic behaviors, straight-line highway driving segments-wiine 40% the training set. This ensured sufficient representation more complex scenarios, including turns, braking events, and evasive maneuvers.
were overrepresented in raw data-wert limited to
26

---

C
Additional Results
Base Simulation Environment. The environments used in our experiments encompass broad spectrum of dynamical behaviors and distribution shifts. As shown in Fig. 8, they include both robotic manipulation and locomotion tasks liks-ranging from the Panda-Lift setup in the RoboSuite environment to established continuous control benchmarks such as Ant, HalfCheetah, Hopper, and Walker.
(a) Panda-Lift
(b) Ant
(c) HalfCheetah
(d) Hopper
(c) Walker
Figure Environments: Panda-Liff, Ant, HalfCheetah, Hopper, and Walker.
C.1 Predictive Performance
We extend our evaluation to four additional environments- HalfChettah-Stationee Panda, and Walker-to further test METAKOOPMAN'S versatility. As shown in Table 2, the model consistently outperforms strong basclines across varied conditions.
Across all environments, METAKOOPMAN achieves the lowest validation loss as illustrated in Figure 9. In the Ant and Walker tasks, it rapidly adapts to changing dynamics and maintains stable performance under noise and contact variability. the SalfCheetah-Stationan environment, preserves robustness without overfitting in the absence of distribution shift. Finally, in the Panda- Damping manipulation setting, META KOOPMAN attains the best overall accuracy with minimal variance, highlighting its scalability and generalization across locomotion and manipulation domains.
(a) Ant-DisahledJoints
(b) HalfCheetah Stationnrrt
Walker-Friction
(d) Panda-Damping
Figure 9; Validation MSE over time for (a) Ant, (b) HalfChectah--Sationn Staataaarrttt (c) Walker, and (d)
Panda-Damping. Shaded regions denote standard deviation across three runs.
27

---

Table Final validation loss (# std) across additional environments. Best performance per environ- ment is bolded.
Environment
MetaKoopman
BayesianMAML
GrBAL
DKO
BayesianNN
EMLP
NeuralODE
Ant-Disia Doits
0.21 0.00
0.27 0.02
0.29 0.00
0.27 0.00
0.69 0.01
0.26 0.04
0.46 1: 0.09
HallCheetah-Saationne
0.02
0.74
Walker-Friction
0.52 0.10
0.54 0.00
0.5 0.07
0.59 +: 0.00
0.67 0.02
0.61 0.03
0.64 +: 0.05
Panda-Dumping
0.035 0.002
0,05 + 0.0002
0.0 0.022
0.06 0.003
0.11 0.044
0.08 0.02
0.07 0.007
C.2 Uncertainty Quantification.
We evaluate the quality of predictive uncertainty by measuring its correlation with actual prediction errors. A strong positive correlation indicates that the model expresses higher uncertainty when its predictions less accurate desirable property for downstream safety-cnn criitycciitical applications.
To assess this, we compute the Pearson correlation coefficient between the predicted uncertainty and mean squared error (MSE) across 100,000 randomly selected states spanning diverse trajectories and time steps. As shown Table 3, META TOOOOMAANoossstere exhibits the highest correlation values across all datasets, indicating well-calibrated uncertainty estimates.
Table Correlation between predicted uncertainty and squared error across datasets. Higher values indicate better uncertainty calibration.
Dataset
EMLP
BNN
Bayesian MAML
MetaKoopman
Truck Dataset
0.58
0.48
0.56
0.68
Ant-Disabledloonts
0.59
0.53
0.58
0.72
HalfChectah-Slee Slope
0.63
0.52
0.62
0.69
D Ablation Studies
We perform series of ablation studies to assess the contribution of key design choices in META MAN. In particular, investigate the role of the tempering mechanism in adaptation and evaluate the effect of varying history lengths on predictive performance. These experiments provide insight into the trade-off's between accuracy and computational efficiency, and help identify configurations that most effectively balance the two.
D.1 Effect of Tempering
We evaluate the impact of Bayesian tempering by comparing METAKOOPMAN with and without this mechanism across three representative environments. As shown in Table 4, incorporating tempering consistently improves predictive accuracy, particularly in settings with pronounced distribution shifts or perturbations such as the truck dynamics and Ant environments.
Table Performance with and without tempering (Avg. MSE).
Environment
Without tempering
With tempering
Truck Dynamies
0.084
0.059
Walker-Friction
0.56
0.52
Ant-Disabledloins
0.235
0.21
28

---

D.2 Effect of History Length
We investigate the effect of trajectory history length on predictive performance across three environ- ments: Hopper-Gravity. HalfCheetah-Slopa and Truck Dynamics. The history length determines the temporal context available to the trajectory encoder. which can influence the model's ss ability to infer latent dynamics. Table 5 presents results for varying history lengths, revealing consistent improvement in accuracy as more context is incorporated. Notably, longer histories yield the best performance, though with diminishing returns returns- highlihhing some sort of trade-off between model expressiveness and computational efficiency.
Table 5: Effect of History Length on Prediction Accuracy (Mean Squared Error).
Environment
1
4
8
16
Hopper-rravityy 0.048
0.046
0.04
0.036
HalfChectah- Slope
0.857
0.846
0.823
0.749
Truck Dynamics
0.0889
0.084
0.075
0.06
29

---

E Motion Planning with Cost-Guided Latent Action Sampling
We integrate METAKOOPMAN into a real-time motion planner for generating dynamically feasible trajectories in truck-trailer system under complex physical constraints. To enable efficient planning scale, we adopt a Cost-Guided Latent Action Sampling (CLAS) approach that leverages the learned variational action encoder to directly sample from the latent control space.
Instead of sampling actions from the original control space and encoding them at test time, CLAS samples directly from the latent Gaussian space learned by the variational encoder (Figure 10). Each sampled latent action passed through the Koopman-baa baasdd
dynamics model via a small number of matrix multi- plications to produce a candidate tobjtrajyctory over a fixed planning horizon.
This approach yields significant gains in computa- tional efficiency. Each trajectory rollout requires only single forward pass through the context encoder, a Bayesian update of the Koopman prior, and a se- quence of matrix multiplications that scale linearly eetiorriii--emmmaaiinnn
Variational Action Encoder
with the number of sampled the need for costly action encoding operations. Con- sequently, CLAS can generate and evaluate hundreds of thousands of dynamically consistent trajectories in parallel.
Mee+1
u4 2
[
Figure 10: Variational action encoder used in the CLAS planner. The encoder learns Gaussian distribution over action embeddings. enabling efficient sampling for trajectory roll- out.
After obtaining the predicted trajectories, each candi- date trajectory is scored according to cost function that incorporates goal proximity, smoothness, and obstacle avoidance. The trajectory with the lowest overall cost is selected for execution. This planning formulation ensures that trajectories are not only feasible under the vehicle's 's dynamics, but also adapted to real-world factors such as low-traction surfaces, trailer articulation limits, and tight spatial constraints. The full planning procedure is detailed in Algorithm 2.
Algorithm 2 Motion Planning with Cost-Guided Latent Action Sampling (CLAS)
Require: Initial state sp, goal state Sgeal, planning horizon H, number trajectory samples N Ensure: Dynamically feasible trajectory goal
1: Initialize empty set of trajectories T +
2: for i = 1 to N do
This loop is fully parallelizable across samples
3: Sample a sequence latent actions "1:
No T)
4: Integrate dynamics: $1.H KoopmanRollout(( so. u 11H
(4)
(6)
5: Compute cost J; +- EvaluateCost(s ($1:1+ Spont)
6: Store (ū $1.H J.) in T
(1) (8)
7: end for
8: Select üi +- arg min Jj from
9: return Trajectory corresponding to ü'
The sampling and rollout steps (lines 3-6) are parallelizable across N trajectories, allowing the method leverage batched computation on GPUs or distributed hardware. This property enables CLAS to evaluate hundreds of thousands of dynamically consistent trajectories in parallel.
30

