  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.
