We introduce a method for training exactly conservative physics-informed neural networks and physics-informed deep operator networks for dynamical systems. The method employs a projection-based technique that maps a candidate solution learned by the neural network solver for any given dynamical system possessing at least one first integral onto an invariant manifold. We illustrate that exactly conservative physics-informed neural network solvers and physics-informed deep operator networks for dynamical systems vastly outperform their non-conservative counterparts for several real-world problems from the mathematical sciences.

翻译：我们提出一种方法，用于训练面向动力系统的精确守恒物理信息神经网络及物理信息深度算子网络。该方法采用基于投影的技术，将神经网络求解器针对任意至少具有一个首次积分的动力系统学到的候选解映射到不变流形上。我们通过数学科学中的多个实际案例证明，面向动力系统的精确守恒物理信息神经网络求解器与物理信息深度算子网络，其性能显著优于非守恒对应方法。