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A Riemannian Framework for Learning Reduced-order Lagrangian Dynamics
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Proposes a Riemannian reduced-order Lagrangian neural network that jointly learns a structure-preserving constrained autoencoder (biorthogonal manifold) and a latent geometric LNN with SPD mass-matrix networks for high-dimensional mechanical systems.
Problem
Learning physically consistent dynamics of high-dimensional Lagrangian systems (rigid and deformable) from data is data-hungry and computationally costly, and existing structure-preserving model order reduction is intrusive or limited to linear subspaces.
Approach
Jointly learn a nonlinear structure-preserving latent embedding via a constrained autoencoder whose biorthogonal weights are optimized on the biorthogonal manifold, together with a latent Lagrangian NN whose mass-inertia matrix is an SPD network trained with Riemannian Adam.
Key results
- On the 16-DoF pendulum, RO-LNN trained with multi-step integration reaches relative position error of 9.39e-3 versus 2.45e-1 for L-OpInf with LNN and 1.31e1 for L-OpInf.
- Full-order LNN on the 16-DoF pendulum yields relative acceleration errors around 1.97e2, orders of magnitude above RO-LNN (3.53e-1 with ODE training).
- Jointly training the constrained AE with latent LNN on the biorthogonal manifold outperforms sequential training and overparametrized training; 4 of 6 overparametrized trainings produced NaNs.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Train RO-LNN with multi-step integration instead of acceleration data (ablation) | RO-LNN · Coupled pendulum (16 DoF) | normalized position prediction error: $(3.07 \pm 1.86)\times 10^{-1} -> (9.39 \pm 3.16)\times 10^{-3}$ | Helped |
| Train RO-LNN with multi-step integration instead of acceleration data (ablation) | RO-LNN · Coupled pendulum (16 DoF) | normalized acceleration prediction error: $(5.87 \pm 4.74)\times 10^{-1} -> (3.53 \pm 2.17)\times 10^{-1}$ | Helped |
| Train RO-LNN with multi-step integration instead of acceleration data (ablation) | RO-LNN · Coupled pendulum (16 DoF) | normalized velocity prediction error: $(1.33 \pm 0.79)\times 10^{0} -> (3.24 \pm 3.43)\times 10^{-1} | Helped |
| Use RO-LNN instead of full-order LNN | Coupled pendulum (16 DoF) · Coupled pendulum (16 DoF) | normalized acceleration prediction error | Helped |
| Use RO-LNN (ODE) instead of L-OpInf (LNN, ODE) | Coupled pendulum (16 DoF) · Coupled pendulum (16 DoF) | normalized position prediction error: $(2.45 \pm 0.96) \times 10^{-1} -> (9.39 \pm 3.16)\times 10^{-3}$ | Helped |
| Train the autoencoder and latent LNN jointly instead of sequentially (ablation) | RO-LNN · Coupled pendulum (16 DoF) | high-dimensional prediction error | Helped |
| Optimize biorthogonal autoencoder weights on the manifold instead of using overparameterization (ablation) | RO-LNN · Coupled pendulum (16 DoF) | prediction error | Helped |
| Use latent dimension 10 instead of 4 (ablation) | RO-LNN · Rope (192 DoF) | position reconstruction error: $2.95\times 10^{-2} \pm 3.24\times 10^{-2} -> 4.75\times 10^{-3} \pm 5.73\times 10^{-3}$ | Helped |
| Increase latent dimension from 10 to 14 (ablation) | RO-LNN · Rope (192 DoF) | position reconstruction error: $\bm{4.75\times 10^{-3} \pm 5.73\times 10^{-3} -> 6.16\times 10^{-2} \pm 6.52\times 10^{-2}$ | Hurt |
| Increase latent dimension from 10 to 14 (ablation) | RO-LNN · Rope (192 DoF) | velocity reconstruction error: $\bm{6.53\times 10^{-3} \pm 1.62\times 10^{-2} -> 5.66\times 10^{-2}\pm 8.48\times 10^{-2}$ | Hurt |
| Add SPD layers to the geometric LNN (ablation) | Geometric LNN · 2-DoF planar pendulum | long-term prediction accuracy | No clear effect |
| Train the geometric LNN with 8 integration steps instead of acceleration data (ablation) | Geometric LNN · 2-DoF planar pendulum | long-term prediction accuracy | Helped |
Surprising findings
- Geometric LNNs with SPD layers did not noticeably outperform those using only Euclidean and exponential-map layers, so SPD layers were dropped for reduced-order experiments.
- Overparametrized biorthogonality regularization made joint training unstable, producing NaN values in 4 of 6 runs.
Limitations
- Prediction quality is bounded by autoencoder reconstruction quality; choosing latent dimension is non-trivial.
- Full-order LNNs and linear OpInf fail to train stably on high-dimensional systems, and acceleration-based training needs expensive second derivatives of the autoencoder.
Experimental setup
Datasets: MuJoCo-simulated 2-DoF pendulum MuJoCo-simulated 16-DoF coupled pendulum 192-DoF rope simulation 600-DoF thin cloth simulation
Benchmarks: 2-DoF pendulum 16-DoF coupled pendulum 192-DoF rope 600-DoF thin cloth
Metrics: relative acceleration error relative velocity error relative position error reconstruction error prediction loss
Baselines: DeLaN full-order LNN L-OpInf L-OpInf with LNN sequentially-trained RO-LNN overparametrized RO-LNN
Builds on: DeLaN constrained autoencoder (biorthogonal) Lagrangian operator inference SPD neural networks (gyroAI / GyroSpd) Geoopt
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