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A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics

Katharina Friedl, Noémie Jaquier, Jens Lundell, Tamim Asfour, Danica Kragic

arXiv:2410.18868v3cs.LG

TL;DR

High-dimensional neural dynamics can be data-intensive, computationally costly, and physically inconsistent. The paper jointly learns a nonlinear structure-preserving latent space and reduced Lagrangian dynamics with a Riemannian geometric network, producing accurate long-term predictions for rigid and deformable systems. Prediction quality remains limited by the autoencoder’s ability to reconstruct the full-order solution space.

  • Problem

    Learning high-dimensional nonlinear dynamics requires costly data and computation, while black-box models may violate physical laws and high-dimensional equations are expensive to solve.

  • Method

    The approach jointly learns a nonlinear structure-preserving latent representation and low-dimensional Lagrangian dynamics using structure-preserving model-order reduction and a latent geometric LNN.

  • Results

    The method accurately predicts long-term dynamics of simulated rigid and deformable systems while conserving reduced energy and inferring interpretable reduced-order parameters.

  • Takeaways & Limitations

    Geometry and physics inductive biases support physically plausible reduced Lagrangian models that describe high-dimensional system behavior with increased data efficiency.

  • Takeaways & Limitations

    Prediction quality depends on autoencoder reconstruction, and selecting the latent dimension is non-trivial because the full-order coupling is unknown.

Abstract

from arXiv · show

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.

1 INTRODUCTION

The paper addresses data-hungry, physically inconsistent learning of high-dimensional nonlinear dynamics by combining physics-informed learning with nonlinear model-order reduction. It jointly learns a structure-preserving latent representation and reduced dynamics, yielding physically plausible long-term predictions for rigid and deformable systems.

  • Black-box neural dynamics models require extensive trajectory data and can violate physical laws such as energy conservation.
  • Model-order reduction provides computationally efficient low-dimensional surrogates, while autoencoders can extract descriptive nonlinear representations from data.
  • Existing non-intrusive reduction is limited by linear reducibility and restricted reduced-parameter expressivity.
  • The proposed method jointly learns a nonlinear structure-preserving latent space and associated low-dimensional dynamics using a constrained autoencoder and latent LNN.
  • Across pendulum, rope, and thin-cloth systems, the approach conserves reduced energy, infers interpretable parameters, and produces accurate long-term high-dimensional predictions.

2 BACKGROUND

The background frames Lagrangian dynamics geometrically on a configuration manifold and presents model-order reduction as approximation on a lower-dimensional embedded manifold. It then connects constrained autoencoders and Lagrangian neural networks as the ingredients for learning reduced, physically consistent dynamics.

  • 2.1 LAGRANGIAN DYNAMICS ON THE CONFIGURATION MANIFOLD: A Lagrangian system consists of a Riemannian configuration manifold and a smooth time-independent Lagrangian defined on its tangent bundle.
  • 2.1 LAGRANGIAN DYNAMICS ON THE CONFIGURATION MANIFOLD: The Lagrangian equals kinetic minus potential energy, and its Euler–Lagrange equations define lifted trajectories on the tangent bundle.
  • 2.2 MODEL ORDER REDUCTION: Model-order reduction approximates high-dimensional solution trajectories with a reduced manifold of dimension d much smaller than the full state dimension.
  • 2.2 MODEL ORDER REDUCTION: Reduction uses an embedding and point and tangent reduction maps satisfying projection properties, with original trajectories approximated by embedded reduced trajectories.
  • 2.2 MODEL ORDER REDUCTION: Autoencoders represent nonlinear reduction maps, while constrained autoencoders guarantee the projection properties needed for reduced Lagrangian systems.
  • 2.3 LAGRANGIAN NEURAL NETWORKS: Lagrangian neural networks impose physics-based inductive bias so learned dynamics conserve energy and produce physically plausible trajectories.

3 GEOMETRIC LAGRANGIAN NEURAL NETWORKS

The geometric LNN models the mass-inertia matrix on the SPD manifold and trains its mixed Euclidean–Riemannian parameters with Riemannian optimization. Its forward architecture combines SPD and Euclidean networks with reduced-order mappings and latent dynamics.