---

E.1 Runtime Analysis
To highlight the efficiency improvements of latent action sampling, We evaluate the runtime efficiency of our method and baselines on the truck dataset, selected for its moderate complexity and relevance online control. All experiments are conducted on an NVIDIA A10 GPU ensure fair comparisons across models. Reported runtimes correspond to the inference phase only and are averaged over 1,000 evaluations to reduce variance from transient hardware effects.
Our method, leverages the variational action encoder to sample directly from the latent Gaussian space, avoiding the need for deterministic encoding of control inputs time. This modification yields substantial computational savings while maintaining predictive performance- it especially well-suited for real-time planning scenarios.
Table 6 summarizes average per-step inference times for all compared models. Notably, METAKOOP MAN W. VARIATIONAL ACTION ENCODER achieves the lowest runtime, outperforming both standard Koopman baselines and more complex probabilistic models such as Bayesian MAML, BNNs and ensembles by wide margin.
Table 6: Inference Runtime Analysis (Average Time per Step in milliseconds).
Method
Runtime (ms)
METAKOOPMAN
0.087
Deep Koopman Operator (DKO)
0.21
Bayesian Neural Networks (BNNs)
1.35
MC Dropout
0.98
Neural ODEs (NODE)
0.22
Ensemble MLP (EMLP)
0,49
GrBAL
0.37
Bayesian MAMI.
0.64
31

---

F Experimental Details
F.1 Simulation Environments
We evaluate META KOOAKKN across five simulated control environments implemented using MuloCo [39], each designed to assess the model's adaptability both distributional shifts and structural perturbations, Distributional shifts include variations in gravity, terrain slope, and surface friction, while structural perturbations are introduced via randomized joint failures. To evaluate generalization beyond the training conditions, meta-training is performed over a constrained range of environmental parameters, and meta-testing spans broader, previously unseen set of conditions, as detailed in Table 7.
Trajectory data is collected using TD3 policy trained for million steps the full meta-test distribution of each environment. This policy is then used to generate trajectories for both meta- training and meta-testing phases. To promote diversity in the data, the agent follows its learned policy while incorporating stochastic action perturbations, where random actions are taken with a fixed probability. To simulate sensor noise, Gaussian observation noise with a standard deviation of 0.001 is added in all environments.
Environment
Meta-Train Train Range
Meta-Test / TD3 Range
Perturbation Type
HalfCheetab-SSe
11100111"
1-15"1
Terrain slope
Walker2d-Friction
(0.3,3.0) scaling
0.2, 0.0) scaling
Surface friction
Hopper- Graviity
[-1.0, 1.0] offset
1.5, 1.5] offset
Gravitational field
Ant- aot-DisabledJoints
Partial joint set
Full joint set
Actuator failure
Table 7: Parameter ranges and shift types for simulated environments.
This environment simulates locomotion on inclined terrain by varying the ground plane angle at the start of each episode, affecting both balance and propulsion.
HalfChectah-Slope.
To model variability in ground contact, the friction coefficients of all ground- facing surfaces are scaled per episode.
Walker2d-Friction.
Hrapir-
The vertical gravitational force is perturbed by offsetting the standard -9.81 value, influencing jump timing and overall dynamics.
Ants Mechanical degradation is simulated by disabling randomly selected joint via nullification of its control signal. Failures are restricted to a subset of joints during meta-training, while meta-testing considers the full joint set.
HalfCheetah-Stian -tatiionnrt This environment retains the standard HalfCheetah-vä configuration without any induced perturbations distribution shifts. It serves stationary baseline to evaluate the robustness of meta-learned models in settings where the underlying dynamics remain consistent. This allows to assess whether adaptation mechanisms degrade performance when shift is present.
F.2 Implementation Details.
This section provides implementation specifics for the proposed method, METAKOOPMAN, as well as concise summaries of the baseline methods used for empirical comparison. We describe architectural choices, adaptation strategies, and training hyperparameters. Common settings shared across experiments are listed at the end to avoid redundancy.
Proposed Method: METAKOOPMAN. META integrates Koopman operator theory with Transformer-based sequence models to capture latent linear dynamics under distributional shift. The architecture employs Transformer encoder-decoder structure, with the context encoder implemented as a standard Transformer encoder and the action encoder as a Transformer decoder. This configuration allows future action embeddings to attend to the historical context effectively.
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---

Input sequences are constructed by concatenating state and action vectors over 16 past time steps, resulting in input tokens of dimension n, + nay where n, and n, denote the state and action dimensions, respectively. These tokens are projected into a hidden dimension of 48 via linear layers. To encode temporal structure, we use Rotary Positional Embedding (RoPE) [36]. The encoder utilizes full self-attention, while the decoder employs a sliding-window causal attention mechanism with window size of 8, enforcing autoregressive structure and limiting attention to recent context.
The decoder processes future action sequences, similarly projected into the hidden dimension. A learnable start token is prepended to the decoder input to initiate sequence generation. This design enables structured conditioning on both past trajectories and future controls, supporting closed-form Bayesian updates in latent space.
Deep Koopman with Control (DKO). The DKO baseline models nonlinear dynamics by learning latent linear system via separate encoders for states and control inputs. The state encoder maps raw states to high-dimension
embedding, which is concatenated with the original state before further
processing. consists of fully connected layers with ReLU activations: [State Dim - 32 + 64 + 128 + 84 + State Embedding Dim] (ReLU applied to all but the output layer). The control encoder processes joint state-action inputs through a parallel MLP: "State*Actinn
Dim + 32 -
64 - 128 + 84 + Control Embedding Dim]. The learned dynamics are governed by linear Koopman operator, with system matrix A initialized from a Gaussian distribution and orthogonalized using singular value decomposition (SVD), and control matrix B that maps control embeddings into the latent space.
Neural Ordinary Differential Equation (Neural ODE). The Neural ODE baseline models continuous- time dynamics by learning the time derivative of the system state as function of the current state and action. The ODE function fz a) is parameterized by fully connected neural network, which takes as input the concatenated state-action vector of dimension fy + Ma- After that, the hidden layers of the network have the widths of: [64, 96, 128, 96, 84, 32], all using ReLU activations. The output layer produces vector of dimension ny. corresponding the predicted time derivative of the state. Time integration is performed using fourth-order Runge- the forchdiffeq library.
Runge-Kutta Kutta solver from
Ensemble Neural Networks. This baseline models system dynamics using an ensemble of ten feedforward networks. Each network takes a concatenated state-action input of dimension m2 + na and passes it through a multilayer perceptron with hidden sizes [32, 64, 96, 64, 32], using ReLU activations throughout. The output is state prediction of dimension ng. Uncertainty is estimated via the variance of the ensemble predictions, capturing epistemic uncertainty arising from model disagreement.
Bayesian Neural Network (BNN). The BNN baseline models predictive uncertainty by placing Gaussian distributions over the weights and biases of the network. The dynamics model imple- mented using Bayesian linear layers, where each layer maintains a variational posterior over its parameters. The input, a concatenated state-action vector of dimension n, + Na. is passed through a multilayer architecture with hidden sizes [32, 64, 128, 84, 32], using ReLU activations, and outputs the predicted next state of dimension ng. All priors are zero-mean unit-variance Gaussians.
The training objective combines mean squared error with a Kullback-Libblrr divergence term to regularize the variational posterior:
L Leata + 33 KL (posterior prior),
with 3 = 10-3, During inference, ten forward passes using Monte Carlo sampling produce mean predictions and diagonal covariance estimates, enabling uncertainty-aware forecasting.
Bayesian Model-Agnostic Meta-Learning (BMAML) The Bayesian MAML (BMAML) model extends gradient-bart based meta- meta-learning by maintaining particle-based ensemble approximate the task posterior. Each particle represents a distinct instantiation of shared dynamics model, parameterized as feedforward neural network with layers [(n, + n,)) - 144 - 96 + n. and ReLU activations, where n., and n. are the dimensions of state and action inputs, respectively. The ensemble is initialized from learned meta-prior and adapted per task using Stein Variational Gradient Descent (SVGD), which preserves uncertainty structure by coordinating updates across particles
33

