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

arXiv stat.ML / 4/21/2026

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Key Points

The paper studies how inductive bias can make data-driven control and target reachability in nonlinear systems sample-efficient, avoiding exponential data growth common under generic smoothness assumptions.
For Hamiltonian systems, it uses symplectic geometry and the intrinsic recurrence behavior on energy level sets to support target reachability.
It combines a recurrence-based argument with “chain policies,” which stitch together locally verified trajectory segments learned from demonstrations.
The authors derive sufficient conditions for reachability and show that the required data scales with explicit geometric and recurrence properties of the Hamiltonian rather than with the full state dimension.
Overall, the work positions physical-law structure as an effective substitute for high-dimensional generic assumptions to achieve better generalization from limited data.

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.