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

2026-04-18

AI Summary

This work addresses the exponential growth in data requirements caused by smoothness assumptions in high-dimensional nonlinear Hamiltonian systems. It proposes a data-efficient, target-oriented reachability control method that uniquely integrates symplectic geometric inductive bias with trajectory recurrence on energy manifolds. By composing demonstrably verified local trajectory segments through a chaining strategy, the approach achieves control without relying on global smoothness. The established sufficient conditions for reachability tie sample complexity to the system’s intrinsic geometric and dynamical properties rather than its state dimensionality, thereby substantially overcoming the data bottleneck that plagues conventional methods in high-dimensional settings.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets
• Security and Privacy: Data transparency and provenance
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Hamiltonian systems
• target reachability
• symplectic geometry
• inductive bias
• data efficiency

Innovation

• symplectic inductive bias
• Hamiltonian systems
• target reachability
• chain policies
• recurrence property