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Learning Compositional Koopman Operators for Model-Based Control

Yunzhu Li, Hao He, Jiajun Wu, Dina Katabi, Antonio Torralba

arXiv:1910.08264v2cs.LGcs.ROmath.OCstat.ML

TL;DR

Models for complex dynamics must be efficient while generalizing to variable numbers of objects, but prior approaches face limitations in one or both respects. The paper combines graph-based object embeddings with compositional linear Koopman dynamics, and reports improved accuracy, adaptation speed, and baseline performance on ropes and soft robots.

  • Problem

    Dynamics models need to be efficient for control and generalize to changing environments and variable numbers of objects, while graph-based models can be costly to optimize.

    Source: §1 · p. 1

  • Method

    The method combines graph neural networks that encode object-centric embeddings with data-identified linear Koopman and control matrices whose blocks are shared by relation.

    Source: §1 · p. 1, §3.2 · p. 5

  • Results

    The method is more accurate than learned physics engines, adapts faster to unknown-parameter environments, and outperforms vanilla deep Koopman and manually designed basis models.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The formulation supports efficient manipulation of deformable objects such as ropes and soft robots and generalization to systems with variable component counts.

    Source: §5 · p. 9

    The block-sharing design assumes that object pairs with the same physical interaction can share the same transition block.

    Source: §3.2 · p. 5

Abstract

from arXiv · show

Finding an embedding space for a linear approximation of a nonlinear dynamical system enables efficient system identification and control synthesis. The Koopman operator theory lays the foundation for identifying the nonlinear-to-linear coordinate transformations with data-driven methods. Recently, researchers have proposed to use deep neural networks as a more expressive class of basis functions for calculating the Koopman operators. These approaches, however, assume a fixed dimensional state space; they are therefore not applicable to scenarios with a variable number of objects. In this paper, we propose to learn compositional Koopman operators, using graph neural networks to encode the state into object-centric embeddings and using a block-wise linear transition matrix to regularize the shared structure across objects. The learned dynamics can quickly adapt to new environments of unknown physical parameters and produce control signals to achieve a specified goal. Our experiments on manipulating ropes and controlling soft robots show that the proposed method has better efficiency and generalization ability than existing baselines.

1 INTRODUCTION

The introduction frames a need for dynamics models that are both efficient for control and generalize across changing systems with variable numbers of components. It proposes compositional Koopman operators and reports stronger accuracy, adaptation speed, and baseline performance in its experiments.

  • Complex-system models need both efficient identification and motor control and the ability to generalize across constantly evolving environments.Source: §1 · p. 1
  • Existing Koopman methods offer efficient linear approximations but focus on single systems, while graph-based dynamics models generalize across object counts but can be slow and sample-inefficient to optimize.Source: §1 · p. 1
  • Compositional Koopman operators combine graph-network object embeddings with data-identified linear Koopman and control matrices for efficient system identification and control synthesis.Source: §1 · p. 1
  • Sharing block-wise transition submatrices among object pairs with the same relation reduces identification parameters independently of system size, addressing quadratic parameter growth.Source: §1 · pp. 1–2
  • The method is significantly more accurate than learned physics engines, adapts faster to unknown-parameter environments, and outperforms vanilla deep Koopman and manual-basis models.Source: §1 · p. 2

2 RELATED WORK

Related work covers Koopman methods, learned physical simulators, and model-based control. The paper combines graph-based object modeling with linear Koopman dynamics, including for efficient control optimization.

  • Koopman methods map nonlinear system states into a space where they propagate linearly, supporting efficient prediction, estimation, and control.Source: §2 · p. 2
  • Neural networks can learn more expressive Koopman observables than hand-crafted basis functions, while this paper uses graph networks to model compositionality and generalize across object counts and shapes.Source: §2 · p. 2
  • Learned physical simulators model object interactions with neural networks and adapt to unknown physics, but their optimization can require substantial samples and compute for online adaptation.Source: §2 · pp. 2–3
  • Unlike approaches that learn a policy or use learned models in model-predictive control, the paper exploits linear Koopman embeddings to formulate control as quadratic programming.Source: §2 · p. 3

3 APPROACH

The approach combines Koopman linearization with graph-based, object-centric representations to model variable-sized physical systems. It uses shared block structure to reduce identification complexity, then learns embeddings and plans controls in the resulting linear dynamics.

3.1 THE KOOPMAN OPERATORS

  • Koopman observation functions map nonlinear states into an invariant subspace where a finite-dimensional linear operator approximates the dynamics.With external inputs, the dynamics are modeled by a Koopman matrix and a control matrix.Source: §3.1 · p. 3

3.2 COMPOSITIONAL KOOPMAN OPERATORS

  • Compositional Koopman operators use object-centric embeddings and shared transition blocks to encode repeated physical interactions across multi-object systems.The block structure is intended to improve generalization and make system identification and control synthesis more efficient.Source: §3.2 · pp. 3–5

3.3 LEARNING THE KOOPMAN EMBEDDINGS USING GRAPH NEURAL NETWORKS

  • A graph neural network maps system states into Koopman embeddings, while linear regression identifies the Koopman and control matrices from observation sequences.Sharing blocks reduces the parameters to identify from N^2m^2 to hm^2, where h is the number of block types and usually does not depend on N.Source: §3.3 · p. 5
  • The method trains its graph encoder and decoder with auto-encoding, prediction, and metric losses, with the metric loss preserving distances used in downstream control.The trained model supports system identification, future prediction, and control synthesis.Source: §3.3 · p. 6

3.4 CONTROL

  • Control is formulated as minimizing a quadratic cost over linear Koopman dynamics, with model predictive control available to address accumulating prediction error.The stated goal-reaching cost penalizes terminal distance from the target and regularizes control signals.Source: §3.4 · p. 6

4 EXPERIMENTS

Across rope and soft-robot simulations, the method outperforms the baselines, predicts dynamics accurately over long horizons, and controls targets accurately. Ablations favor the block-wise Koopman structure, while results also vary with data quantity and embedding dimension.