---

using an RBF kernel. During inference, future state trajectories are predicted by averaging rollouts from all particles. The model outputs both the predictive mean and variance, capturing epistemic uncertainty in manner suited for few-shot adaptation.
Gradient-Based Adaptive Learner (GrBAL) The Gradient- Adaptive Learner (GrBAL) enables fast online adaptation of neural dynamics model by leveraging meta-learned initial parameters optimized for fast adaptation. The model is a feedforward network with layers [(n. + na)) 56 + 108 56 > narl where n, and are the dimensions of the state and action respectively, and ReLU activations are used throughout. GrBAL performs inner-loop adap- tation each task by computing gradients of the prediction loss over recent trajectory segments and updating the model weights accordingly. This is implemented using differentiable inner-loop optimization with higher-order gradients, allowing backpropagation through the adaptation process. During inference, the adapted model predicts future state trajectories from the last observed state and planned action sequence. GrBAL is designed support rapid and effective adaptation in dynamic environments, particularly when system dynamics change abruptly due to disturbances or unmodeled factors.
Common Implementation Details. To ensure comparability, all models are configured to have approximately 60k trainable parameters, with the exception of Bayesian MAML and Ensemble Neural Networks, which consist of multiple networks and therefore contain significantly more parameters. All models are trained using the AdamW optimizer with learning rate of 3 10-3 and weight decay of 1 X 10-2, A StepLR scheduler decays the learning rate by factor of 0.3 at one-third intervals the course of 300 training epochs. All models trained with batch size of 128 and fixed prediction horizon of 32 future time steps. Meta-learning models are provided with an additional context window of 16 time steps for adaptation. The primary training objective is the Mean Squared Error (MSE) between predicted and ground truth future states, except for LEOTKOOPMAN, which minimizes the negative liklliclddoood (NLL). Gradient clipping with maximum norm of 1.0 is applied to all feedforward networks to improve training stability. A summary of baseline capabilities with respect to adaptation, uncertainty estimation, and structured dynamies modeling is provided in Table 8.
Method
Adaptation
Uncertainty
Structured Dynamics
METAKOOPMAN
Bayesian MAML
GrBAL
Deep Koopman (DKO)
Ensemble NN
Bayesian NN
Neural ODE
Table Comparison of baseline capabilities in terms of adaptation, uncertainty quantification, and incorporation of structured dynamics.
34


