On Distributional Reinforcement Learning in Chaotic Dynamical Systems

Researchers propose that distributional reinforcement learning offers superior performance in chaotic dynamical systems by measuring return distributions under the 1-Wasserstein metric rather than optimizing scalar expected values. This approach reduces variance and improves gradient conditioning in systems with exponential sensitivity to initial conditions, providing theoretical foundations for applying RL to climate, fluid dynamics, and multi-agent scenarios.

Chaotic dynamical systems present a fundamental obstacle for reinforcement learning algorithms: minute variations in initial conditions cascade into exponentially divergent trajectories, destabilizing the bootstrap targets that RL methods rely upon. Traditional RL approaches optimize expected scalar returns, effectively averaging across these diverging paths and conflating trajectory-level instability with the learning objective itself. This research introduces a paradigm shift by demonstrating that return distributions—rather than point estimates—evolve more predictably under the 1-Wasserstein metric, a mathematical measure of probability distribution distance.

The theoretical insight centers on statistical stability: while individual trajectories diverge chaotically, the statistical properties of return distributions maintain sufficient regularity to support effective learning. By aligning optimization with this distributional structure rather than scalar expectations, the method achieves better-conditioned gradient updates and reduced variance in learning dynamics. This represents a principled mathematical explanation for why distributional RL methods empirically outperform value-based approaches in chaotic regimes.

The implications span multiple domains where chaotic dynamics dominate: climate modeling, turbulent fluid flows, and multi-agent systems all exhibit the exponential sensitivity that currently hampers RL application. Improved learning stability in these areas could accelerate the development of control strategies for complex physical systems. For the broader ML community, this work bridges dynamical systems theory with reinforcement learning, offering geometric insights into how optimization objectives interact with system chaos. The research provides actionable theoretical guidance for practitioners designing RL systems for inherently unstable environments.

• →Distributional RL measures return distributions under 1-Wasserstein metric, which evolves more smoothly than individual chaotic trajectories
• →Chaotic systems cause high-variance bootstrap targets and poor gradient conditioning in standard RL methods through exponential sensitivity to initial conditions
• →The method applies to climate, fluid dynamics, and multi-agent systems where reliable RL has been previously difficult
• →Return distributions maintain statistical stability even when underlying trajectories diverge exponentially
• →This provides theoretical foundation explaining why distributional methods empirically outperform scalar value-based approaches in chaotic domains