Symplectic Inductive Bias for Data-Driven Target Reachability in Hamiltonian Systems

TL;DR

This paper introduces a symplectic inductive bias leveraging Hamiltonian structure and recurrence properties to improve data efficiency in learning control policies for nonlinear Hamiltonian systems.

Contribution

It proposes a novel approach combining symplectic geometry and chain policies to achieve target reachability with data requirements depending on geometric properties, not dimension.

Findings

Data requirements depend on geometric and recurrence properties of the Hamiltonian.

The approach enables target reachability using locally certified trajectory segments.

Leverages physical laws as inductive bias for nonlinear control.

Abstract

Inductive bias refers to restrictions on the hypothesis class that enable a learning method to generalize effectively from limited data. A canonical example in control is linearity, which underpins low sample-complexity guarantees for stabilization and optimal control. For general nonlinear dynamics, by contrast, guarantees often rely on smoothness assumptions (e.g., Lipschitz continuity) which, when combined with covering arguments, can lead to data requirements that grow exponentially with the ambient dimension. In this paper we argue that data-efficient nonlinear control demands exploiting inductive bias embedded in nature itself, namely, structure imposed by physical laws. Focusing on Hamiltonian systems, we leverage symplectic geometry and intrinsic recurrence on energy level sets to solve target reachability problems. Our approach combines the recurrence property with a recently proposed class of policies, called chain policies, which composes locally certified trajectory segments extracted from demonstrations to achieve target reachability. We provide sufficient conditions for reachability under this construction and show that the resulting data requirements depend on explicit geometric and recurrence properties of the Hamiltonian rather than the state dimension.