Reviews:
[{"review_id": "IBCYBFPkzs", "reviewer": "Reviewer_WRyM", "summary": "The paper introduces MetaKoopman, a Bayesian meta-learning framework for learning dynamical systems under distribution shifts. It consists of meta-learning a conjugate prior over Koopman operators, and performing closed-form posterior updates from short trajectory segments. Experiments demonstrate that MetaKoopman consistently improves multi-step prediction accuracy, adapts effectively to new dynamics, and enables safer closed-loop control when deployed in the real-world on a heavy-duty autonomous truck.", "questions": "See weaknesses.", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Meta-learning is a relevant problem to the community and the presented approach of meta-learning nonlinear dynamics is well-motivated and sound. The paper is clear, well written, and supplemented by extensive experiments including a real-world deployment of the proposed algorithm.  \n\nThe main weaknesses, in my opinion, are the following:\n- From a technical point of view, it is not clear what are the contributions of the paper. Would be good to comment what are the challenges in proving Lemma 4.2 and if there is some additional technical contributions compared to the classical Koopman model results.\n- As explicitly stated, MetaKoopman models uncertainty of the Koopman operator but of the input embeddings. I believe this is reasonable in several settings, but could be problematic if embeddings are not trained robustly enough. Thus, I think a more detailed discussions on how the encoder is trained and meta-learned is needed to support the provided results.\n- Connected to the point above, it would have been nice to see failure modes of the proposed approach (as well as of other baselines) as a function of the distribution shift, to illustrate the robustness over a spectrum of disturbances.", "was_revised": true, "final_justification": "I lean towards acceptance. The authors adequately addressed my concerns providing new experimental results that strengthen the paper contribution.", "rebuttal": "We sincerely thank the reviewer for the thoughtful and constructive feedback. We are encouraged by the reviewer’s positive remarks on the paper’s clear writing, extensive experimental evaluation, and real-world truck deployment, and we appreciate the time you invested in this review. We address the reviewer's concerns below:\n\n---\n\n> * **W1: From a technical point of view, it is not clear what the specific contributions are, the challenges in proving Lemma 4.2 and what is new compared to classical Koopman results.**\n\nWe thank the reviewer for the opportunity to clarify how our approach builds on prior Koopman work and makes new contributions:\n\n1. **Novel probabilistic formulation.**  Where earlier methods ([1, 2]) learn a single, fixed Koopman operator, we place a distribution over all plausible operators and meta-learn this distribution. Online, this distribution is updated via closed-form Bayesian inference, enabling rapid adaptation to distribution shifts (e.g., our truck switching between asphalt and ice) while explicitly capturing uncertainty in the system dynamics through uncertainty in the operator’s parameters—capabilities that classical Koopman models lack.\n\n2. **Analytic propagation of that uncertainty.**  Lemma 4.2 shows how this parameter uncertainty maps—in closed form—into a posterior predictive distribution over future states, providing principled uncertainty quantification in the predictions.\n\n3. **Real-time planning & first low-friction deployment.**  A variational action encoder lets the planner sample actions directly in latent space, dramatically reducing rollout time. To the best of our knowledge, this is also the first time these models are demonstrated on low-friction autonomous driving.\n\nA detailed list of our contributions is presented in lines 62–77 of the paper, which we hope addresses the reviewer’s concern.\n\n---\n\n> * **W2: MetaKoopman captures uncertainty in the Koopman operator rather than input embeddings. Therefore, a more detailed explanation of the encoder’s training and meta-learning process is necessary to validate the results.**\n\nWe thank the reviewer for the insightful question. We clarify that both the context and action encoders are trained jointly with the Koopman prior in the same meta-learning loop in Alg. 1 (i.e., no separate pretraining).\n\nBelow, we outline how these components interact during training and explain why this design leads to strong predictive performance and well-calibrated uncertainty.\n\n**How the pieces are trained—one loop, three parameter blocks**\n\n1. **Embed**\n   - A context encoder $G_{\\theta_\\text{ctx}}$ turns a $q$-step history trajectory $(x_{t-q:t}, u_{t-q:t})$ into latent matrix $\\mathcal{Z}$.\n   - An action encoder $H_{\\theta_\\text{act}}$ maps planned future controls $u_{t+1:t+h}$ into latent action embeddings.\n\n2. **Adapt**  \n   - The history embeddings $\\mathcal{Z}$ drive the *closed-form* Bayesian update (Eq.~10) that adapts the Koopman prior  $\\theta_k = \\\\{M, V, \\nu, \\Psi, \\beta\\\\} $ into task-specific parameters:  $ \\theta_k'(\\mathcal{D}^{\\text{tr}}_{\\mathcal{T}}) = \\text{Adapt}(\\theta_k,\\mathcal{Z}). $\n\n3. **Predict**  \n   - The adapted operator $\\mathcal{K}(\\theta'_k)$ and the future-action embeddings are combined using Lemma 2 to forecast future state predictions.\n\n4. **Optimise**  \n   - The **only** objective we use is the task-level negative log-likelihood in Eq.~15. Gradients from this objective flow through the analytic Koopman update and into both encoders, so $G$ and $H$ are trained to produce embeddings that enable a single-step Bayesian adaptation to generalise well. No extra fine-tuning stages or auxiliary losses are used.\n\n**Empirically,** MetaKoopman achieves strong predictive performance across a range of distribution shifts (Tables 1 and 2), without requiring test-time encoder updates.\n\n**The model's uncertainty accounts for suboptimal embeddings by design.** We use the posterior predictive variance to measure when the Koopman operator has difficulty in adapting to the past trajectory segment. This behavior is consistent with our observed correlation between predictive uncertainty and error (Table 3), indicating that the model expresses higher uncertainty when adaptation is less reliable.\n\nWe will highlight this training structure and its implications for robustness more clearly in the future version of the manuscript.\n\n---\n\n> * **W3:  It would have been nice to see failure modes of the proposed approach (as well as of other baselines) as a function of the distribution shift, to illustrate the robustness over a spectrum of disturbances.**\n\nThank you for this excellent suggestion to evaluate robustness under increasing distribution shift. In response, we conducted a targeted experiment in the **Hopper** environment to assess how each method handles significant deviations in gravity.\n\nModels were trained on gravity values sampled within a narrow range of $\\pm 1.5$ around the nominal setting. For evaluation, we tested each method under a series of increasing gravity shifts: $[0, 1, 2, 3, 4]$ added to the default gravity, simulating progressively more challenging out-of-distribution dynamics.\n\nTo enable fair comparison, the multi-step prediction MSEs are **normalized within each column**: values are reported relative to the method’s own performance at shift = 0. **Note:** The first-row values are all `1.0` by construction, but the *underlying absolute MSE differs across methods*.\n\n| Gravity Shift | MetaKoopman | BMAML     | GrBAL     | DKO       | BNN       | EMLP      | NODE      |\n|---------------|-------------|-----------|-----------|-----------|-----------|-----------|-----------|\n| +0            | **1**       | **1**     | **1**     | **1**     | **1**     | **1**     | **1**     |\n| +1            | 1.021       | 1.085     | **1.017** | 1.195     | 1.082     | 1.195     | 1.099     |\n| +2            | **1.455**   | 1.534     | 1.555     | 2.080     | 2.122     | 2.080     | 1.388     |\n| +3            | **1.866**   | 1.918     | 2.645     | 4.938     | 2.419     | 4.938     | 2.061     |\n| +4            | **2.368**   | 2.903     | 2.850     | 4.538     | 3.586     | 4.538     | 3.661     |\n\nAs shown, **MetaKoopman** demonstrates the most gradual increase in error across all shift levels, outperforming all baselines in most rows. This highlights its ability to generalize under physical perturbations, benefiting from its closed-form Bayesian adaptation and meta-learned structure. We thank the reviewer so much for the suggestion and will include this additional evaluation in the updated manuscript.\n\n---\n\nWe sincerely thank the reviewer for the thoughtful and constructive feedback. We hope our responses have adequately addressed the main concerns and clarified our contributions. We are happy to provide any further clarification or respond to any additional questions the reviewer may have.\n\n---\n\n##### [1] Shi, Haojie, and Max Q-H. Meng. \"Deep Koopman operator with control for nonlinear systems.\" IEEE Robotics and Automation Letters 7.3 (2022): 7700-7707.\n\n##### [2] Takeishi, Naoya, Yoshinobu Kawahara, and Takehisa Yairi. \"Learning Koopman invariant subspaces for dynamic mode decomposition.