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This paper introduces Lagrangian Descriptors (LDs) as a novel diagnostic framework for evaluating how well neural networks model Hamiltonian systems, going beyond simple trajectory prediction accuracy. By weighting probability density functions with LD values, the method embeds geometric information about orbits and separatrices into a statistical framework. Benchmarking symplectic architectures (SympNet, H\'enonNet, GHNN) and Reservoir Computing on the Duffing oscillator and the three-mode nonlinear Schrödinger equation reveals that while symplectic nets preserve energy, Reservoir Computing better reproduces phase-space topology, despite lacking explicit physical constraints.
Reservoir Computing, despite lacking explicit constraints, can outperform symplectic architectures in capturing the phase-space topology of Hamiltonian systems.
We propose Lagrangian Descriptors (LDs) as a diagnostic framework for evaluating neural network models of Hamiltonian systems beyond conventional trajectory-based metrics. Standard error measures quantify short-term predictive accuracy but provide little insight into global geometric structures such as orbits and separatrices. Existing evaluation tools in dissipative systems are inadequate for Hamiltonian dynamics due to fundamental differences in the systems. By constructing probability density functions weighted by LD values, we embed geometric information into a statistical framework suitable for information-theoretic comparison. We benchmark physically constrained architectures (SympNet, H\'enonNet, Generalized Hamiltonian Neural Networks) against data-driven Reservoir Computing across two canonical systems. For the Duffing oscillator, all models recover the homoclinic orbit geometry with modest data requirements, though their accuracy near critical structures varies. For the three-mode nonlinear Schr\"odinger equation, however, clear differences emerge: symplectic architectures preserve energy but distort phase-space topology, while Reservoir Computing, despite lacking explicit physical constraints, reproduces the homoclinic structure with high fidelity. These results demonstrate the value of LD-based diagnostics for assessing not only predictive performance but also the global dynamical integrity of learned Hamiltonian models.