Dissipative Lagrangian Neural Networks for the Interpretable Modelling of Passive Mechanical Systems

Rob Seabourne; Alice Cicirello

cs.LGOct 11, 2026

Dissipative Lagrangian Neural Networks for the Interpretable Modelling of Passive Mechanical Systems

Authors: Rob Seabourne, Alice Cicirello

Organizations: Department of Engineering, University of Cambridge Trumpington Street, Cambridge CB2 1PZ, UK

Abstract

In this paper, we propose the Dissipative Lagrangian Neural Network (dLNN), a continuous-time system identification technique that separately learns the conservative and dissipative dynamics of a mechanical system directly from data. A key contribution is a novel parametrisation of dissipative forces that (i) is structurally constrained to be dissipative, enforcing passivity of the learned model, (ii) cannot absorb conservative mechanics, and (iii) retains sufficient expressivity to represent a broad family of state-dependent damping and regularised friction forces. The model is validated on prediction tasks on two real-world mechanical systems: trajectory prediction of a chaotic double pendulum and inverse-dynamics modelling of a six-degree-of-freedom industrial robot. Across these tasks, the dLNN demonstrates improved predictive performance over existing Lagrangian architectures while remaining competitive with a black-box neural model. The imposed physical structure also improves generalisation to unseen states, with the benefits becoming particularly pronounced when training data is limited. Finally, we introduce a symbolic distillation procedure that extracts closed-form expressions from the learned physical components. Applied to the industrial robot, this produces interpretable symbolic models that retain high predictive accuracy while reducing the computational overhead associated with evaluating neural Lagrangian models.

Figures & tables

Figure 1: Architecture of the Dissipative Lagrangian Neural Network (dLNN). The conservative network learns the positive-definite mass matrix $\mathbf{M}_{\theta}(\mathbf{q})$ and potential energy $V_{\theta}(\mathbf{q})$ , while the dissipative network learns the positive-definite dissipation matrix $\mathbf{D}_{\theta}(\mathbf{q},\dot{\mathbf{q}})$ . These components define a common physical model that can be evaluated for either forward or inverse dynamics.

System	DoF	Training task	Evaluation task	Training data	Test data
Simulated double pendulum	2	$(\mathbf{q}_{t},\dot{\mathbf{q}}_{t})\rightarrow\ddot{\mathbf{q}}_{t}$	$(\mathbf{q}_{0},\dot{\mathbf{q}}_{0})\rightarrow(\mathbf{q}_{1:T},\dot{\mathbf{q}}_{1:T})$	$90\,\mathrm{Hz}\times 11.7\,\mathrm{s}\times 445$	$90\,\mathrm{Hz}\times 5\,\mathrm{s}\times 350$
Experimental double pendulum	2	$(\mathbf{q}_{t},\dot{\mathbf{q}}_{t})\rightarrow(\mathbf{q}_{t+1},\dot{\mathbf{q}}_{t+1})$	$(\mathbf{q}_{0},\dot{\mathbf{q}}_{0})\rightarrow(\mathbf{q}_{1:T},\dot{\mathbf{q}}_{1:T})$	$90\,\mathrm{Hz}\times 11.7\,\mathrm{s}\times 5$	$90\,\mathrm{Hz}\times 5\,\mathrm{s}\times 350$
Industrial robot arm	6	$(\mathbf{q}_{t},\dot{\mathbf{q}}_{t},\ddot{\mathbf{q}}_{t})\rightarrow\boldsymbol{\tau}_{t}$	$(\mathbf{q}_{t},\dot{\mathbf{q}}_{t},\ddot{\mathbf{q}}_{t})\rightarrow\boldsymbol{\tau}_{t}$	$10\,\mathrm{Hz}\times 4020\,\mathrm{s}$	$10\,\mathrm{Hz}\times 360\,\mathrm{s}$

Table 1: Summary of the three benchmark datasets and associated training and evaluation tasks. Double-pendulum dataset sizes are reported as sampling rate

\times

trajectory duration

\times

number of trajectories; robot dataset sizes are reported as sampling rate

\times

total duration.

Panel of Figure 2 — Figure 2: Double-pendulum benchmark system. ( Left ) Experimental double-pendulum apparatus used to collect the real-world trajectory data. ( Right ) Corresponding mechanical model, showing the absolute angles $\alpha$ and $\beta$ , rod lengths $a$ and $b$ , rod masses $m_{a}$ and $m_{b}$ , and intermediate point mass $m_{p}$ .

