Latent Dynamics via Lagrangian Mechanics

• Latent dynamics via Lagrangian mechanics is a framework that models physical systems in a reduced-order space while preserving variational and geometric structures.
• The approach integrates neural networks with discrete variational principles to learn system evolution from limited positional data, ensuring conservation laws and symplecticity.
• Empirical validations demonstrate that these models achieve stable control and precise prediction in high-dimensional, dissipative environments, outperforming traditional methods.

Latent dynamics via Lagrangian mechanics refers to a class of methodologies that construct, identify, or control the underlying dynamical laws of physical systems within a latent or reduced-order space, where the evolution adheres to the variational and geometric structure dictated by Lagrangian or extended Lagrangian principles. These approaches leverage neural networks as parameterizations while preserving symmetries, conservation laws, and dissipation, and have demonstrated efficacy even when observations are limited to partial, high-dimensional, or positional measurements. Recent advances formalize these methods through discrete variational calculus, latent submanifold geometries, and even doubling of coordinates to restore reversibility in dissipative evolution.

1. Discrete Variational Principles in the Latent Space

The foundation of latent Lagrangian dynamics is the direct discretization of the action principle at the trajectory level. The discrete Lagrange–d'Alembert principle extends Hamilton's principle to non-conservative systems, encoding both conservative and external forcing components in the system’s update equations. For trajectory segments $\{q_n\}_{n=0}^N$ sampled at step $h$, this principle gives rise to the forced discrete Euler–Lagrange (DEL) equations, where all variational terms are faithfully approximated over position pairs using, for instance, the midpoint rule: $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$

$$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$

Resulting DEL residuals can be minimized in a least-squares sense over position triplets, enabling learning directly from positional measurements without differentiating observed data or requiring velocities. In the absence of external forces, such discretizations reproduce variational integrators, guaranteeing symplecticity and favorable energy behavior (Hansen et al., 26 May 2025).

2. Structured Latent Model Architectures

Latent Lagrangian models parameterize the system evolution through the composition of two neural function classes:

• Lagrangian networks: $L_\theta(q, \dot q)$ or its discrete counterpart, which may be either fully unstructured (generic feed-forward networks) or structured via classical mechanics, e.g. $$L_\theta = \frac{1}{2}\dot q^\top M_\theta(q)\dot q - U_\theta(q)$$, with $M_\theta(q)$ (mass matrix/metric) and $U_\theta(q)$ (potential) given by specifically constrained network architectures
• Force networks: $F_\theta(q, \dot q)$, expressed either as unconstrained neural nets or as Rayleigh dissipative terms $F_\theta(q, \dot q) = -K_\theta(q)\dot q$

For high-dimensional observations, an autoencoder $h$0 and decoder $h$1 allow for learning dynamics in a low-dimensional “latent” space, where all variational dynamics are imposed in the compressed variable $h$2 (Hansen et al., 26 May 2025, Friedl et al., 9 Feb 2026). Regularization on the Hessian of $h$3 ensures non-degeneracy of the Lagrangian.

3. Learning and Identifiability from Position-Only Data

The variational structure allows identification of the full dynamic evolution (and its decomposition into conservative and dissipative parts) using only sequences of positions. The DEL residual, formed entirely from positional triplets, is minimized as the physics-based loss: $h$4 across all training trajectories. When observations are images or high-dimensional measurement vectors, the physics loss is backpropagated through the latent encoder, jointly training the encoder and dynamical network in latent space. No explicit velocity measurements or finite differencing of data is necessary, unlike approaches based on continuous-time neural ODEs or “generalized” Lagrangian NNs (Hansen et al., 26 May 2025).

4. Latent Space Geometry, Projection, and Control

Reduced-order (latent) Lagrangian frameworks, as developed in (Friedl et al., 9 Feb 2026), formalize the latent space $h$5 as a geometric manifold of dimension $h$6, equipped with a latent metric $h$7, learned by a symmetric positive-definite (SPD) neural net. Encoders $h$8 and decoders $h$9 define a submersion-embedding pair, with the composition $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$0 acting as a projection. The reduced Euler–Lagrange equations in $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$1 determine the latent evolution: $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$2 where $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$3 is the Christoffel/Coriolis structure from $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$4.

Closed-loop control policies are defined in latent space (e.g., latent PD+ tracking), then lifted to the original space via the decoder and tangent pullbacks, guaranteeing that actuation remains within the embedded submanifold and that the resulting closed-loop system inherits local exponential input-to-state stability: $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$5 where $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$6 is the error, $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$7 and $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$8 represent modeling and alignment errors, respectively (Friedl et al., 9 Feb 2026).

