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Efficient Dynamics Modeling in Interactive Environments with Koopman Theory

Arnab Kumar Mondal, Siba Smarak Panigrahi, Sai Rajeswar, Kaleem Siddiqi, Siamak Ravanbakhsh

TL;DR

This work leverages Koopman theory to linearize nonlinear interactive-environment dynamics in a high-dimensional latent space, enabling parallelized, stable long-horizon prediction for model-based planning and model-free RL. By employing a diagonal Koopman operator and decoupled state-action encoders, the model achieves efficient training via convolution-based time unrolling and offers gradient-control guarantees through eigenvalue initialization. Empirically, it demonstrates competitive or superior long-horizon state and reward prediction compared with MLP/Transformer/DSSM baselines, while significantly speeding up training, and provides promising results for model-based planning (TD-MPC) and model-free RL in continuous control tasks. The approach highlights practical benefits for data-efficient RL, scalable planning, and robust gradient dynamics, with clear avenues for extending to stochastic dynamics and broader RL algorithms.

Abstract

The accurate modeling of dynamics in interactive environments is critical for successful long-range prediction. Such a capability could advance Reinforcement Learning (RL) and Planning algorithms, but achieving it is challenging. Inaccuracies in model estimates can compound, resulting in increased errors over long horizons. We approach this problem from the lens of Koopman theory, where the nonlinear dynamics of the environment can be linearized in a high-dimensional latent space. This allows us to efficiently parallelize the sequential problem of long-range prediction using convolution while accounting for the agent's action at every time step. Our approach also enables stability analysis and better control over gradients through time. Taken together, these advantages result in significant improvement over the existing approaches, both in the efficiency and the accuracy of modeling dynamics over extended horizons. We also show that this model can be easily incorporated into dynamics modeling for model-based planning and model-free RL and report promising experimental results.

Efficient Dynamics Modeling in Interactive Environments with Koopman Theory

TL;DR

This work leverages Koopman theory to linearize nonlinear interactive-environment dynamics in a high-dimensional latent space, enabling parallelized, stable long-horizon prediction for model-based planning and model-free RL. By employing a diagonal Koopman operator and decoupled state-action encoders, the model achieves efficient training via convolution-based time unrolling and offers gradient-control guarantees through eigenvalue initialization. Empirically, it demonstrates competitive or superior long-horizon state and reward prediction compared with MLP/Transformer/DSSM baselines, while significantly speeding up training, and provides promising results for model-based planning (TD-MPC) and model-free RL in continuous control tasks. The approach highlights practical benefits for data-efficient RL, scalable planning, and robust gradient dynamics, with clear avenues for extending to stochastic dynamics and broader RL algorithms.

Abstract

The accurate modeling of dynamics in interactive environments is critical for successful long-range prediction. Such a capability could advance Reinforcement Learning (RL) and Planning algorithms, but achieving it is challenging. Inaccuracies in model estimates can compound, resulting in increased errors over long horizons. We approach this problem from the lens of Koopman theory, where the nonlinear dynamics of the environment can be linearized in a high-dimensional latent space. This allows us to efficiently parallelize the sequential problem of long-range prediction using convolution while accounting for the agent's action at every time step. Our approach also enables stability analysis and better control over gradients through time. Taken together, these advantages result in significant improvement over the existing approaches, both in the efficiency and the accuracy of modeling dynamics over extended horizons. We also show that this model can be easily incorporated into dynamics modeling for model-based planning and model-free RL and report promising experimental results.
Paper Structure (35 sections, 2 theorems, 31 equations, 10 figures, 7 tables, 2 algorithms)

This paper contains 35 sections, 2 theorems, 31 equations, 10 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Koopman Theory for Dynamical Systems
  4. Approximate Koopman with Control Input
  5. Proposed Model
  6. Linear Latent Dynamics Model
  7. Diagonalization, Efficiency and Stability of the Koopman Operator
  8. Gradients Through Time
  9. Initialization of the Eigenspectrum
  10. Experiments
  11. Long-Range Forward Dynamics Modeling with Control
  12. Training stability
  13. State and Reward Prediction
  14. Koopman Dynamics Model for RL and Planning
  15. Evaluation Metric
  16. ...and 20 more sections

Key Result

Theorem 3.1

For every time step $k\in \{1, .., \tau\}$ in the discrete dynamics, the norm of the gradient of any loss at $k$-step given by $\mathcal{L}_k$ with respect to latent representation at time step $t$ given by $x_t$ is a scaled version of the norm of the gradient of the same loss by $x_{t+k}$, where th and similarly, for all $l \leq k$, the norm of the gradient of $\mathcal{L}_k$ with respect to the

Figures (10)

  • Figure 1: A comparison of our Koopman-based linear dynamics model with a non-linear MLP-based dynamics model. The Diagonal Koopman formulation allows for modeling longer horizons efficiently with control over gradients. Here BPTT stands for Backpropagation Through Time.
  • Figure 2: A schematic of the latent Koopman dynamics model. Both actions and initial state embedding are encoded into a latent space in complex ($\mathbb{C}$) domain before passing through the Koopman dynamics block. (see \ref{['appdx:jax_implementation']} for an efficient Jax implementation of the model)
  • Figure 3: Forward state and reward prediction error in Offline Reinforcement Learning environments. We consider five dynamics modeling techniques and perform this prediction task over a horizon of 100 environment steps. The results are over 3 runs. Our Koopman-based method is competitive with the best performing GRU baseline while being $2\times$ faster. See \ref{['appdx:detailed_results']} for exact numerical values.
  • Figure 4: Training speed in iterations/second ($\uparrow$) for the state prediction task using different dynamics model on halfcheetah-expert-v2. Each iteration consists of one gradient update of the entire model using a mini-batch of 256 in A100 GPU. See \ref{['tab:time']} for exact numerical values.
  • Figure 5: Comparison of our Koopman-based dynamics model (with a horizon of 20) and an MLP-based dynamics model of vanilla TD-MPC pmlr-v162-hansen22a. The results are over 5 random seeds for each environment. Higher Mean $\&$ IQM and lower Optimality Gap is better.
  • ...and 5 more figures

Theorems & Definitions (3)

  • Theorem 3.1
  • Theorem A.1
  • proof