离散时间物理的深层能量建模
Deep Energy-Based Modeling of Discrete-Time Physics
摘要：
现实世界中的物理现象通常由基于能量的建模理论描述，例如哈密顿力学或朗道理论，它们产生各种物理定律。神经网络的最新发展已经能够通过学习底层连续时间微分方程来模拟能量守恒律。然而，这在离散时间中可能是不可能的，这在实际学习和计算中经常是这种情况。此外，在以前的神经网络模型中忽略了其他物理定律。在这项研究中，我们提出了一个深能量为基础的物理模型，承认一个特定的微分几何结构。从这个结构中，能量守恒或耗散定律和质量守恒律自然地遵循。为了确保离散时间的能量行为，我们还提出了一个自动离散差分算法，使神经网络采用离散梯度法。
英文摘要：
Physical phenomena in the real world are often described by energy-based modeling theories, such as Hamiltonian mechanics or the Landau theory, which yield various physical laws. Recent developments in neural networks have enabled the mimicking of the energy conservation law by learning the underlying continuous-time differential equations. However, this may not be possible in discrete time, which is often the case in practical learning and computation. Moreover, other physical laws have been overlooked in the previous neural network models. In this study, we propose a deep energy-based physical model that admits a specific differential geometric structure. From this structure, the conservation or dissipation law of energy and the mass conservation law follow naturally. To ensure the energetic behavior in discrete time, we also propose an automatic discrete differential algorithm that enables neural networks to employ the discrete gradient method.
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