Lagrangian and Hamiltonian Neural Networks With a Dissipative System

V. Rayamajhi; J. Singal

cs.LGSep 25, 2026

Lagrangian and Hamiltonian Neural Networks With a Dissipative System

Authors: V. Rayamajhi, J. Singal

Organizations: University of Richmond

Abstract

We investigate the applicability of Lagrangian and Hamiltonian Neural Network models to a dissipative system that has explicit time dependence in its Lagrangian, Hamiltonian, and total energy. To do so we consider these neural network models for simulated systems of a harmonic one-dimensional, one-component oscillator with damping, as well as without damping for comparison. We find that both the Lagrangian and Hamiltonian approaches are able to predict the empirical physical behavior of the damped oscillator systems and to effectively ``learn'' to varying degrees the underlying Lagrangians and Hamiltonians, as has previously been shown to be the case with undamped oscillator systems. These investigations elucidate important properties of Lagrangian and Hamiltonian mechanics, including properties that are not manifest when considering systems without explicit time dependence.

Figures & tables

Figure 1: A schematic of the basic principle of a Lagrangian NN and a Hamiltonian NN as implemented in this work, contrasted with those of purely empirical NNs which would predict the same quantities. As discussed in § 1 , Lagrangian and Hamiltonian NNs can learn the Lagrangian and Hamiltonian functions of a system, which are not themselves directly observable but are important for understanding its properties, by using derivatives of the Lagrangian and Hamiltonian functions with respect to the inputs to predict other observables. There are more hidden layers, and the hidden layers contain many more neurons, than depicted here for simplicity. Network architectures are given in Table 1 .

Panel of Figure 2 — Figure 2: The actual and predicted acceleration $\ddot{q}_{\rm\text{SHO}}(t)$ versus time ( upper ) and actual and predicted Lagrangian $L_{\rm\text{SHO}}$ vs time ( lower ) for the undamped oscillator, for the cases of no noise ( left ) and full simulated noise ( right ) in the “observed” data. As discussed in § 5 the Lagrangian is also ambiguous to the presence of a particular function of the generalized coordinates, which explains the apparent discrepancy between the predicted and actual Lagrangians.

Panel of Figure 3 — Figure 3: The actual and predicted acceleration $\ddot{q}_{\rm\text{D}}(t)$ versus time ( upper ) and actual and predicted Lagrangian $L_{\rm\text{D}}$ vs time ( lower ) for the damped oscillator, for the cases of no noise ( left ) and full simulated noise ( right ) in the “observed” data. As discussed in § 5 the Lagrangian is also ambiguous to the presence of a particular function of the generalized coordinates, which explains the apparent discrepancy between the predicted and actual Lagrangians.

Panel of Figure 4 — Figure 4: Actual and predicted velocity $\dot{q}_{\rm\text{SHO}}(t)$ versus time ( upper ), actual and predicted $\dot{p}_{\rm\text{SHO}}(t)$ versus $\dot{q}_{\rm\text{SHO}}(t)$ ( middle ), and actual and predicted Hamiltonian $H_{\rm\text{SHO}}$ vs time ( lower ) for the undamped oscillator, For the cases of no noise ( left ) and full simulated noise ( right ) in the “observed” data.

Panel of Figure 5 — Figure 5: Actual and predicted velocity $\dot{q}_{\rm\text{D}}(t)$ versus time ( upper panels ), actual and predicted linear $\dot{p}_{\rm\text{D},linear}(t)$ versus $\dot{q}_{\rm\text{D}}(t)$ ( middle panels ), and actual and predicted Hamiltonian versus time ( lower panels ) for the damped oscillator in the Hamiltonian NN, for the cases of no noise ( left ) and full simulated noise ( right ) in the “observed” data. The Hamiltonian NN uses the canonical momentum $p_{\rm\text{D}}(t)$ as input and provides the canonical $\dot{p}_{\rm\text{D}}(t)$ . The conversion between $\dot{p}_{\rm\text{D},linear}(t)$ and $\dot{p}_{\rm\text{D}}(t)$ is given by equation 31 .

Panel of Figure 6 — Figure 6: Actual and predicted velocity $\dot{q}_{\rm\text{D}}(t)$ versus time ( upper panels ), actual and predicted linear $\dot{p}_{\rm\text{D},linear}(t)$ versus $\dot{q}_{\rm\text{D}}(t)$ ( middle panels ), and actual and predicted non-dissipative system energy $E_{\rm D,tot}$ versus time ( lower panels ) for the damped oscillator in the Energy NN, for the cases of no noise ( left ) and full simulated noise ( right ) in the “observed” data. The Energy NN uses the linear momentum $p_{\rm\text{D,linear}}(t)$ as input and provides $\dot{p}_{\rm\text{D},linear}(t)$ directly.

Panel of Figure 7 — Figure 7: Calculated true default Lagrangian functions for the undamped ( top row) and damped with time as an input parameter ( bottom row) given by equations 14 and 18 , respectively, along with the visually approximated alternate $L^{\prime}$ functions of the form of equation 33 for each case. The forms of the functions $F(q)$ and the values for the constants $A^{\prime}$ and $C$ for the plotted $L^{\prime}$ functions are given in Table 2 . As discussed in § 5 these $L^{\prime}$ functions can be compared to the Lagrangians found by the LNN models in Figures 2 and 3 to see that the LNN models are able to successfully recover a correct Lagrangian for the systems, even if not the simplest default Lagrangian. We note that the damping cases may require additional higher order terms for a more precise fit to the recovered Lagrangian functions.

	Undamped, No Noise	Undamped, Noise	Damped, No Noise	Damped, Noise
$F(q)$	$\frac{1.3\,{\rm N}}{\omega_{0}}\,q$	$\frac{0.13\,{\rm N}}{\omega_{0}}\,$ q $-\frac{0.09\,{\rm N\,s}}{\omega_{0}^{2}}\frac{dq}{dt}\,$	$\frac{0.9\,{\rm N}}{\omega_{0}}\,q+\frac{0.045\,{\rm N}}{\omega_{0}^{2}\,{\rm m}}q^{2}+\frac{0.09\,{\rm N}}{\omega_{0}^{2}\,{\rm m^{3}}}q^{4}$	$\frac{-7.15\,{\rm N\,s}}{\omega_{0}^{2}}\,\frac{dq}{dt}\,-\frac{5.23\,{\rm N}}{\omega_{0}^{2}\,{\rm m}}q^{2}+\frac{0.07\,{\rm N}}{\omega_{0}^{2}\,{\rm m^{3}}}q^{4}$
$A^{\prime}$	0.15	0.62	-0.25	0.03
$C$	-0.035 J	0 J	0 J	-0.05 J

Table 2: LNN

F(q)

functions for

L^{\prime}=A^{\prime}\left(L+\frac{dF}{dt}\right)+C

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

Abrari Noor Hasmi, Haralampos Hatzikirou, Hadi Susanto

Department of Mathematics, Khalifa University of Science and Technology, PO Box 127788, Abu Dhabi, United Arab Emirates · Public Health and Epidemeology Department, Khalifa University of Science and Technology, PO Box 127788, Abu Dhabi, United Arab Emirates.

Symplectic RNNs Hamiltonian Neural Networks Hamiltonian Dynamics Reservoir Computing

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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.

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