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Latent-Lagrangian Neural Networks for Reduced Order Modeling of Non-autonomous Nonlinear Dynamical Systems
Anand Kumar Agrawal, Anders Thorin
TL;DR
Modeling nonlinear dynamics can require difficult characterization, and static latent-energy approaches do not capture time-dependent dynamics. This paper learns a reduced-dimensional latent Lagrangian using energy networks, force supervision, and virtual-work consistency; it validates the framework on two nonlinear dynamical systems by comparing predicted responses and learned energy topologies with true vector fields.
Problem
Accurately characterizing constitutive response and energy dissipation can be difficult, and latent-energy models that omit kinetic energy do not address time-dependent dynamics.
Source: §1 · pp. 1–2Method
An autoencoder learns latent coordinates alongside separate kinetic- and potential-energy networks, trained with force supervision and virtual-work consistency.
Source: §1 · p. 3Results
The framework was validated on 2-DOF and 5-DOF nonlinear systems by comparing predicted responses and learned energy topologies with those from the true vector fields.
Source: §4 · p. 10Takeaways & Limitations
The authors present the framework as a physically grounded approach to reduced-order modeling of high-dimensional nonlinear dynamical systems.
Source: §5 · p. 19Precise long-term trajectory prediction in chaotic systems is limited by sensitive dependence on initial conditions.
Source: §4.2 · p. 19
Abstract
from arXiv · showhide
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.
1. Introduction
Accurately modeling nonlinear dynamics can require difficult characterization, while data-driven models may lack physical structure and generalize poorly. The paper introduces a latent-space Lagrangian model that combines separate energy networks with force supervision and virtual-work consistency.
1. Introduction
- First-principles models require difficult constitutive and dissipation characterization, while conventional neural mappings can miss temporal dependencies and generalize poorly.The paper motivates combining data-driven learning from input–output data with consistency to physical laws.Source: §1 · pp. 1–2
- Prior approaches leave gaps for this setting: LNNs were demonstrated on autonomous systems, while LEBNNs address statics and omit kinetic energy.LEBNN's separate force and displacement autoencoders also break the energy–force relationship.Source: §1 · p. 2
- A prior Lagrangian ROM relies on a prescribed linear POD basis and finite-difference derivatives, with limited forcing and energy-network architectural control.The cited approach projects onto a fixed POD subspace, identifies a linear Lagrangian ROM, then adds structure-preserving terms for missing nonlinear operators.Source: §1 · p. 2
- L-LNN learns latent coordinates and separate potential- and kinetic-energy networks, allowing modular architecture design in a reduced-dimensional Lagrangian model.An autoencoder maps generalized coordinates to latent coordinates, from which latent velocities and accelerations are obtained through derivative relationships.Source: §1 · pp. 2–4
- Virtual-work consistency preserves the variational relationship between latent-space and original-space forces, unlike LEBNN's separately learned latent forces.The paper describes this consistency as preserving the energy–force duality under differentiation.Source: §1 · pp. 2, 4
- Force supervision avoids an ODE solver during training, while the composite loss combines force-prediction and autoencoder-reconstruction errors.Position-dependent mass and stiffness matrices are represented by Hessians of the latent kinetic- and potential-energy networks, and damping uses a position-dependent Rayleigh form.Source: §1 · pp. 3–4
2. Proposed method: Latent-Lagrangian Neural Networks
Latent-LNN jointly learns a reduced coordinate representation and latent kinetic and potential energies to model system dynamics. Its training combines force matching with reconstruction and uses virtual-work consistency to relate latent forces to the original coordinates.
