[Paper Review] Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning

Michael Lutter, Christian Ritter+1 more|arXiv (Cornell University)|Jul 10, 2019

Model Reduction and Neural NetworksPhysics and Astronomy38 references80 citations

TL;DR

DeLaN encodes Euler-Lagrange mechanics into a neural network to learn robot dynamics with physical plausibility, enabling robust extrapolation and online learning for control.

ABSTRACT

Deep learning has achieved astonishing results on many tasks with large amounts of data and generalization within the proximity of training data. For many important real-world applications, these requirements are unfeasible and additional prior knowledge on the task domain is required to overcome the resulting problems. In particular, learning physics models for model-based control requires robust extrapolation from fewer samples - often collected online in real-time - and model errors may lead to drastic damages of the system. Directly incorporating physical insight has enabled us to obtain a novel deep model learning approach that extrapolates well while requiring fewer samples. As a first example, we propose Deep Lagrangian Networks (DeLaN) as a deep network structure upon which Lagrangian Mechanics have been imposed. DeLaN can learn the equations of motion of a mechanical system (i.e., system dynamics) with a deep network efficiently while ensuring physical plausibility. The resulting DeLaN network performs very well at robot tracking control. The proposed method did not only outperform previous model learning approaches at learning speed but exhibits substantially improved and more robust extrapolation to novel trajectories and learns online in real-time

Motivation & Objective

• Motivate learning physical models for embodied systems to enable robust extrapolation from few samples and safe online updates.
• Propose a network topology that enforces physics via the Euler–Lagrange equation while remaining generic to system structure.
• Enable end-to-end training of the model with differentiable representations of inertia and non-conservative forces.
• Demonstrate real-time learning and control performance on simulated and physical robots using online data.

Proposed method

• Represent the inertia matrix H(q) as L(q)L(q)^T with L having positive diagonal elements to ensure symmetry and positive definiteness.
• Model the non-conservative forces g(q) with a separate neural network head.
• Formulate f^{-1}(q, qdot, qddot; θ, ψ) using the learned L and g to satisfy the Euler–Lagrange equation.
• Train end-to-end by minimizing a loss between predicted and true torques, with an additional regularization term to address non-uniqueness of L.
• Compute derivatives analytically to enable real-time backpropagation through the L and g components.
• Use a specialized Lagrangian layer that yields a single forward pass to obtain H, dH/dt, and partials needed for control.

Experimental results

Research questions

• RQ1 Can a neural network-based representation of the Lagrangian dynamics (through H(q) and g(q)) provide physically plausible and accurate models for robotic systems?
• RQ2 Does enforcing Lagrangian structure improve extrapolation to unseen trajectories and velocities compared to standard feed-forward models?
• RQ3 Is online learning of the dynamics feasible in real-time control loops using DeLaN without sacrificing stability or plausibility?
• RQ4 How does DeLaN perform relative to analytic physics models and purely data-driven networks in simulated and real robots?

Key findings

• DeLaN learns the dynamics components inertial, Coriolis/centrifugal, and gravity from data, matching ground-truth components in experiments.
• Extrapolation to novel trajectories and higher speeds is significantly better with DeLaN than a standard feed-forward network, especially when training data are limited.
• In online control, DeLaN achieves robust tracking and can adapt to new trajectories starting from random initialization.
• On a physical Barrett WAM, DeLaN’s tracking performance is competitive with analytic models and can learn dynamics under complex cable-driven effects.
• Across simulations and real robot experiments, DeLaN demonstrates improved sample efficiency and extrapolation capabilities compared to baseline models.