  • 3.1 LEARNING POSITIVE-DEFINITE MASS-INERTIA MATRICES ON THE SPD MANIFOLD: The mass-inertia matrix is the Riemannian metric for quadratic-kinetic-energy Lagrangian systems and therefore lies on the SPD manifold.
  • 3.1 LEARNING POSITIVE-DEFINITE MASS-INERTIA MATRICES ON THE SPD MANIFOLD: The proposed SPD network replaces Euclidean Cholesky parameterization with layers formulated directly using the Riemannian geometry of SPD matrices.
  • 3.1 LEARNING POSITIVE-DEFINITE MASS-INERTIA MATRICES ON THE SPD MANIFOLD: The SPD architecture maps Euclidean network outputs into the SPD manifold through a tangent-space exponential map before applying SPD layers and activations.
  • 3.1 LEARNING POSITIVE-DEFINITE MASS-INERTIA MATRICES ON THE SPD MANIFOLD: The geometric LNN models kinetic energy with an SPD-network mass-inertia matrix and potential energy with a Euclidean multilayer perceptron.
  • 3.2 MODEL TRAINING AND PARAMETER OPTIMIZATION: Riemannian optimization updates parameters on their product of Euclidean and SPD manifolds using manifold-aware gradients, exponential maps, and parallel transport.
  • 3.2 MODEL TRAINING AND PARAMETER OPTIMIZATION: The complete reduced-order architecture combines constrained-autoencoder reduction mappings, latent geometric LNN dynamics, and an SPD mass-inertia network.

4 LEARNING REDUCED-ORDER LAGRANGIAN DYNAMICS

The proposed reduced-order LNN jointly learns a nonlinear, structure-preserving latent space and low-dimensional Lagrangian dynamics for high-dimensional physical systems. It combines constrained autoencoding, geometric parameter learning, and multi-step or acceleration-based training to reconstruct and predict full-order behavior.

  • The method jointly learns a latent dynamical system and reduced Lagrangian parameters while preserving the original system's structure.The latent space and reduced dynamics are inferred through structure-preserving model-order reduction and a latent geometric LNN.
  • Lagrangian reduced-order model: The lifted embedding maps reduced configurations and velocities into the full tangent bundle, enabling structure-preserving reduction of Lagrangian trajectories.The reduction operates on TQ because Lagrangian trajectories are lifted curves containing positions and velocities.
  • Lagrangian reduced-order model: The reduced Lagrangian is constructed through the embedding pullback, and its Euler-Lagrange equations define the reduced dynamics.In the intrusive formulation, reduced parameters derive from known full-order dynamics; here, unknown reduced parameters are learned instead.
  • Learning the embedding and point reduction: A constrained autoencoder represents the nonlinear point reduction and embedding, with differentials providing the corresponding tangent reduction and pushforward maps.The architecture uses encoder-decoder networks and constrains layer pairs so the projection properties hold.
  • Learning the embedding and point reduction: Riemannian optimization on biorthogonal matrix manifolds minimizes reconstruction error while maintaining the autoencoder constraints during joint latent-space and dynamics learning.The method treats each weight pair as an element of a biorthogonal manifold and uses smooth invertible activations for the second constraint.
  • Model training: Training combines autoencoder reconstruction, latent LNN prediction, and reconstructed-prediction errors using either acceleration supervision or multi-step Euler integration.Multi-step training avoids the Hessian computations required by acceleration-based loss and directly evaluates multiple predicted steps.

5 EXPERIMENTS

Experiments evaluate geometric LNNs on a 2-DoF pendulum and geometric RO-LNNs on high-dimensional rigid and deformable systems. Across these settings, the approach supports accurate long-term prediction, with multi-step training, joint geometric learning, and suitable latent dimensions improving performance.

  • 5.1 LEARNING LAGRANGIAN DYNAMICS WITH GEOMETRIC LNNS: Geometric LNNs outperform DeLaNs in acceleration and long-term trajectory prediction, especially in low-data regimes, while eight-step multi-step training gives the most accurate predictions.SPD layers do not noticeably improve performance over Euclidean and exponential-map layers.
  • 5.2.1 COUPLED PENDULUM (16 DOFS): The 16-DoF pendulum is reduced to four latent dimensions, and multi-step-trained RO-LNNs consistently outperform models trained only on acceleration.Even the best full-order LNN has errors orders of magnitude higher than the RO-LNN and cannot produce stable velocity and position predictions.
  • 5.2.1 COUPLED PENDULUM (16 DOFS): Jointly training the autoencoder and latent LNN outperforms sequential training, while the biorthogonal-manifold model has lower errors and greater training stability than the overparametrized alternative.Four of six overparametrized joint trainings produced NaN values.
  • 5.2.2 ROPE (192 DOFS): The RO-LNN accurately predicts the 192-dimensional rope dynamics over a 25-step horizon, whereas full-order LNN and L-OpInf training did not yield stable solutions.The rope experiment uses a ten-dimensional latent space and circular actuation at one end while the other end remains fixed.
  • 5.2.2 ROPE (192 DOFS): Latent dimensions initially reduce reconstruction and prediction errors, but errors rise beyond a certain dimension because higher-dimensional latent LNNs become harder to train.Thus, latent dimension trades off representational expressivity against the limitations of LNNs in higher dimensions.
  • 5.2.3 THIN CLOTH (600 DOFS): For a 600-DoF thin cloth, the RO-LNN accurately predicts configurations over both 25-step and 2500-step horizons, including constraints and actuation effects.The model uses a ten-dimensional latent space and is trained on 8000 samples.