2. Proposed method: Latent-Lagrangian Neural Networks
- The method trains an autoencoder and kinetic- and potential-energy networks together, with a loss incorporating force discrepancy and autoencoder reconstruction.The section presents a pseudocode algorithm and a forward-pass schematic for the composite network training.Source: §2 · p. 4, §2.6 · pp. 6–7
2.1. Latent-Space Kinematics via Autoencoder
- The autoencoder maps generalized coordinates to latent coordinates and reconstructs them; chain-rule derivatives yield latent velocities and accelerations with inertial and geometric terms.The encoder Jacobian depends on the generalized coordinates over time.Source: §2.1 · pp. 4–5
2.2. Latent-Space Lagrangian Dynamics via Potential and Kinetic Energy Networks
- Separate networks represent latent kinetic and potential energy, and the Euler–Lagrange equation derives the latent dynamics.The kinetic energy may use a structured, coordinate-dependent mass matrix with a quadratic velocity form or an unconstrained network.Source: §2.2 · p. 5
2.3. Incorporating Damping and Final Force Expression
- Damping uses a position-dependent combination of kinetic- and potential-energy Hessians, with learnable scalar weights α and β.The approach is similar to Rayleigh damping.Source: §2.3 · p. 6
2.4. Force relations via virtual work consistency
- Virtual-work consistency projects the latent-space force to the original coordinates using the transpose of the encoder Jacobian.The projection equates virtual work under the corresponding latent and original-space displacements.Source: §2.4 · p. 6
2.5. Formulation of the Loss Function
- The networks and damping parameters are trained jointly using a loss that combines reconstruction and force discrepancies, weighted by λ.The loss uses the decoder’s reconstructed coordinates and the predicted force in the original coordinates.Source: §2.5 · p. 6
2.6. Graphical and algorithmic summaries
- The training algorithm computes latent-space force and updates the autoencoder, energy networks, and damping parameters.The algorithm’s stated inputs include generalized coordinates, velocities, accelerations, and true forces.Source: §2.6 · p. 7
3. Validation methodology
Validation tests the learned model on two nonlinear systems and compares decoded trajectories with the true dynamics, as well as learned and true energy topologies. Training data comes from simulated forced responses, with the regularization weight affecting 5-DOF training stability.
3. Validation methodology
- Validation compares decoded learned-system responses with the true system and compares learned and true kinetic- and potential-energy topologies.The workflow solves the learned dynamics and the ground-truth system for comparison.Source: §3 · p. 8
3.1. Nonlinear systems used for validation
- The 2-DOF case combines large deflections and nonconvex potential energy, producing nonlinear dynamics with stable and unstable equilibria.Its low dimensionality also permits visual comparison of learned and true energy topologies.Source: §3.1 · pp. 8–9, Figure 2 · p. 9
- The 5-DOF Duffing-like system tests scalability and robustness for model reduction using a latent space smaller than its five degrees of freedom.Its springs have cubic nonlinearity, and the last mass receives an external load.Source: §3.1 · p. 9, Figure 3 · p. 9
3.2. Data-generation and training
- Training data comprises simulated displacement, velocity, acceleration, and force trajectories from 50 2-DOF and 70 5-DOF force-frequency combinations.Both datasets use 10-second simulations with a fourth-order Runge–Kutta method at 0.01-second steps; 80% is used for training.Source: §3.2 · pp. 9–10
- For the 2-DOF system, prediction accuracy and convergence showed no significant sensitivity to λ = 1, 0.05, or 0.005.The reported results use λ = 1, selected arbitrarily.Source: §3.2 · p. 10
- For the 5-DOF system, λ = 0.005 produced the most stable and reliable performance across tested values, affecting convergence and robustness to model uncertainty.Robustness was assessed through repeated training with different random seeds.Source: §3.2 · p. 10
4. Performance Validation: Results and Discussion
The validation tests response prediction, learned energy structure, and robustness on 2-DOF and 5-DOF nonlinear systems. The results show accurate trajectories, including with a reduced latent dimension, alongside energy-shape deviations in the reduced 5-DOF case.
