cs.CEOct 3, 2026

Latent-Lagrangian Neural Networks for Reduced Order Modeling of Non-autonomous Nonlinear Dynamical Systems

Authors: Anand Kumar Agrawal, Anders Thorin

Organizations: Université Paris-Saclay, CEA List, Palaiseau, France

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Abstract

This work proposes a latent Lagrangian-based framework for reduced-order modelling of forced nonlinear dynamical systems. In contrast with conventional Lagrangian or Hamiltonian neural networks, our approach learns a set of latent coordinates sufficient to capture the dynamics conjointly with two neural networks for the latent kinetic and latent potential energies, and leverages force supervision to eliminate the need for an ODE solver during training. Consistency of physical laws in the latent space is ensured through the principle of virtual work. Results show that the model effectively learns the subtle dynamics induced by the system's nonlinearity and non-convex potential energy, and generalizes well to unseen forces and initial conditions. These observations confirm the physical relevance of the proposed approach, and its interest for model reduction.

Figures & tables

FeatureNeural netNeural ODEHNNDeLaNLNNLEBNNLOpInf-SpM [ 22 ]Latent-LNN (ours)
Can model dynamical systems✓✓✓✓✓✗✓✓
Learns differential equations✗✓✓✓✓✗✓✓
Learns exact conservation laws✗✗✓✓✓✓ ∗✓✓
Learns from arbitrary coords.✓✓✗✓✓✓✓✓
Learns arbitrary Lagrangians✗✗✗✗✓no TT✓✓
Separate control over architectures of TT , VV✗✗✗✗✗no TT✗✓
Table 1: Comparison of learning-based physics models. TT denotes kinetic energy, VV potential energy.
Figure 1: Graphical representation of Latent-LNN
 fz←HTz¨+(αHT+βHV)z˙+∇zV+∂2T∂z˙∂zz˙−∂T∂zf_{z}\leftarrow H_{T}\ddot{z}+(\alpha H_{T}+\beta H_{V})\dot{z}+\nabla_{z}V+\frac{\partial^{2}T}{\partial\dot{z}\partial z}\dot{z}-\frac{\partial T}{\partial z} 
Algorithm 1 Latent Energy-Based Neural Network Training with Force Supervision
Figure 2: 2-DOF system used to assess and compare the proposed architecture. Springs are linear, but the geometry is nonlinear (large displacements). From [ 20 ] .
Figure 3: Schematic of the 5-DOF nonlinear system. The last mass is subject to an external load. Springs have a nonlinear behaviour (cubic nonlinearity).
Hyperparameter2-DOF5-DOF
Initialization (PE)Xavier (Uniform) [ 10 ]Xavier (Uniform)
Encoder: hidden depth and width1 and 641 and 64
Decoder: hidden depth and width1 and 641 and 64
PE, hidden depth and width3 and 1283 and 128
KE, hidden depth and width1 and 1281 and 64
Learning rate(initial)1E-041E-04
Table 2: Values of Hyperparameters
Figure 4: 2-DOF system: displacements predicted through learned system vs the true displacements. The relative errors eqie_{q_{i}} computed as per Algorithm 2 are 0.012 for q1q_{1} and 0.016 for q2q_{2} .
Figure 5: 2-DOF system: velocities predicted through learned system vs the true velocities. The relative errors eq˙ie_{\dot{q}_{i}} computed as per Algorithm 2 is 0.019 for q˙1\dot{q}_{1} and 0.020 for q˙2\dot{q}_{2} .
Figure 6: Comparison of energy topology. The learned energies are converted in terms of original coordinates q1,q2q_{1},q_{2} and q˙1,q˙2\dot{q}_{1},\dot{q}_{2} to make a direct comparison of the contour lines relevant.
Figure 7: Variance of Relative Errors in Displacement and Velocity Responses (2-DOF System)
Figure 8: 2-DOF system: displacements predicted through system trained with noisy data vs the true displacements. The relative errors eqie_{q_{i}} computed as per Algorithm 2 is 0.051 for q1q_{1} and 0.062 for q2q_{2} .
Figure 9: 2-DOF system: velocities predicted through system trained with noisy data vs the true velocities. The relative errors eq˙ie_{\dot{q}_{i}} computed as per Algorithm 2 are 0.097 for q˙1\dot{q}_{1} and 0.083 for q˙2\dot{q}_{2} .
Figure 10: 5-DOF system: displacements predicted through learned system vs true displacements. The relative error eqie_{q_{i}} computed as per Algorithm 2 are respectively 0.12, 0.10, 0.079, 0.043 and 0.023, for ii from 1 to 5.
Figure 11: 5-DOF system: velocities predicted through learned system vs true velocities. The relative errors eq˙ie_{\dot{q}_{i}} computed as per Algorithm 2 are respectively 0.21, 0.16, 0.094, 0.033 and 0.018, for ii from 1 to 5.
Figure 12: 5-DOF system: Kinetic Energy Comparison
Figure 13: 5-DOF system: Potential Energy Comparison
Figure 14: Variance of Relative Errors in Displacement and Velocity Responses (5-DOF System)
Figure 15: 5-DOF system: displacements predicted through system trained with noisy data vs the true displacements. The relative errors eqie_{q_{i}} computed as per Algorithm 2 are 0.22, 0.12, 0.12, 0.13 and 0.096, for ii from 1 to 5.
Figure 16: 5-DOF system: velocities predicted through system trained with noisy data vs the true velocities. The relative errors eq˙ie_{\dot{q}_{i}} computed as per Algorithm 2 are 0.73, 0.18, 0.34, 0.16, 0.035, for ii for 1 to 5.
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