Poisson Hamiltonian Neural Networks: A New Paradigm of Deep Learning That Preserves Physical Structures

The Poisson Hamiltonian Neural Network (PHNN) introduced in this article is an important direction for the integration of deep learning and physical modeling. Its core idea is to solve the problem that traditional deep learning cannot guarantee conservation laws when learning physical system dynamics by preserving the Poisson structure and Hamiltonian of physical systems, thereby achieving more stable and accurate long-term predictions.

When deep learning is applied to learning physical system dynamics, traditional methods often face the challenge of failing to guarantee physical conservation laws (e.g., energy drift). Hamiltonian mechanics provides a framework for describing system evolution, which naturally ensures energy and momentum conservation through Hamiltonian equations. However, traditional neural networks lack these physical constraints and only pursue minimization of prediction errors, making it difficult to accurately simulate physical systems.

PHNN generalizes classical Hamiltonian mechanics and describes system dynamics based on Poisson brackets in Poisson geometry. Its network architecture consists of two components:

1. Hamiltonian network: A scalar-valued function H_θ(x) that takes system state as input and outputs the Hamiltonian;
2. Poisson tensor network: Parameterizes the skew-symmetric Poisson tensor J(x) = L(x)L(x)^T - L(x)^T L(x) through a lower triangular matrix L(x), ensuring the basic requirements of the Poisson structure.

During PHNN training, trajectories are predicted via numerical integration, minimizing the difference between predictions and observations. The loss function includes trajectory loss, derivative loss, and conservation quantity loss. In the inference phase, the model structure ensures physical conservation laws, so there is no energy drift in long-term integration, making it suitable for physical simulations that require long-term predictions.

PHNN shows potential in multiple fields:

• Celestial mechanics: Learning planetary system dynamics, preserving angular momentum and energy conservation, aiding long-term orbit prediction;
• Molecular dynamics: Ensuring energy conservation, providing efficient and physically correct simulations;
• Control systems: Learning dynamic models of systems like robots, providing a reliable foundation for controller design.

PHNN faces the following challenges:

1. Computational complexity: Numerical integration leads to higher training and inference costs than standard neural networks;
2. Data requirements: Learning Poisson structures and Hamiltonians requires more training data;
3. High-dimensional systems: The representation and learning of Poisson tensors for high-dimensional systems remain open problems.

PHNN represents an important direction in physics-aware machine learning, which not only improves the stability of long-term predictions but also provides new tools for understanding complex physical systems. In the future, we look forward to more innovative methods that integrate physical priors into neural networks, which have important exploration value for fields such as physical simulation and robot control.