\" Advances in neural information processing systems 30 (2017)."}, {"review_id": "UKYg2kDQUx", "reviewer": "Reviewer_azPt", "summary": "This paper introduces a hierarchical Bayesian framework for modelling dynamical systems via Koopman operators under distributional shift. Koopman operators model state transitions of dynamical systems by elevating the state-action space to a higher dimensional feature space where the dynamics are linear and can be effectively approximated by matrices after truncation. Instead of learning a single Koopman operator model, the paper focuses on learning a computationally efficient probability distribution over Koopman operators conditioned on trajectory data. Once learned, the model can be adapted online to changes in the trajectory distribution due to, e.g., changes in the terrain a vehicle is driving over. The paper proposes parameterising a prior over Koopman operators via a matrix normal inverse Wishart (MNIW) distribution, so that the posterior distribution is conjugate and available in closed form. States and actions are encoded via a context encoder, which maps state-action sequences to the sequences of observables (feature space vectors), capturing temporal dependences between steps. The method can also be used for multi-step prediction via an action encoder for the sequences of control/action signals. Experiments involving a range of synthetic and real data are presented, including evaluation on the control of a real-world autonomous truck driving in icy road conditions.", "questions": "* How are the parameters of the encoders learned? Are they learned within the same loop in Algorithm 1 or pre-trained via a separate process?\n\n* How is the learning objective for the variational action encoder formulated?", "limitations": "The main limitations are discussed in Section 6.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n* The paper addresses an important problem in the modelling of dynamical systems deployed in real-world situations.\n* Comprehensive experimental evaluation against relevant baselines in simulated RL benchmark environments and on real truck trajectory data\n* Real-world deployment on an autonomous truck-trailer system using the proposed method for uncertainty-aware motion planning\n* Significant performance improvements with respect to traditional baselines\n\n### Weaknesses\n* **Missing related work.** The paper by Singh et al. (2025, available on ArXiv since 2024, reference below) seems to be addressing the same problem of adaptation of Koopman operators for robust control, though with a different technique. It'd be worth discussing it in the related work section. In addition, the current related work section focuses on Koopman operators and meta-learning for control, missing the whole body of literature on *robust control*, which the current framework could be fit within. Such discussion would help to contextualise the method and discuss the potential advantages/disadvantages of using Koopman-based frameworks for robustness.\n\n* Some technical details are missing, which I elaborate further below.\n    - The predictive distribution in Eq. 13 (Lemma 4.2) should be conditioned on the Koopman operator $\\mathcal{K}$, which is treated as a random variable by the proposed method.\n    - It is also not clear what the superscript means in Eq. 14, though it can be inferred from context that it indexes Koopman operator samples, which again are not explicitly conditioned on, over a rolling window.\n    - I missed an explanation of how the context encoder and action encoder parameters are learned, whether it happens within Algorithm 1's loop or via separate pre-training.\n    - The distribution of trajectories $\\rho(\\mathcal{T})$ is supposedly sampling trajectories under \"distinct operating conditions\", which is a bit confusing. It'd make sense to treat the operating conditions as random variables as well, in case it is a requirement that trajectories are sampled from different environments. Otherwise, in principle, a trajectories distribution may represent trajectories from the same environment.\n    - The learning objective for the variational action encoder, one of the claimed contributions, should be stated in the main text, which currently only discusses its KL regularisation.\n\n* As a minor adjustment, above Eq. 15, it should state the meta-objective is to minimise the *negative* log-likelihood (or maximise the log-likelihood) to match the objective described in the equation.\n\n#### References\n* Singh R, Sah CK, Keshavan J. Adaptive Koopman embedding for robust control of nonlinear dynamical systems. *The International Journal of Robotics Research*. 2025;0(0). doi:10.1177/02783649251341907 (ArXiv: https://arxiv.org/abs/2405.09101)", "was_revised": true, "final_justification": "My main concerns have been addressed, and I believe the feedback from the reviews and discussions in the rebuttal phase should significantly strengthen the paper if appropriately taken into account in the paper's revision.", "rebuttal": "We thank the reviewer for the thoughtful and constructive feedback. We appreciate their recognition of our work in tackling real-world modeling under distribution shifts, conducting comprehensive evaluations across synthetic and real benchmarks, demonstrating strong empirical gains over baselines, and deploying the approach on an autonomous truck. We also value their engagement with both theoretical and practical aspects of the work. Below, we address the reviewer’s comments and clarify several technical points.\n\n---\n\n> ### **W1: Missing Related Work (Singh et al., 2025) [1]**\n\nWe thank the reviewer for pointing us to the relevant and important work by Singh et al. (2025), which addresses online adaptation of Koopman models in the context of robust control. We agree that it is closely related and will incorporate a discussion of it in the final version.\n\nWhile both approaches tackle the broader challenge of adapting Koopman-based models to shifting system dynamics, our method differs in several aspects:\n\n- **Bayesian adaptation:** Singh et al. introduce an online neural correction module that learns residual dynamics. In contrast, our MetaKoopman model performs closed-form Bayesian updates to the Koopman operator using a meta-learned Matrix Normal–Inverse Wishart prior, enabling fast and analytically grounded adaptation.\n\n- **Uncertainty quantification:** Singh et al. do not explicitly model uncertainty. However, our MetaKoopman models predictive uncertainty by producing a posterior predictive distribution, which captures both epistemic and aleatoric uncertainty—important for decision-making under dynamic uncertainty, and not explicitly modeled in Singh et al.\n\n- **Meta-learning framework:** Rather than training an online residual module from scratch, we meta-learn a prior over Koopman operators across a distribution of tasks. This supports rapid adaptation from a few trajectory points without iterative optimization.\n\nWe thank the reviewer for highlighting the relevant related work by Singh et al. We will update our manuscript to link their work to ours, and include a discussion of robust control methodologies in the related works section.\n\n---\n\n\n> ### **W2: Some technical details are missing, which I elaborate further below.**\n>\n> * **The predictive distribution in Eq. 13 (Lemma 4.2) should be conditioned on the Koopman operator $\\mathcal{K}$ which is a random variable.**\n\nThank you for this insightful comment. You are correct that $\\mathcal{K}$ is treated as a random variable in our formulation. As shown in Eq.18 of the appendix, the posterior predictive distribution is obtained by marginalizing over both $\\mathcal{K}$ and $\\Sigma$. This marginalization naturally leads to the posterior mean $\\mathcal{M}$, as reflected in Eq.26. We will revise Eqs.13 and 14 accordingly to express the predictive distribution in terms of $\\mathcal{M}$ in the final version. Thank you again for your close reading and for pointing this out.\n\n> * **It is also not clear what the superscript means in Eq. 14, though it can be inferred from context that it indexes Koopman operator samples**\n\nWe appreciate the reviewer’s question regarding the superscript notation in Equation 14. To clarify, the superscript on $\\mathcal{K}^i$ denotes matrix powers, not sampled Koopman operators — specifically, it refers to recursively applying the Koopman dynamics $i$ times. However, we agree that this notation could be misinterpreted. In response, we have revised the formulation and now present a recursive update rule that more clearly captures multi-step uncertainty propagation:\n\n$$\n\\mu_{t+1} = M \\mu_t, \\quad \n\\Sigma_{t+1} = M \\Sigma_t M^\\top + \\Psi \\left(1 + \\mu_t^\\top V \\mu_t \\right)\n$$\n\nwhere $\\mu_t$ and $\\Sigma_t$ are the mean and the covariance at timestep $t$ respectively. \nWe will incorporate this clarification and the updated expression in the final version. We thank the reviewer for this helpful observation.\n\n> * **How the context encoder and action encoder parameters are learned, whether it happens within Algorithm 1's loop or via separate pre-training.