Parameter	Double pendulum	Robot
Primary network size (width $\times$ depth)	$128\times 4$	$256\times 4$
gLNN/dLNN force network size (width $\times$ depth)	$64\times 3$	$64\times 3$
Learning rates, $\mathrm{LR}1\rightarrow\mathrm{LR}2$	$10^{-2}\rightarrow 10^{-4}$	$10^{-2}\rightarrow 10^{-3}$
Training time	$10\,\mathrm{min}$	$2\,\mathrm{min}$
Mass-matrix offset, $\epsilon$	$10^{-2}$	$10^{-1}$
Regularisation	None	Dropout $p=0.4$

Table 2: Neural-network architectures and training parameters used for the double-pendulum and industrial robot benchmarks.

Panel of Figure 3 — Figure 3: Trajectory-prediction performance on the simulated double-pendulum benchmark across increasing levels of friction. ( Left ) Mean divergence time $t_{\mathrm{div}}$ . ( Right ) Mean energy mean-squared error (MSE). Friction is expressed as a percentage of the nominal experimental friction parameters.

Panel of Figure 4 — Figure 4: Trajectory-prediction performance on the reduced experimental double-pendulum benchmark. ( Left ) Mean divergence time $t_{\mathrm{div}}$ . ( Right ) Mean energy mean-squared error (MSE). Error bars show performance range over three random initialisations.

Panel of Figure 5 — Figure 5: Example trajectory predictions from the experimental double-pendulum benchmark, illustrating the predictive performance of the dLNN and the non-physical instability observed in a small subset of nODE predictions. ( Top ) Measured and dLNN-predicted lower-arm angle $\beta$ for trajectory 166, with initial state $(\alpha_{0},\dot{\alpha}_{0},\beta_{0},\dot{\beta}_{0})=(43.7^{\circ},-446.8^{\circ}\,\mathrm{s}^{-1},-43.1^{\circ},-196.2^{\circ}\,\mathrm{s}^{-1})$ . ( Bottom ) Measured and nODE-predicted lower-arm angle $\beta$ for trajectory 99, with initial state $(-60.4^{\circ},814.6^{\circ}\,\mathrm{s}^{-1},34.4^{\circ},-202.4^{\circ}\,\mathrm{s}^{-1})$ . Dotted vertical lines indicate the divergence time $t_{\mathrm{div}}$ .

Model	Overall $R^{2}$ (%)
dLNN	88.8
MLP	88.1
gLNN	74.4
Linear	62.0
LNN	57.5

Table 3: Inverse-dynamics prediction performance on the industrial robot benchmark. The reported

R^{2}

is averaged across the six robot joints.

Figure 6: Measured and dLNN-predicted joint torques $\tau_{1}$ – $\tau_{6}$ across the six sequential test trajectories of the industrial robot benchmark.

Component	Candidate functions	Coordinate indices
Mass-matrix entry $M_{ij}$	$1,\;q_{k},\;q_{k}q_{\ell}$	$k,\ell\in\{\min(i,j)+1,\ldots,6\}$
Potential energy $V$	$1,\;q_{k},\;q_{k}q_{\ell},\;q_{k}q_{\ell}q_{m}$	$k,\ell,m\in\{2,\ldots,6\}$
Dissipative force $Q_{i}$	$\operatorname{sgn}(\dot{q}_{i}),\;\dot{q}_{i}$	–

Table 4: Symbolic libraries used for distillation of the six-joint robot model. Polynomial libraries include all distinct terms, including powers and cross terms.

Model	$R^{2}$ (%)	CPU time, $B=1$ (ms)	CPU time, $B=64$ (ms)
dLNN	88.8	0.494	3.162
Hybrid ( $M,V$ symbolic; $Q$ neural)	88.4	0.155	0.491
Fully Symbolic	81.5	0.101	0.280
MLP	88.1	0.191	0.806

Table 5: Inverse dynamics prediction accuracy and CPU evaluation time for the dLNN, distilled models and black-box MLP on the industrial robot benchmark. Evaluation times are reported for batch sizes

B=1

and

B=64

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Quantity	Symbol	Value	Unit
Upper rod length	$a$	$0.185$	$\mathrm{m}$
Lower rod length	$b$	$0.172$	$\mathrm{m}$
Upper rod mass	$m_{a}$	$0.044$	$\mathrm{kg}$
Lower rod mass	$m_{b}$	$0.019$	$\mathrm{kg}$
Intermediate point mass	$m_{p}$	$0.022$	$\mathrm{kg}$
Gravitational acceleration	$g$	$9.81$	$\mathrm{m\,s^{-2}}$

Appendix

Table 6: Physical and friction parameters of the double-pendulum model, selected to reproduce the experimental apparatus.

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Lagrangian and Hamiltonian Neural Networks With a Dissipative System

V. Rayamajhi, J. Singal

University of Richmond

Lagrangian Neural Networks Hamiltonian Neural Networks Hamiltonian Dynamics Dynamical Systems

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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.

Abstract

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