Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data (2012.02334v6)

Abstract: The last few years have witnessed an increased interest in incorporating physics-informed inductive bias in deep learning frameworks. In particular, a growing volume of literature has been exploring ways to enforce energy conservation while using neural networks for learning dynamics from observed time-series data. In this work, we survey ten recently proposed energy-conserving neural network models, including HNN, LNN, DeLaN, SymODEN, CHNN, CLNN and their variants. We provide a compact derivation of the theory behind these models and explain their similarities and differences. Their performance are compared in 4 physical systems. We point out the possibility of leveraging some of these energy-conserving models to design energy-based controllers.

• The paper benchmarks ten neural network models incorporating physics-informed inductive biases to learn dynamics while conserving energy.
• Experimental results show explicit constraint models, specifically CLNN and CHNN, generally outperform implicit models, especially in complex dynamics.
• Angle-aware designs and the use of simpler coordinate systems in explicit models improve accuracy and computational efficiency for rotational dynamics.

Benchmarking Energy-Conserving Neural Networks

In the paper "Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data," the authors analyze and compare ten recently proposed neural network models that incorporate physics-informed inductive biases to learn dynamics while enforcing energy conservation. These models, classified primarily into Lagrangian and Hamiltonian neural networks, offer computational frameworks to infer unknown dynamics from observed trajectory data. This work meticulously evaluates the models within the context of four physical systems—namely, the N-pendulum variants and a gyroscopic system.

The survey provides a comprehensive overview of both implicit and explicit constraint enforcement in dynamic modeling and demonstrates how these techniques can enhance the fidelity of learned motion trajectories. Models covered include the Hamiltonian Neural Network (HNN) and its variants, such as HNN-structure and CHNN, as well as Lagrangian Neural Network (LNN) variants like LNN-structure and CLNN. Reflecting on the equations of motion derived from basic principles, these models leverage structures inherent to Hamiltonian and Lagrangian dynamics to efficiently capture and simulate system behaviors in a limited data environment.

Noteworthy contributions of this paper include an analysis of data efficiency and long-term prediction capabilities, as well as exploring the balance between explicit and implicit constraint handling. Including angle-aware designs by embedding angular coordinates on $\mathbb{S}^1$, the paper illustrates improved accuracy and predictability for systems characterized by rotational dynamics—a typical challenge in many physical simulations.

The experimental results indicate that explicit constraint models, particularly CLNN and CHNN, consistently outperform implicit models. Especially in scenarios involving complex dynamics or multi-dimensional systems—such as gyroscopic dynamics—these models show significant advantages. The explicit models benefit from a simpler coordinate system (Cartesian coordinates), which leads to computational efficiencies and more stable training sequences. Despite the heightened implementation burden, these benefits present promising avenues for future research in energy-conserving models aiming to simulate intricate physical systems accurately.

While Lagrangian and Hamiltonian models offer distinct advantages depending on the specific nature of the dynamic system, the structured variants of each potentially yield better performance by leveraging energy-conserving formulations. These formulations are propagated through differentiable ODE solvers, which enable precise modeling of complex, multi-body interactions.

Given the demonstrated effectiveness of these frameworks, future work may focus on extending applicability to additional domains or refining computational models to offer greater efficiency in divergent physical environments. For practitioners designing controllers in real-world applications, the research highlights the potential integration of energy-conserving principles based on learned dynamics, allowing for seamless and effective control strategies.

In conclusion, "Benchmarking Energy-Conserving Neural Networks for Learning Dynamics from Data" presents valuable insights into neural network models for dynamics learning, paving the way for improved simulations of physical systems that incorporate the foundational laws of physics in machine learning contexts. The ongoing exploration of these models may further enhance our understanding and control of dynamic systems across various scientific and industrial applications.