A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics

TL;DR

This work introduces a geometry-informed framework to learn reduced-order Lagrangian dynamics for high-dimensional rigid and deformable systems by fusing nonlinear model order reduction with a Riemannian, structure-preserving Lagrangian network. It jointly learns a nonlinear latent embedding of the configuration space and reduced dynamics, with mass-inertia matrices modeled on the SPD manifold using Riemannian optimization, and a constrained autoencoder enforcing projection properties. The reduced-order model preserves energy and yields accurate, long-horizon predictions across diverse systems (16 DOF pendulum, 192 DOF rope, 600 DOF cloth), outperforming full-order LNNs and linear MOR baselines, especially in data-scarce regimes. The approach offers a scalable path to physically-consistent control and simulation of complex deformable objects by leveraging intrinsic geometry and energy preservation in learned reduced dynamics.

Abstract

By incorporating physical consistency as inductive bias, deep neural networks display increased generalization capabilities and data efficiency in learning nonlinear dynamic models. However, the complexity of these models generally increases with the system dimensionality, requiring larger datasets, more complex deep networks, and significant computational effort. We propose a novel geometric network architecture to learn physically-consistent reduced-order dynamic parameters that accurately describe the original high-dimensional system behavior. This is achieved by building on recent advances in model-order reduction and by adopting a Riemannian perspective to jointly learn a non-linear structure-preserving latent space and the associated low-dimensional dynamics. Our approach enables accurate long-term predictions of the high-dimensional dynamics of rigid and deformable systems with increased data efficiency by inferring interpretable and physically-plausible reduced Lagrangian models.

Figures (16)

• Figure 1: Flowchart of the forward dynamics of the proposed reduced-order lnn. The reduction mappings and embeddings of the Lagrangian rom are depicted in blue and parametrized via a constrained ae with biorthogonal layers (right). The rom dynamics are learned via a latent geometric lnn depicted in orange. The mass-inertia matrix is parametrized via a spd network (left).
• Figure 2: $2$-dof pendulum: Left: Median acceleration prediction error for different lnn and training set sizes $\mathcal{D}_\text{train}$ over $10$ test trajectories. Shaded regions represent first and third quartiles. Middle, right: Relative error of numerically-integrated position and velocity predictions with respect to the ground-truth trajectory over a prediction horizon $H_{\text{test}}=2000$.
• Figure 3: $16$-dof pendulum: Comparisons of the position and velocity predictions from the rolnn trained on acceleration () and via multi-step integration () with the ground truth (). The corresponding ae reconstructions () and () are depicted for completeness. The model is updated with a new initial condition $(\bm{q}_0, \dot{\bm{q}}_0)^{\mathsf{T}}$ every $0.025s$ ($H_{\text{test}}=25$).
• Figure 4: Median and quartiles of the errors of jointly-trained (on-biorthogonal-manifold () vs overparametrized ()) and sequentially-trained () models. Note that the latter have their own $y$-axis in the left three plots, which is at least an order of magnitude higher.
• Figure 5: Predicted rope position () and ground truth () at selected timesteps for a prediction horizon $H_{\text{test}}=25$. The grey circle depicts the circular trajectory of the end of the rope.