4 EXPERIMENTS

  • Experiments evaluate Rope, Soft, and Swim systems with varying object counts, including held-out systems with 10 to 14 objects.The test protocol adapts models to unknown physical parameters using short observation sequences.Source: §4 · pp. 6–8

4.1 SIMULATION

  • Our method significantly outperforms all baselines in simulation across testing environments, accurately predicting dynamics for more than 100 steps.Rope errors reflect delayed force propagation, while Soft errors mainly reflect local-component misalignment.Source: Figure 3 · p. 8, §4.1 · pp. 8–9

4.2 CONTROL

  • Our model achieves smaller control errors than KPM across environments and accurately moves rope and soft-body parts to their targets.Soft and Swim control sequences receive feedback after 32 of 64 steps.Source: Figure 4 · p. 8, §4.2 · p. 9

4.3 ABLATION STUDY

  • More system-identification data improves simulation, while embedding dimension 16 gives the best simulation results among tested dimensions.The authors suggest the Rope system's intrinsic Koopman invariant-space dimension may be around 16 per object.Source: Figure 4 · p. 8, §4.3 · p. 9
  • Block structure yields smaller errors across settings; Diag impairs control, while None has comparable simulation errors but larger control errors.Block uses relation-based submatrices; Diag shares diagonal blocks and sets off-diagonal blocks to zero, while None imposes no structure.Source: Table 1 · p. 9, §4.3 · p. 9

5 CONCLUSION

The conclusion presents compositionality as a route to manipulating deformable objects and generalizing across systems with variable numbers of components. It proposes integrating Koopman theory with neural networks to model larger, more complex systems.

5 CONCLUSION

  • Compositional Koopman operators with graph neural networks manipulate ropes and soft robots and generalize to variable numbers of components.The authors frame integration of Koopman theory and neural networks as a direction for modeling larger, more complex systems.Source: §5 · p. 9
  • The formulation combines deep Koopman operators with graph neural networks to leverage compositional structure in deformable objects.The conclusion identifies ropes, soft robots, granular materials, cloth, and modeling clay as examples of compositional objects.Source: §5 · p. 9
  • Traditional methods make manipulation of challenging deformable objects difficult, motivating the proposed formulation's new direction.The conclusion names ropes and soft robots as examples.Source: §5 · p. 9

A ENVIRONMENT AND MODEL DETAILS

The experiments assign interaction types according to relation type and object properties, with environment-specific interaction categories for ropes and soft robots.

Interaction types

  • Interactions count as different when their types or participating objects' physical properties differ.This criterion is used in the experiments to distinguish interaction relations.Source: A ENVIRONMENT AND MODEL DETAILS · p. 13

Rope

  • Rope interactions distinguish top from non-top masses and include eight types of interactions between different objects.The eight types combine object categories with nearby or two-hop relations.Source: A ENVIRONMENT AND MODEL DETAILS · p. 13

Soft

  • Soft interactions use four quadrilateral types and eight connection directions, yielding 36 relation types between different objects.The relation types also encode the receiver object's type.Source: A ENVIRONMENT AND MODEL DETAILS · p. 13

Swim

B ADDITIONAL EXPERIMENTS

Additional experiments compare the method with Bayesian Optimization (BO) under different assumptions about physical-model knowledge and test the metric loss. BO takes much longer in the known-model setting, while the metric loss improves distance preservation without a large prediction-accuracy sacrifice.

B ADDITIONAL EXPERIMENTS

  • 0.43 vs. 180 seconds (Ours vs. BO) in the Rope environment when the ground-truth model is known.Times are averaged over 100 trials.Source: B ADDITIONAL EXPERIMENTS · p. 13
  • Simulation error 0.046 vs. 0.084 and control error 0.854 vs. 2.547 (Ours vs. BO) when the Rope system is approximated by a chained spring-mass model.BO takes much more time and still does not give a satisfying result when the ground-truth model is uncertain.Source: B ADDITIONAL EXPERIMENTS · p. 13
  • Metric loss concentrates the log distance ratio nearer 0 while leaving prediction performance comparable, and the metric-loss model performs better control.The Rope ablation compares models trained with and without metric loss.Source: B ADDITIONAL EXPERIMENTS · pp. 13–14
  • With ground-truth physical parameters, IN and PN slightly outperform the method because its internal linear structure limits expressiveness.The authors note that physical parameters are not always known in real-world adaptation settings.Source: B ADDITIONAL EXPERIMENTS · p. 14, Figure 6 · p. 14

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