**\n\nWe thank the reviewer for the insightful question. We clarify that both the context and action encoders are trained jointly with the Koopman prior in the same meta-learning loop in Alg. 1 (i.e., no separate pretraining). Gradients from the likelihood objective propagate through the analytic Koopman update into both encoders, enabling them to learn embeddings that support effective single-step Bayesian adaptation. This end-to-end training requires no additional fine-tuning stages or auxiliary losses, resulting in a streamlined and straightforward training pipeline.\n\n> * **The distribution of trajectories is supposedly sampling trajectories under \"distinct operating conditions\", which is a bit confusing.**\n\nThank you for raising this point. In our setting—similar to the formulation in [2]—the term “distinct operating conditions” refers to trajectories generated under different latent parameters affecting the system’s dynamics, such as road surface friction, or slopes. Each trajectory corresponds to a short temporal segment from an environment with a specific configuration of these parameters. A detailed discussion of this setup is provided in Lines 117–122 of the Preliminaries section.\n\nIn this formulation, we treat each task $\\mathcal{T}$ as a random variable indicating which environment a given trajectory segment is drawn from. These variations are not explicitly parameterized, but are implicitly encoded in the trajectory data itself—consistent with common practice in meta-learning.\n\nTo evaluate whether our method can also handle tasks drawn from a static environment (i.e., without distribution shift), we included the HalfCheetah-Stationary benchmark in our experiments. In this setting, MetaKoopman maintained strong performance and outperformed adaptive baselines, indicating that the method is robust and not overly reliant on spurious task variation.\n\nWe have revised the preliminaries section to clarify the interpretation of $\\mathcal{T}$ and better convey this intuition in the corresponding section. \n\n> * **The learning objective for the variational action encoder should be stated more clearly in the main text**\n\nThank you for highlighting this. The variational action encoder is trained using a combination of the KL regularization term (Eq.17) and the negative log-likelihood (NLL) objective (Eq.15) derived from our Koopman-based trajectory model.\n\nTo clarify: the model is trained end-to-end to minimize the following loss:\n\n$$\n\\mathcal{L}\\_{\\text{total}} = \\mathcal{L}\\_{\\text{NLL}} + \\mathrm{KL}\\left(q\\_\\phi(\\tilde{u} \\mid u) || \\mathcal{N}(0, I)\\right),\n$$\n\nwhere $\\mathcal{L}\\_{\\text{NLL}}$ corresponds to the negative log-likelihood of the predicted future trajectories under the Koopman dynamics (Eq.15), and the KL term encourages the latent actions $\\tilde{u}$ to match a standard Gaussian prior for efficient sampling at test time.\n\nThis joint objective ensures that the learned latent space supports both accurate forecasting and scalable planning via the CLAS mechanism. We revised both Section 4.4 and Appendix E to explicitly state this training objective and the interaction between its components.\n\n--- \n\n> ### **W3: As a minor adjustment, above Eq. 15, it should state the meta-objective is to minimise the negative log-likelihood (or maximise the log-likelihood) to match the objective described in the equation.**\n\nWe thank the reviewer for raising this point. We would like to note that lines 197–198, right above Eq.~15, mention this: *\"The meta-objective is to minimize the log likelihood of the ground truth under the posterior predictive distribution.”* Nevertheless, we will give this clearer spotlight in the paper. We thank the reviewer for the helpful and attentive feedback.\n\n---\n---\n\nWe now address the reviewer's *questions* below:\n\n> **Q1: How are the parameters of the encoders learned? Are they learned within the same loop in Algorithm 1 or pre-trained via a separate process?**\n\nWe thank the reviewer for raising this question. We have addressed this in the 'Weaknesses' section and hope our response clarifies the issue.\n\n\n> **Q2: How is the learning objective for the variational action encoder formulated?**\n\nWe appreciate the reviewer for raising this important point. We hope that our answer in the 'Weaknesses' section clarifies the answer.\n\n---\n\n\nWe hope that our responses have addressed your concerns and provided additional clarity on the key points raised. We would be happy to elaborate further if there are any remaining questions. \n\n---\n\n##### [1] Singh, Rajpal, Chandan Kumar Sah, and Jishnu Keshavan. \"Adaptive Koopman embedding for robust control of complex nonlinear dynamical systems.\" The International Journal of Robotics Research.\n\n##### [2] Nagabandi, Anusha, et al. \"Learning to Adapt in Dynamic, Real-World Environments through Meta-Reinforcement Learning.\" International Conference on Learning Representations."}, {"review_id": "uChZy5ALrE", "reviewer": "Reviewer_wjtk", "summary": "This paper presents MetaKoopman, a novel Bayesian meta-learning framework for modeling nonlinear dynamical systems under distribution shift. The core idea is to learn a Matrix Normal-Inverse Wishart (MNIW) prior over the Koopman operator, enabling closed-form Bayesian inference for fast adaptation using short trajectory segments. The model is evaluated on both simulated environments (MuJoCo) and real-world autonomous truck trials in adverse winter conditions, showing strong improvements in prediction accuracy, uncertainty calibration, and robustness to distribution shift.", "questions": "1. Have the authors considered using a probabilistic context encoder or ensemble-based embeddings?\n2. Can MetaKoopman handle high-dimensional control inputs or sparse rewards in reinforcement learning settings?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths\n1. The method combines structured Koopman modeling with closed-form Bayesian updates, avoiding costly inner-loop optimization at test time. Clear derivation and use of conjugate priors (MNIW) allow efficient inference and uncertainty estimation.\n2. While Koopman-based models have been used before, this work is the first to meta-learn a prior over them, addressing dynamic environments and shifts.\n3. Contributions are well broken down: Bayesian operator, variational action encoder, large-scale dataset, etc.\n\nWeaknesses\n1. Although the truck setting is compelling, it remains unclear how well this method generalizes to higher-dimensional robotic tasks (e.g., manipulation, aerial vehicles).\n2. MSE and closed-loop safety in two scenarios are useful, but the breadth of downstream planning metrics (e.g., success rate, collisions, time to goal) is not fully explored.", "was_revised": true, "final_justification": "This work has solid theory and proof. My concerns have also been addressed properly. I keep my positive rating.", "rebuttal": "We sincerely thank the reviewer for their thoughtful and constructive feedback. We appreciate the recognition of the novelty in meta-learning a prior over the Koopman operator, as well as the value of the closed-form updates for efficient inference and principled uncertainty quantification. Below, we address the reviewer’s concerns and questions in detail.\n\n---\n\n> * **W1: Although the truck setting is compelling, it remains unclear how well this method generalizes to higher-dimensional robotic tasks (e.g., manipulation, aerial vehicles).**\n\nThank you for highlighting this important point. While our real-world deployment focuses on a truck–trailer system, the MetaKoopman framework is designed to be domain-agnostic and scalable to high-dimensional systems. Our experiments were deliberately designed to test this scalability.\n\nIn the experiments section of the paper, we include results on Ant, a high-dimensional 8-DoF agent with 113-dimensional observations and complex contact dynamics. As shown in Table 2 (Appendix C.1), MetaKoopman achieves the lowest error across all baselines.\n\nMotivated by the reviewer’s suggestion, we have further extended our evaluation to include a simulated 7-DoF robotic manipulation task under varying damping conditions—**Panda Lift** from *robosuite* [1]. Our evaluation now comprises a real-world autonomous driving dataset alongside six simulated environments, covering three distinct application modalities: self-driving, locomotion, and manipulation. The results for the manipulation task are presented below. \n\n| **metakoopman** | **bmaml**    | **grbal**     | **DKO**       | **BNN**       | **EMLP**     | **NODE**      |\n|-----------------|--------------|---------------|---------------|---------------|--------------|---------------|\n| **0.035 ± .001** | 0.05 ± .002  | 0.07 ± .004   | 0.06 ± .002   | 0.11 ± .004   | 0.08 ± .012  | 0.07 ± .008   |\n\n**Table 1** Prediction error (averaged over 3 seeds) across 32 timesteps. **MetaKoopman** continues to outperform prior methods, demonstrating its scalability and strong generalization capabilities beyond the locomotion and driving domains.\n\n\nWe thank the reviewer for their excellent suggestion to enhance our evaluation, and we remain open to addressing any further comments or concerns they may have. \n\n\n---\n\n> * **W2: MSE and closed-loop safety in two scenarios are useful, but the breadth of downstream planning metrics (e.g., success rate, collisions, time to goal) is not fully explored.