6 DISCUSSION

The paper’s geometric architecture jointly preserves Lagrangian and manifold structure to learn interpretable reduced dynamics, while its predictions remain limited by autoencoder reconstruction and latent-dimension selection.

  • 6 DISCUSSION: The architecture uses geometry and physics as inductive biases to conserve reduced energy, infer interpretable parameters, and model high-dimensional rigid and deformable systems.Geometry is used for nonlinear energy-preserving embeddings and for manifold-constrained joint optimization, which improves performance especially in low-data regimes.
  • 6 DISCUSSION: Prediction quality is bounded by autoencoder reconstruction: inadequate representation of the full-order solution space causes systematic deviation from ground truth.The latent dimension is therefore crucial but difficult to select because the degree of coupling in the full-order model is unknown.
  • SPD geometry: The SPD manifold uses exponential and logarithmic maps, geodesic distance, and parallel transport to connect Euclidean tangent-space computations with manifold-valued parameters.These operations support optimization of the mass-inertia representation and preserve the geometry of positive-definite matrices.
  • Biorthogonal geometry: Biorthogonal autoencoder weights form a manifold constrained by Ψ^TΦ = I_d, with tangent projection, retraction, and approximate parallel transport enabling Riemannian optimization.The biorthogonal manifold has dimension 2nd − d^2, and retraction provides a first-order approximation to the exponential map.
  • SPD network: The mass-inertia matrix is modeled directly on the SPD manifold with an affine-invariant metric to avoid the swelling effect associated with Cholesky-based Euclidean distances.The SPD network is optimized with Riemannian methods, while ReEig, Euclidean, exponential-map, and gyrocalculus layers provide manifold-aware network operations.

D CONSTRAINED AE: IMPLEMENTATION DETAILS

The constrained autoencoder enforces projection-compatible activations and trains biorthogonal weights on their manifold, while acceleration-based training requires second-order derivatives and contrasts with penalty-based enforcement.

  • D CONSTRAINED AE: IMPLEMENTATION DETAILS: The encoder and decoder activations satisfy σ− ∘ σ+ = id, ensuring the constrained autoencoder retains the required projection properties.The activations are implemented as smooth, rotation-symmetric variants of leaky ReLU functions.
  • D CONSTRAINED AE: IMPLEMENTATION DETAILS: Autoencoder weights are optimized on a product of Euclidean and biorthogonal manifolds, directly enforcing the biorthogonality condition during training.The weight pairs (Φ_l, Ψ_l) belong to B_nl,nl−1, while biases remain Euclidean parameters.
  • D CONSTRAINED AE: IMPLEMENTATION DETAILS: Training on acceleration data differentiates encoder and decoder outputs with respect to their inputs, using layerwise analytical derivatives and the chain rule.The dataset includes position, velocity, and acceleration observations; irrelevant velocity or acceleration terms can be removed from the loss.
  • D CONSTRAINED AE: IMPLEMENTATION DETAILS: Compared with Riemannian optimization, penalty-based overparameterization reaches its intended minimum only when every layer’s weights satisfy biorthogonality.The additional regularization loss also introduces another learning rate and more hyperparameter combinations.
  • D CONSTRAINED AE: IMPLEMENTATION DETAILS: Figure 7 summarizes the forward pass from acceleration-trained reduced-order dynamics through reduction mappings, latent geometric LNNs, and an SPD mass-inertia network.The constrained autoencoder supplies the reduction mappings and embeddings, while the latent geometric LNN learns the reduced dynamics.

E TRAINING ON ACCELERATION

The proposed reduced-order LNN is trained on acceleration by minimizing the corresponding loss, with position and velocity predictions obtained through Euler forward integration.

  • E TRAINING ON ACCELERATION: The complete forward dynamics of the proposed RO-LNN trained on acceleration are illustrated in Figure 7 through minimization of loss (12).The figure depicts the reduction mappings, latent geometric LNN, and SPD mass-inertia network in the forward pipeline.
  • E TRAINING ON ACCELERATION: Position and velocity predictions are obtained via Euler forward integration in the experimental implementation.

