Lagrangian Neural Networks (2003.04630v2)

Abstract: Accurate models of the world are built upon notions of its underlying symmetries. In physics, these symmetries correspond to conservation laws, such as for energy and momentum. Yet even though neural network models see increasing use in the physical sciences, they struggle to learn these symmetries. In this paper, we propose Lagrangian Neural Networks (LNNs), which can parameterize arbitrary Lagrangians using neural networks. In contrast to models that learn Hamiltonians, LNNs do not require canonical coordinates, and thus perform well in situations where canonical momenta are unknown or difficult to compute. Unlike previous approaches, our method does not restrict the functional form of learned energies and will produce energy-conserving models for a variety of tasks. We test our approach on a double pendulum and a relativistic particle, demonstrating energy conservation where a baseline approach incurs dissipation and modeling relativity without canonical coordinates where a Hamiltonian approach fails. Finally, we show how this model can be applied to graphs and continuous systems using a Lagrangian Graph Network, and demonstrate it on the 1D wave equation.

• The paper introduces Lagrangian Neural Networks that learn arbitrary Lagrangians without requiring canonical coordinates, broadening the scope for modeling diverse physical systems.
• Empirical evaluations on systems like the double pendulum demonstrate near-exact energy conservation, showcasing the practical advantages of the LNN approach.
• The approach outperforms conventional Hamiltonian methods and opens new avenues for integrating physical principles into machine learning across various domains.

Lagrangian Neural Networks: A Novel Approach to Modeling Physical Systems

The paper presents a compelling exploration of Lagrangian Neural Networks (LNNs), proposing these networks as a robust alternative for modeling physical systems characterized by unknown or complex canonical momenta. The primary motivation for this research is the limitation observed in conventional Hamiltonian Neural Networks (HNNs), which necessitate the coordinates of the system to be canonical, thereby restricting their applicability to a subset of physical problems. In stark contrast, LNNs circumvent this limitation by learning arbitrary Lagrangians, thereby enhancing their versatility in modeling a broader class of systems.

Key Contributions

The authors delineate several contributions of their work:

1. Lagrangian Neural Networks: The central contribution is the introduction of LNNs capable of learning arbitrary Lagrangians. This framework does not impose any restraints on the functional form of learned energies, thus allowing the network to operate effectively even when traditional canonical coordinates are inaccessible or infeasible to compute.
2. Empirical Evaluation: The paper empirically tests the efficacy of LNNs on well-known physical systems, such as the double pendulum and a relativistic particle, highlighting their superiority in conserving energy compared to baseline models. The results demonstrate notable evidence of LNNs maintaining near-exact energy conservation over extensive temporal simulations.
3. Comparison with Related Approaches: The research contrasts LNNs with other contemporary models like Deep Lagrangian Networks (DeLaNs) and HNNs, underscoring the flexibility and generality of LNNs. Unlike DeLaNs, which are constrained to modeling specific dynamical systems such as those involving rigid bodies, LNNs impose no such limitations.
4. Potential for Graph and Continuous Systems: The theoretical foundation extends to proposing a Lagrangian Graph Network for modeling systems with discrete connectivity or continuous media, asserting the broader applicability of LNNs in these contexts.

Implications and Future Developments

The implications of adopting LNNs in physical sciences and engineering are substantial, where conservation laws play a pivotal role in predictive accuracy and reliability. The ability to employ arbitrary coordinates significantly reduces constraints on input data, making LNNs particularly valuable in scenarios where traditional dynamical variables are not readily apparent or easily quantifiable.

Theoretically, the development of LNNs presents a paradigm shift in how neural networks can be used to embed physical principles directly within their architectures. This approach has the potential to inspire future research aimed at integrating similar physical priors into ML models for a range of disciplines, possibly extending beyond physics to include areas where symmetries or conserved quantities are intrinsic.

Looking forward, several avenues for future research and development emerge:

• Scaling and Efficiency: While the implementation of LNNs using frameworks like JAX demonstrates feasibility, further work is necessary to optimize these algorithms for high-dimensional systems involving complex geometries and interactions.
• Generalization Across Domains: Investigating the adaptability of LNNs across domains and various types of dynamical systems would broaden their impact, making them indispensable in a plethora of scientific and industrial applications.
• Integration with Other ML Models: In the context of hybrid models combining LNNs with other ML architectures (e.g., reinforcement learning or graph neural networks), potential gains in both performance and interpretability could be explored.

In conclusion, Lagrangian Neural Networks offer a promising and flexible framework that can expand the toolkit available for researchers and practitioners modeling dynamic physical systems. The paper lays a foundational step toward harnessing the power of neural networks in congruence with the fundamental laws of physics, paving the way for innovative applications and methodologies in the field of computational modeling.