**\n\nWe agree that evaluating a broader set of downstream planning metrics would further strengthen the empirical case for our method. In our submission, we focused on two safety-critical real-world scenarios—adaptive braking on ice and evasive lane change on snow—where predictive accuracy under distribution shift directly impacts feasibility and safety.\n\nTo that end, we conducted five independent full-scale trials per scenario. MetaKoopman succeeded in all five, while non-adaptive baselines failed to stop safely or misjudged braking feasibility. Although these outcomes are consistent, we acknowledge that this sample size may be small to establish statistical significance. Scaling up these tests is constrained by the cost and logistics of deploying a 37.5-ton vehicle under controlled winter conditions. We are happy to include these results in the paper if the reviewer thinks so. Additionally, we are open to conducting an extra reinforcement learning experiment to include further downstream metrics in the camera-ready version, should the reviewer recommends it.\n\n\n---\n---\n\nBelow, we answer the reviewer's questions:\n> * **Q1: Have the authors considered using a probabilistic context encoder or ensemble-based embeddings?**\n\nWe appreciate the reviewer raising this point. Yes, we have considered using a probabilistic context encoder, and we explicitly discuss this in Section 6 (Discussion) and Appendix A.1 (lines 797–805). However, motivated by our application, we aim to develop systems that are both rapidly adaptable and explicitly aware of uncertainty in the dynamics. For this reason, we chose to focus uncertainty modeling on the Koopman operator rather than the encoder.\n\nThis modeling choice is guided by the fact that, in practice—especially under distribution shift—**uncertainty in the system dynamics dominates**. When test-time conditions deviate (e.g., due to changes in surface friction or payload), it is typically the dynamics—not the representation—that drive prediction quality. In our setting, the encoder is trained across a diverse task distribution, and its embeddings are generally stable and informative. Focusing uncertainty on the Koopman operator therefore offers the greatest benefit for both performance and calibration, while maintaining real-time efficiency.\n\nNonetheless, we agree that extending MetaKoopman with distributional encoders (e.g., [2, 3, 3]) is a promising direction, particularly for regimes with partial observability or severe data sparsity, and we plan to explore this in future work.\n\n---\n\n> * **Q2: Can MetaKoopman handle high-dimensional control inputs or sparse rewards in reinforcement learning settings?**\n\nYes, similar to our discussion in Weakness 1 — where we evaluated MetaKoopman on high-dimensional tasks such as Ant and HalfCheetah — we believe that the approach can be extended to handle high-dimensional control inputs as well.\n\nRegarding sparse rewards, while our primary focus is on dynamics modeling and planning, we believe MetaKoopman can be seamlessly integrated into model-based reinforcement learning frameworks. Its ability to adapt quickly and model uncertainty in dynamics makes it especially promising for model-based and meta-RL methods.\n\n\n---\n\nWe sincerely thank the reviewer for their valuable feedback. We hope that our responses have adequately addressed the concerns raised and helped clarify the key points. We would be more than happy to provide further elaboration should any questions remain.\n\n---\n\n##### [1] Zhu, Yuke, et al. \"robosuite: A modular simulation framework and benchmark for robot learning.\" arXiv preprint arXiv:2009.12293 (2020).\n\n##### [2] Fraccaro, Marco, et al. \"A disentangled recognition and nonlinear dynamics model for unsupervised learning.\" Advances in neural information processing systems 30 (2017).\n\n##### [3] Yildiz, Cagatay, Markus Heinonen, and Harri Lahdesmaki. \"Ode2vae: Deep generative second order odes with bayesian neural networks.\" Advances in Neural Information Processing Systems 32 (2019).\n\n##### [4] Han, Minghao, Jacob Euler-Rolle, and Robert K. Katzschmann. \"DeSKO: Stability-assured robust control with a deep stochastic Koopman operator.\" International conference on learning representations (ICLR). 2022."}, {"review_id": "0qeWl2sifa", "reviewer": "Reviewer_cqV4", "summary": "This paper proposes MetaKoopman, a framework that combines Koopman operator theory, Bayesian inference, and meta-learning for dynamics modeling under distribution shift. The approach enables fast adaptation and uncertainty-aware forecasting, with evaluation on simulated environments and a real-world autonomous truck dataset.", "questions": "1. What is the purpose of Lemma 4.1 and 4.2 in the context of the paper? Are they supporting a particular theoretical claim? It is unclear whether formalizing them as lemmas is necessary.\n\n2. The real-world dataset is described as being collected across a wide variety of scenarios (e.g., snow, ice, mixed-friction). While this is valuable, it raises concerns about whether the test-time scenarios are actually out-of-distribution. The meta-learner might act more like a state switcher than a generalization engine. How do the authors ensure that test conditions are truly novel and that the adaptation mechanism is not simply memorizing specific dynamics from training?\n\n3. In Table 1, the metric shown is referred to as \"validation loss.\" Could the authors clarify what \"validation\" means in this context? Why is it not test error?\n\n4. In Line 302 the text mentions that \"Transformer-based embedding function\" is kept fixed after training. However, there is no mention of a Transformer model in the section 4.\n\n5. No hyperparameter sensitivity analysis.", "limitations": "yes", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths:\n1. The proposed system demonstrates clearly performance on the experimental datasets.\n\n2. The authors collect a long-duration, multi-scenario real-world dataset from a full-scale autonomous truck system and evaluate the method under realistic operational conditions. This greatly enhances the practical relevance of the work. If released, the dataset would also be a valuable contribution to the community.\n\n3. The paper is well-written and generally easy to follow, with a clear explanation of the system design.\n\nWeaknesses:\n1. While the combination of Koopman operators, Bayesian inference, and meta-learning is technically sound, the framework feels loosely assembled. The motivation and necessity for integrating these specific components is not clearly justified. Each module appears replaceable. For instance, one could use an ODE, SDE, or a Transformer instead of the Koopman operator, or HyperNetworks in place of meta-learning. As such, the novelty is somewhat marginal and the contribution seems incremental.\n\n2. The paper lacks a thorough review and discussion of existing work on handling time-series distribution shifts. Numerous relevant methods have been proposed in meta-learning, continual learning, test-time adaptation, and uncertainty-aware modeling, yet the paper does not engage with this literature, nor are they included in the comparisons.\n\n3. The evaluation is limited in both baselines and datasets. As said before, several strong methods in the distribution shift literature are missing in the baselines. Additionally, the only real-world dataset used is from a single truck platform, with no validation on datasets from different domains or modalities. This limits the generality and persuasiveness of the claims.\n\n4. Figure 1 is not well integrated into the text. Its visual design is not intuitive, making it difficult to understand the proposed workflow. Figure 3 presents the comparison in a visually entangled way, making it hard to discern differences among methods.", "was_revised": true, "final_justification": "While the method is well presented, more real-world datasets and comparative methods with distribution shifts are needed for validation. The current version still requires additional work. I am inclined to keep my current score.", "rebuttal": "We thank the reviewer for their constructive feedback. We are encouraged by the positive remarks regarding the clarity, real-world dataset, and system design. We address the concerns raised below:\n\n---\n\n> * **W1: While the combination of Koopman operators, Bayesian inference, and meta-learning is technically sound, the framework feels loosely assembled...**\n\nWe thank the reviewer and would like to clarify the motivation of our design. Our objective is to perform real-time planning in safety-critical real-world conditions. This means we require an efficient model that is uncertainty-aware and can adapt to distribution shifts.\n\n* Koopman operators offer a way of modeling nonlinear dynamics via linear evolution in a latent space, enabling efficient long-horizon prediction. Importantly, Transformers are not an alternative to Koopman modeling; In fact, in our case, they are used to learn temporally informed latent embeddings where Koopman dynamics apply.