F.1.1 SIMULATION AND DATA GENERATION

The experiments use MUJOCO simulations of a two-degree-of-freedom pendulum under unactuated and sine-tracking conditions, with trajectories generated using RK4 integration.

  • F.1.1 SIMULATION AND DATA GENERATION: The pendulum experiment uses a 2-DoF MUJOCO model with hinge-connected cylindrical links and no dissipative effects.Each link has radius 0.025 m, length 0.4 m, and mass 0.1 kg.
  • F.1.1 SIMULATION AND DATA GENERATION: Trajectories are simulated with an RK4 solver at Δt = 10^-3 s over the interval [0,T] s.The datasets contain position, velocity, acceleration, and torque samples across multiple trajectories.
  • F.1.1 SIMULATION AND DATA GENERATION: The unactuated pendulum dataset samples each initial joint configuration from [0,30]° with zero initial velocity and records trajectories for T = 2 s.All models in the main pendulum experiment are trained on this dataset.
  • F.1.1 SIMULATION AND DATA GENERATION: The sine-tracking dataset controls each joint toward a sinusoidal reference trajectory using inverse-dynamics torque control.Initial positions and velocities match the reference trajectory, and each trajectory is recorded for T = 3.5 s.

F.1.2 ARCHITECTURES & TRAINING

The experiments use geometric LNNs and RO-LNNs built from energy networks, structure-preserving autoencoders, and Riemannian optimization across pendulum, rope, and cloth settings.

  • The geometric LNN parametrizes kinetic energy through a mass-inertia SPD network and potential energy through a Euclidean multilayer perceptron.
  • The study compares acceleration-based and multi-step training, with the latter integrating latent acceleration predictions over H steps before decoding.
  • RO-LNN jointly learns a constrained autoencoder and latent geometric LNN, using a four-dimensional latent space for the high-dimensional pendulum.

G.1.1 ADDITIONAL RESULTS ON THE EXPERIMENT OF SECTION 5.1

Additional pendulum results show that geometric LNNs improve trajectory prediction and energy behavior relative to DeLaN, while multi-step training is especially important for the sine-tracking dataset.

  • Geometric LNNs outperform DeLaNs in position and velocity prediction, while multi-step-trained models outperform acceleration-trained models over long horizons.
  • All LNNs approximately conserve predicted total energy on the unactuated pendulum, although DeLaNs exhibit the largest fluctuations.
  • On the sine-tracking dataset, geometric LNNs outperform Cholesky-based models, particularly in low-data regimes, and SPD layers yield slightly lower acceleration errors.
  • Multi-step integration is crucial for accurate long-term trajectory prediction on the sine-tracking dataset.

G.1.3 INFLUENCE OF SPD LAYERS

SPD layers provide limited and inconsistent benefits relative to simpler geometric components, while multi-step training improves prediction, efficiency, and high-dimensional reduced-order learning.

  • Influence of SPD layers: Increasing SPD-layer depth does not clearly improve expressivity, and architectural choices show no consistent influence across training-set sizes.
  • Influence of SPD layers: SPD networks can create numerical optimization problems through repeated matrix exponentials and logarithms, with ReEig variants sometimes failing to converge.
  • Runtimes: Adding demanding GyroSpd+++ReEig layers doubles training time, whereas the well-performing identity-basepoint geometric LNN trains almost as fast as DeLaN.
  • Acceleration vs multi-step training: Multi-step training increases data efficiency and improves long-term predictions, while avoiding the large Hessian computations required by acceleration-based training.
  • Reduced-order learning: Riemannian constrained autoencoders consistently outperform regular autoencoders and improve reconstruction more than overparametrization in low-data regimes.
  • Computational speedup of ROM w.r.t. FOM: The ROM evaluates 3000 Euler-integration steps in 1.57 s versus 113.59 s for the FOM, while accurately modeling the high-dimensional rope dynamics.

G.4 ADDITIONAL RESULTS ON THE 600-DOF CLOTH EXPERIMENT OF SECTION 5.2.3

The cloth experiment evaluates RO-LNN reconstruction and prediction at short and full horizons, including latent-energy behavior during extrapolation beyond the observed actuation interval.

  • RO-LNN predictions are compared with autoencoder reconstructions and ground truth for both Htest = 25 and Htest = 3000 horizons.
  • The cloth evaluation reports reconstruction and 25-step prediction errors averaged over 10 testing trajectories for a d = 10 RO-LNN.
  • For the final 0.05 s beyond the 0.25 s actuation interval, ground-truth states and torques are held at their last recorded values.
  • The predicted latent energy is compared with ground-truth energy projected into the latent space, supporting long-term prediction of the deformable cloth dynamics.
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