MetaKoopman

Truck Trials

Real-world winter test demonstrating the operational environment and the nonlinear dynamics MetaKoopman is designed to model and adapt to.

Field footage from the truck trials. These tests highlight the distributional challenges MetaKoopman must adapt to: slippery surfaces, varying loads, and rapidly shifting dynamics.

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.

Methodology

Background: Koopman Operators

Koopman operator theory represents nonlinear dynamics linearly in a lifted latent space.
States and controls are mapped into a higher-dimensional latent representation where the Koopman operator acts linearly.
This makes complex dynamics more interpretable and a natural fit for prediction, planning, and control.
Classical Koopman models are typically trained offline and do not model uncertainty or online adaptation.

MetaKoopman: Core Idea

Place a Matrix Normal–Inverse Wishart (MNIW) prior distribution over the Koopman operator, learning a distribution over dynamics rather than a single matrix.
As new trajectory segments arrive, perform a closed-form Bayesian update to obtain a posterior over the Koopman operator.
Use this posterior to generate uncertainty-aware predictions of future latent states and their projections to the physical state space.
Explicitly model both epistemic (model) and aleatoric (noise) uncertainty, enabling fast adaptation under distribution shifts.

MetaKoopman Framework Overview

High-level architecture: context and action encoders feed a conjugate Bayesian update over the Koopman operator, which is then used for multi-step prediction in latent and physical state spaces.

Step 1 — Context Encoding: Encode past trajectories of states and actions into latent representations.
Step 2 — Bayesian Meta-Update: Update the MNIW prior over the Koopman operator using recent data, yielding a posterior distribution over dynamics.
Step 3 — Prediction & Planning: Roll out future latent states under the posterior Koopman operator and project them into the physical state space.
Training Objective: Minimize the Negative Log-Likelihood (NLL) of trajectories under the predictive posterior, encouraging calibrated, adaptable Koopman dynamics.

Experiments

Real-World Data Collection

The truck dataset is collected using a full-scale, 37.5-ton autonomous truck–trailer platform. The dataset includes comprehensive 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 a 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.

Truck & Trailer: A 37.5-ton tractor–semitrailer operating close to its physical traction limits, capturing realistic heavy-vehicle dynamics under extreme winter conditions.
μ-Split: Sections of roadway with asymmetric friction, producing abrupt left–right force imbalances that challenge stability, braking performance, and model adaptation.
Test maneuvers: A diverse set of maneuvers, including emergency braking, evasive lane changes, high-speed cornering, and transitions across changing surface types.

Truck & Trailer System

Full-scale articulated truck–trailer platform tested on snow, ice, and μ-split surfaces.

µ-Split Scenario

Asymmetric friction stressing rapid adaptation to shifting tire forces.

Simulated Environments

Beyond the real truck data, we evaluate MetaKoopman on a suite of MuJoCo-based control tasks that induce structured distribution shifts:

HalfCheetah-Slope: varying ground slope causes persistent changes in dynamics.
Hopper-Gravity: altered gravitational acceleration changes stability and control.
Walker-Friction: contact friction ranges from slippery to sticky terrain.
Ant-DisabledJoints: randomly disabled legs model structural failures.
Panda-Damping: time-varying joint damping perturbs manipulation dynamics.

HalfCheetah-Slope
Varying ground slope.

Hopper-Gravity
Changing gravity.

Walker-Friction
Varying contact friction.

Ant-DisabledJoints
Random joint failures.

Panda-Damping
Time-varying damping.

Predictive Performance under Distribution Shifts

We report validation MSE (mean ± standard deviation) across methods and environments. MetaKoopman consistently achieves the best predictive accuracy on both the truck dataset and the simulated benchmarks. Insert the numerical values from the paper into the cells marked “see paper”.

Environment	MetaKoopman	BayesianMAML	GrBAL	DKO	BayesianNN	EMLP	NeuralODE
Truck Dataset	0.0596 ± 0.0021	0.0888 ± 0.0100	0.0917 ± 0.0023	0.2042 ± 0.1674	0.2923 ± 0.0120	0.2589 ± 0.0053	0.0926 ± 0.0022
Hopper-Gravity	0.0369 ± 0.0019	0.0475 ± 0.0027	0.0926 ± 0.0016	0.0913 ± 0.0051	0.1332 ± 0.0116	0.1683 ± 0.0000	0.0793 ± 0.0078
HalfCheetah-Slope	0.7490 ± 0.0117	0.8808 ± 0.0104	1.2800 ± 0.0028	1.2927 ± 0.0196	1.7008 ± 0.0374	1.4395 ± 0.0210	1.9981 ± 0.2123
Walker-Friction	0.52 ± 0.10	0.54 ± 0.00	0.57 ± 0.07	0.59 ± 0.00	0.67 ± 0.02	0.61 ± 0.03	0.64 ± 0.05
Ant-DisabledJoints	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 ± 0.09
Panda-Damping	0.035 ± 0.002	0.05 ± 0.0002	0.07 ± 0.002	0.06 ± 0.003	0.11 ± 0.004	0.08 ± 0.02	0.07 ± 0.007

Uncertainty Calibration

To assess uncertainty quality, we compute the Pearson correlation between predicted variance and squared error over a large held-out set. Higher correlation means uncertainty peaks where the model tends to make larger errors. MetaKoopman yields the best calibration across all evaluated datasets.

Dataset	MetaKoopman	EMLP	BayesianNN	BayesianMAML
Truck Dataset	0.68	0.58	0.48	0.56
HalfCheetah-Slope	0.69	0.63	0.52	0.62
Ant-DisabledJoints	0.72	0.59	0.53	0.58

Further ablations on history length, tempering, and inference runtime — including comparisons to Deep Koopman, neural ODEs, Bayesian neural networks, ensembles and GrBAL — are reported in the main paper and appendix.

Truck Trial Scenarios

Finally, we visualize two representative winter-test maneuvers on the full-scale truck–trailer system.

Braking on Polished Ice

Emergency braking on a polished-ice section, stressing longitudinal dynamics and MetaKoopman’s ability to adapt to extremely low friction.

Lane Change Maneuver

High-demand lane change on mixed-friction (µ-split) roads, probing the model’s ability to anticipate lateral dynamics and distribution shifts for safe motion planning.

BibTeX

If you find MetaKoopman useful in your work, please consider citing:

Copy citation:

@inproceedings{selimmetakoopman,
  title     = {MetaKoopman: Bayesian Meta-Learning of Koopman Operators for Modeling Structured Dynamics under Distribution Shifts},
  author    = {Selim, Mahmoud and Bhat, Sriharsha and Johansson, Karl Henrik},
  booktitle = {The Thirty-ninth Annual Conference on Neural Information Processing Systems}
}