Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning (1907.04490v1)

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

• The paper introduces Deep Lagrangian Networks that integrate Euler-Lagrange mechanics into neural models to enforce physical constraints.
• The approach is validated on both simulated and physical robotics, achieving superior extrapolation and sample efficiency over traditional methods.
• The methodology reduces data requirements by embedding physics as a prior, enhancing performance in real-time control and dynamic systems.

Overview of Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning

The paper "Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning" by Michael Lutter, Christian Ritter, and Jan Peters introduces an innovative approach to deep learning by embedding physical principles, specifically Lagrangian mechanics, directly into the modeling process. This integration aims to significantly enhance the ability of deep networks to extrapolate from limited data, thereby addressing a crucial requirement for learning physics-based models in real-world applications.

The research focuses on applying deep network structures to represent mechanical systems' equations of motion, leveraging the Euler-Lagrange formulation to impose physical constraints on the learning process. The primary motivation is to overcome challenges posed by high sample complexities and poor generalization in purely data-driven models, especially in domains like robotics where real-time learning and system safety are paramount.

Key Contributions

The paper makes two substantial contributions to model-based control learning:

1. Network Topology Incorporating Lagrangian Mechanics: The authors propose a novel network topology—Deep Lagrangian Networks (DeLaN)—that encodes the Euler-Lagrange equations. This approach guarantees that the learned models comply with physical laws, unlike previous methods which depend on engineered physics features specific to the system's physical embodiment. DeLaN only requires specifying the system state and input, making it versatile across different mechanical systems.
2. Experimental Validation and Generalization: DeLaN is extensively evaluated on simulated and physical robotic systems—a 2-degree-of-freedom robot and the Barrett WAM, respectively. The results demonstrate that DeLaN not only learns the dynamics models efficiently from scratch but also exhibits superior extrapolation capabilities to novel trajectories and higher dynamics such as increased velocities. This outperforms both traditional analytic models and other learning-based approaches in terms of tracking control and sample efficiency.

Implications and Future Directions

The paper's methodology showcases how deep learning models can benefit from integrating domain-specific knowledge, particularly physics, to enhance learning performance in controlled environments. This represents a substantial shift from traditional model-agnostic learning to more structured and informed model learning.

Practically, this approach reduces the dependency on large datasets, which is a significant challenge in domains like robotics where data collection can be costly and time-consuming. Theoretically, it opens pathways to further explore how other physical laws might be incorporated into machine learning frameworks to improve model robustness and generalization.

Looking forward, an area of future exploration extends to broadening the class of systems that can be modeled by incorporating additional non-conservative forces or dealing with systems with more complex dynamics, such as those encountered when working with highly flexible or soft robots. Further research may also involve combining these physics-informed networks with reinforcement learning to optimize control policies under more uncertain and varied conditions.

Overall, the integration of deep learning with fundamental physics provides an exciting avenue for enhancing autonomous system capabilities, with the potential to influence areas beyond robotics, such as bioinformatics and materials science, where physical models play a critical role.