4.1. 2-DOF Nonlinear dynamical system with nonconvex potential energy
- For the 2-DOF system, displacement relative errors were 0.012 and 0.016, while velocity errors were 0.019 and 0.020; errors at 0.01 s and 0.02 s matched to two decimal places.Source: Figure 4 · p. 11, Figure 5 · p. 11, §4.1 · p. 12
- The 2-DOF learned potential contours reproduce multiple equilibrium wells and kinetic contours reflect isotropic inertia, although learned potential-energy values differ slightly from the true values.The authors attribute the value discrepancy to training on energy gradients rather than energy values; the equations of motion depend on those gradients.Source: §4.1 · pp. 12–13, Figure 6 · p. 12
4. Performance Validation: Results and Discussion
- Across three random seeds, relative-error variability was of order 10^-5 for 2-DOF and 10^-4 for 5-DOF; with 5%-of-signal-standard-deviation Gaussian noise, response agreement remained satisfactory despite increased errors.For noisy 5-DOF training, the authors added excitation at the second mass because responses from the first two masses were comparable to the noise level in the original dataset.Source: §4.1 · pp. 13–14, §4.2 · pp. 17–18
4.2. 5-DOF Nonlinear dynamical system with cubic nonlinearity
- 12.1%, 10.5%, 7.9%, 4.3%, and 2.3% are the respective 5-DOF displacement errors; velocity errors are 21% and 16.4% for q̇1 and q̇2, and below 10% for the remaining components.The comparison uses nonzero, nonsymmetric initial displacements and non-harmonic forcing distinct from training conditions.Source: §4.2 · pp. 14, 16, Figure 10 · p. 15, Figure 11 · p. 15
- In the 5-DOF case, learned kinetic energy preserves the true energy’s scale, general quadratic form, and convexity, while dimensional reduction distorts the contours, especially for potential energy.The authors suggest that the kinetic network’s quadratic bias in latent velocity may explain its smaller deviations.Source: §4.2 · pp. 16–17
- The authors caution that precise long-term trajectory prediction in chaotic systems is limited by sensitive dependence on initial conditions; the stated objective there is reproducing long-term statistical and dynamical properties.Source: §4.2 · p. 19
5. Conclusion
The work develops a physically grounded latent-space Lagrangian framework for reduced-order modeling of high-dimensional nonlinear systems and validates it on two nonlinear dynamical systems.
5. Conclusion
- The framework learns a lower-dimensional latent Lagrangian using an autoencoder and separate neural networks for potential and kinetic energy.The networks are trained jointly with a common loss derived from the Euler–Lagrange equations and the principle of virtual work.Source: §5 · p. 19
- The method predicted the behavior of two nonlinear dynamical systems while training on fewer than 60 ten-second time series for each test case.Source: §5 · p. 19
- The authors identify modular energy-network architectures and virtual-work consistency between true and latent force fields as features associated with improved control and interpretability.Source: §5 · p. 19
- The encouraging results on prototypical problems motivate future application to more challenging nonlinear mechanical systems.Source: §5 · p. 20
A.1. 2-DOF nonlinear spring-mass system
The 2-DOF nonlinear spring–mass system is specified by two generalized coordinates, initial conditions, spring and damping parameters, and harmonic external forces.
A.1. 2-DOF nonlinear spring-mass system
- The system’s motion is described using generalized coordinates q1(t) and q2(t).Source: §A.1 · p. 21
- The section presents expressions for the system’s potential and kinetic energies.Source: §A.1 · p. 21
- The model parameters include particle mass, directional spring stiffnesses, free spring lengths, and viscous damping coefficients.Source: §A.1 · p. 21
- External forcing functions are harmonic, with separate amplitudes and frequencies for the two directions.The initial displacements and velocities are specified as initial conditions.Source: §A.1 · p. 21
A.2. 5-DOF nonlinear spring–mass system
The 5-DOF nonlinear spring–mass system is described by five generalized coordinates, with cubic restoring-force nonlinearity and specified initial conditions.
A.2. 5-DOF nonlinear spring–mass system
- The 5-DOF system is represented by generalized coordinates q1(t), q2(t), q3(t), q4(t), and q5(t).Source: §A.2 · p. 22
- Expressions for the system’s kinetic and potential energies are presented.Source: §A.2 · p. 22
- Its nonlinear stiffness contribution is governed by α∗, the nonlinear stiffness coefficient controlling the cubic restoring force.Source: §A.2 · p. 22
- The system description includes an external force vector and specified initial conditions.Source: §A.2 · pp. 22–23