As a fundamental mathmatical tool in many engineering disciplines, coupled differential equation groups are being widely used to model complex structures containing multiple physical quantities. Engineers constantly adjust structural parameters at the design stage, which requires a highly efficient solver. The rise of deep learning technologies has offered new perspectives on this task. Unfortunately, existing black-box models suffer from poor accuracy and robustness, while the advanced methodologies of single-output operator regression cannot deal with multiple quantities simultaneously. To address these challenges, we propose PINO-CDE, a deep learning framework for solving coupled differential equation groups (CDEs) along with an equation normalization algorithm for performance enhancing. Based on the theory of physics-informed neural operator (PINO), PINO-CDE uses a single network for all quantities in a CDEs, instead of training dozens, or even hundreds of networks as in the existing literature. We demonstrate the flexibility and feasibility of PINO-CDE for one toy example and two engineering applications: vehicle-track coupled dynamics (VTCD) and reliability assessment for a four-storey building (uncertainty propagation). The performance of VTCD indicates that PINO-CDE outperforms existing software and deep learning-based methods in terms of efficiency and precision, respectively. For the uncertainty propagation task, PINO-CDE provides higher-resolution results in less than a quarter of the time incurred when using the probability density evolution method (PDEM). This framework integrates engineering dynamics and deep learning technologies and may reveal a new concept for CDEs solving and uncertainty propagation.

翻译：作为众多工程学科中的基础数学工具，耦合微分方程组被广泛应用于描述包含多物理量的复杂结构。工程师在设计阶段需要不断调整结构参数，这要求具备高效的求解器。深度学习技术的兴起为这一任务提供了新思路。然而，现有黑箱模型存在精度和鲁棒性不足的问题，而先进的单输出算子回归方法无法同时处理多个物理量。针对这些挑战，我们提出了PINO-CDE——一种用于求解耦合微分方程组的深度学习框架，并配套提出方程归一化算法以提升性能。基于物理信息神经算子理论，PINO-CDE采用单一网络同时处理耦合微分方程组中的所有物理量，而现有文献往往需要训练数十甚至数百个网络。我们通过一个玩具示例和两个工程应用（车辆-轨道耦合动力学和四层建筑可靠性评估中的不确定度传播）验证了PINO-CDE的灵活性与可行性。车辆-轨道耦合动力学案例表明，PINO-CDE在效率与精度上分别优于现有软件和基于深度学习的传统方法。在不确定度传播任务中，PINO-CDE可在不到概率密度演化方法四分之一的计算时间内提供更高分辨率的结果。该框架融合了工程动力学与深度学习技术，为求解耦合微分方程组及不确定度传播提供了新思路。