Phase space integrity in neural network models of Hamiltonian dynamics: A Lagrangian descriptor approach

Abstract

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, HénonNet, 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 Schrödinger 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.

Figures (22)

• Figure 1: LDs and LD-weighted PDF for the Duffing oscillator. Top panels: 3D visualization of LD fields (left) and the corresponding PDF (right). Bottom panels: 2D projections of the same data. Singular features in the LD align with separatrices, while the weighted PDF highlights these regions as dynamically dominant. The LD functional in Eq. \ref{['eq:LD_arc_length']} is used with $c=0.7$ and an integration time of $\tau=4$. The PDF employs the weighting function $g(x) = 1/x$.
• Figure 2: Training (left) and validation (right) loss for the NN models in the Duffing case. SympNet (red dotted), HénonNet (green dash–dotted), GHNN (purple dash–dotted), and Reservoir Computing (RC, orange dashed). Note that RC training does not involve iterative optimization; the output weights are obtained by solving a regularized normal equation.
• Figure 3: Forward and backward predictions of the Duffing equation starting from $t_0=0$. Top row: $(q(0), p(0))=(-0.71, -0.51)$ (inside homoclinic orbit); Middle row: $(q(0), p(0))=(-0.71, -0.61)$ (near homoclinic orbit); Bottom row: $(q(0), p(0))=(-0.1, -0.71)$ (outside homoclinic orbit). From left to right, the columns show: the pointwise error in position $q(t)-\hat{q}(t)$, the pointwise error in momentum $p(t)-\hat{p}(t)$, the normalized prediction error (Eq. \ref{['eq:error_Duffing']}) as function of the integration time $\tau$, and normalized LD (Eq. \ref{['eq:LD_Duffing']}) as function of the integration time $\tau$. Predictions are shown over the interval $t\in[-10,10]$. The reference solution (blue solid) is compared with SympNet (red dotted), HénonNet (green dash-dotted), GHNN (purple dash-dotted), and RC (orange dashed). All NNs were trained on the dataset corresponding to the 200-trajectory case.
• Figure 4: Same as Fig. \ref{['fig:time_trace_Duff']}, but with $p(0)>0$. Top row: $(q(0), p(0))=(-0.71, 0.51)$ (inside homoclinic orbit); Middle row: $(q(0), p(0))=(-0.71, 0.61)$ (near homoclinic orbit); Bottom row: $(q(0), p(0))=(-0.1, -0.71)$ (outside homoclinic orbit).
• Figure 5: The prediction error $\hat{e}$ (left) and normalized prediction error $e$ (right) of Duffing trajectories across initial conditions : (A-B) SympNet, (C-D) HénonNet, (E-F) GHNN, (G-H) RC. The white dots indicate the initial conditions of the trajectories shown in Figs. \ref{['fig:time_trace_Duff']} and \ref{['fig:time_trace_Duff_plus']}.