5. Physics-Preserving Dissipative Latent Dynamics

Beyond standard Lagrangian structure, latent dynamics models have been extended to incorporate irreversible dissipation through the Morse–Feshbach doubling construction (Sundararaghavan et al., 2024). Here, for any dissipative ODE, the dynamics are encoded as the observable subsystem of a higher-dimensional conservative (Hamiltonian) flow:

• Introduce latent “mirror” variables $$L_\Delta(q_n, q_{n+1}) = h\,L\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$9, doubling the state space: $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$0.
• Define a coupled Lagrangian $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$1 so that dissipation in $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$2 is compensated by anti-dissipation in $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$3: $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$4
• The corresponding “dissipative Lagrangian” $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$5 suffices to recover the equations of motion for $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$6. Neural nets learn $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$7 from data, and the system evolves reversibly in the doubled space; dissipative evolution in $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$8 is a projection of conservative evolution in $$F_\Delta^\pm(q_n, q_{n+1}) = \frac{h}{2} F\left(\frac{q_n + q_{n+1}}{2}, \frac{q_{n+1} - q_n}{h}\right)$$9. This construction guarantees strong extrapolation and even exact invertibility in the presence of dissipation (e.g., Fickian diffusion) (Sundararaghavan et al., 2024). The mirror variables are not observed but provide a rigorous latent-manifold interpretation of energy transfer in dissipative systems.

6. Empirical Validation and Performance

Experiments across these frameworks demonstrate superior performance compared to baseline neural ODE and Lagrangian NN approaches, particularly in regimes with strong dissipation, limited observability, or high-dimensional measurement:

• DFLNN (Hansen et al., 26 May 2025) outperforms NODE and GLNN on stiff dissipative systems (double pendulum, charged particle in $L_\theta(q, \dot q)$0-field), with stable conservative rollouts when external forces are disabled. On image-based pendulum tracking, DFLNN achieves Dice scores ($L_\theta(q, \dot q)$1) an order of magnitude below NODE and GLNN. For real human motion capture, DFLNN maintains plausible extrapolations beyond the training horizon.
• Reduced-order latent Lagrangian networks (Friedl et al., 9 Feb 2026) stably regulate and track simulated 15-DoF pendula and 351-DoF underactuated RGB-D “plush puppet” tasks. RO-LNN-based controllers ensure near-zero steady-state errors and outperform classical PD and intrusive model-order–reduction controllers.
• Mirror-lifted Lagrangian NNs (Sundararaghavan et al., 2024) show strong long-term extrapolation and precise inversion for dissipative phenomena. Diffusion fields are time-reversible within numerical error when evolved within the augmented conservative framework.

7. Limitations and Open Problems

Current latent Lagrangian frameworks face several limitations:

• Regularization cost may be nontrivial for enforcing Lagrangian invertibility on large datasets (Hansen et al., 26 May 2025).
• Architectural choices (depth, structure of Lagrangian and force networks, expressivity in SPD nets) remain largely unexplored; most empirical studies rely on relatively shallow architectures.
• Extensions to multi-step variational discretizations, symmetry discovery, and high-dimensional PDEs via latent Lagrangian models are open research frontiers (Hansen et al., 26 May 2025).
• Formally introducing and leveraging the latent “mirror” variables in dissipative Lagrangian NNs is an unresolved direction; it may enable probabilistic trajectory backward inference (Sundararaghavan et al., 2024).
• Scaling up latent dimensions while maintaining geometric structure and stability underactuation persists as a technical and theoretical challenge (Friedl et al., 9 Feb 2026).
• For control, the separation of dynamic-model and projection-alignment errors necessitates further investigation, especially for complex, high-dimensional tasks.

A plausible implication is that latent Lagrangian mechanics—especially when generalized to nonlinear, underactuated, or partially observed systems—will catalyze advances in physically interpretable AI for robotics, computational physics, and data-driven engineering sciences, though efficiency, scalability, and integration with generative learning remain significant research challenges.

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References:

• "Learning mechanical systems from real-world data using discrete forced Lagrangian dynamics" (Hansen et al., 26 May 2025).
• "Reduced-order Control and Geometric Structure of Learned Lagrangian Latent Dynamics" (Friedl et al., 9 Feb 2026).
• "Lagrangian Neural Networks for Reversible Dissipative Evolution" (Sundararaghavan et al., 2024).