\n\n* Regarding ODE/SDE-based alternatives, we agree they are valuable. However, their dependence on numerical solvers often leads to inefficiencies and challenges in uncertainty propagation. As shown in Table 1, 2, MetaKoopman consistently outperforms Neural ODEs. We also evaluated against physics-based ODE models (not included in the paper for space and scope reasons), which we are happy to share upon request.\n\n* Bayesian inference enables closed-form updates over the Koopman operator, directly modeling dynamics uncertainty — a capability absent in prior Koopman methods that used point-estimate operators. To our knowledge, this is the first tractable probabilistic Koopman with explicit dynamics uncertainty quantification.\n\n* Meta-learning is critical for adaptation. While hypernetworks are one class of meta-learners, they are typically computationally intensive and do not easily support uncertainty quantification. In contrast, our approach meta-learns a conjugate prior, enabling fast updates without test-time optimization—a key requirement for planning/control.\n\nEach component in MetaKoopman addresses a specific modeling challenge: Koopman for structure and prediction, Bayesian inference for uncertainty quantification, and meta-learning for adaptation.\n\n---\n\n> * **W2: The paper lacks a thorough review and discussion of existing work on handling time-series distribution shifts… the paper does not engage with this literature, nor are they included in the comparisons.**\n\nThank you for bringing up time-series distribution shifts. We focused on literature related to controllable systems of the form $f(x(t),u(t))$ instead of systems of the form $f(t)$ since they have more complex nuances. However, we agree that some of this work may offer complementary perspectives and will include a brief discussion in the revised version.\n\nOn the meta-learning side, we agree that it is essential. In Section 2, we discuss several meta and bayesian meta-learning methods, highlighting that MetaKoopman differs by offering closed-form adaptation and principled uncertainty quantification. Our evaluation (Tables 1–3, 6) includes both meta-learners and probabilistic baselines (e.g., BNNs, ensembles) to ensure a comprehensive comparison.\n\nFor continual learning, our setup differs: we model piecewise-stationary regimes with rapid shifts (e.g., changes in surface friction), rather than continuous data streams. Nonetheless, we acknowledge the relevance and will include a discussion on this distinction.\n\n---\n\n> * **W3: The evaluation is limited in baselines and datasets...the only real-world dataset used is from a truck platform, with no validation from different domains or modalities. This limits the generality and persuasiveness of the claims.**\n\nThank you for raising this point. The real world dataset presented in the paper is one of the world's first demonstrations of autonomous driving with heavy trucks on ice. We believe this is a significant contribution to embodied AI.\n\nOur experiments cover **seven environments** and a diverse set of distribution shifts: physical (e.g., gravity in *Hopper-Gravity*, slope in *HalfCheetah-Slope*, friction in *Walker-Friction*), structural (e.g., joint failures in *Ant-DisabledJoints*), and real-world non-stationary settings (e.g., snow, ice, and mu-split in the truck dataset). We also include a new robotic manipulation task under varying damping conditions. The results for the manipulation is shown below, highlighting MetaKoopman's strong generalization across **driving, locomotion, and manipulation** domains.\n\n|**metakoopman**|**bmaml**|**grbal**|**DKO**|**BNN**|**EMLP**|**NODE**|\n|-|-|-|-|-|-|-|\n|**0.035±.001**|0.05±.002|0.07±.004|0.06±.002|0.11±.004|0.08±.012|0.07±.008|\n\n**Table 1** Prediction error (averaged over 3 seeds) across 32 timesteps.\n\nRegarding baselines, we compare against **six widely used and representative methods**:\n\n- **Bayesian MAML** (meta-learning and uncertainty),\n- **GrBAL** (meta-learning),\n- **Deep Koopman** (structure-based),\n- **Neural ODEs** (continuous-time modeling),\n- **Bayesian NNs, Ensemble models** (uncertainty).\n\n**MetaKoopman** outperforms these baselines in terms of **prediction accuracy** (Tables 1–2), **uncertainty calibration** (Table 3), **inference speed** (Table 6), and **safety-critical planning outcomes** (Section 5.2, Figs. 4–5). A summary of comparative capabilities is provided in **Table 8**.\n\nThat said, we agree broader validation is important and would be glad to add more experiments should the reviewer have specific suggestions.\n\n\n\n\n---\n\n> * **W4: Figure 1 is not well integrated into the text... Figure 3 presents the comparison in a visually entangled way**\n\nWe appreciate the reviewer’s feedback on Figures 1 and 3 and welcome the opportunity to clarify their intent:\n\n**Figure 1** provides a high-level overview of the *MetaKoopman* pipeline:\n\n1. **Latent context encoding** maps nonlinear state-action trajectories into a latent space where dynamics evolve linearly;\n2. **Closed-form Bayesian inference** adapts a distribution over the Koopman operator using recent trajectory data — our central contribution;\n3. **Uncertainty-aware prediction** uses this adapted distribution to forecast future states with calibrated uncertainty.\n\nThis sequence is described in Section 4 and visualized step-by-step in the figure (p. 4). We will revise the figure to improve clarity, as recommended.\n\n**Figure 3** illustrates the impact of adaptation on predictive accuracy using the truck dataset. It compares the prediction error **before and after** the update to the Koopman operator. As shown, the error decreases from **1.37 to 0.06**, validating the effectiveness of the adaptation. We will make Figure 3 easier to follow, as recommended.\n\nWe thank the reviewer for the suggestions and are confident that the revised figures will improve readability.\n\n\n\n---\n---\n\n**We now answer the reviewer's questions:**\n\n> * **Q1: What is the purpose of Lemma 4.1 and 4.2 in the context of the paper? Are they supporting a particular theoretical claim? It is unclear whether formalizing them as lemmas is necessary.**\n\nThank you for raising this question. **Lemma 4.1 and Lemma 4.2 form the theoretical foundation of the closed-form Bayesian inference procedure at the core of MetaKoopman.**\n\n- **Lemma 4.1** shows that under a MNIW prior, the posterior over $\\mathbf{K}$ and $\\boldsymbol{\\Sigma}$ remains in the same conjugate family, enabling fast, analytic updates at test time—without gradient-based optimization like prior works.\n\n- **Lemma 4.2** derives the posterior predictive over future latent states, supporting uncertainty-aware multi-step forecasting and planning.\n\nWe present these as formal lemmas to emphasize their foundational role in our framework.\n\n---\n\n> * **Q2: The real-world dataset is collected across a variety of scenarios...How do the authors ensure that test conditions are novel and that the adaptation is not memorizing dynamics from training?**\n\nWe appreciate this question. Each task is a trajectory segment reflecting latent conditions (e.g., surface, slope) that evolve unpredictably over time. We note that challenging scenarios (including both the polished ice and mu-split (Appendix Fig. 7)) were only used in the test dataset, to make sure the model did not learn these nuances but instead adapted to them.\n\n---\n\n> * **Q3: In Table 1, the metric shown is referred to as ‘validation loss.’ Could the authors clarify what ‘validation’ means? Why is it not test error?**\n\nThank you for the close reading. The metric in Table 1 reflects the final test loss on held-out trajectories. We agree that using \"validation\" was misleading and will revise the terminology for clarity in the updated version.\n\n---\n\n> * **Q4: In Line 302 the text mentions that 'Transformer-based embedding function' is kept fixed. However, there is no mention of a Transformer in Section 4.**\n\nThank you for raising this point. Both the context and action encoders use Transformer architectures, allowing them to model delayed state-action sequences. While Section 4 describes the framework in a model-agnostic way, full architectural details are provided in Appendix F.2. This is also referenced at the end of Section 4.1 (Line 303). We agree this could be made clearer and will revise the main text to improve clarity.\n\n---\n\n> * **Q5: No hyperparameter sensitivity analysis.**\n\nThank you for raising this point. Appendix D (Table 5) presents a sensitivity analysis on history length, showing consistent gains up to a saturation point. Table 4 ablates the tempering mechanism, where meta-learned tempering improves performance under distribution shift. We’re happy to include further ablations upon request and will add more results in the next version.\n\n---\n\nThank you for the thoughtful feedback. We hope our responses have addressed the main concerns and clarified our contributions. We're happy to provide any clarification or answer any further questions the reviewer might have."}]

Author rebuttal:

Meta-review: This paper proposes meta learning of a koopman operator for control of
a nonlinear under domain shift.

The strengths of the paper include proposing a novel method, testing
it on a real semi truck, and doing an extensive simulation comparison
to other methods.
Decision: Accept (poster)