采用两步优化器的深度配点法与深度能量法求解薄板弯曲问题

With the advancement of computing power and machine learning algorithms, deep learning methods have been widely applied in a wide range of fields. In this manuscript, we develop the deep collocation method and the deep energy method fitted to engineering computation and further apply them to solve the Kirchhoff thin plate bending problems. The deep collocation method adopts the physics-informed neural networks, incorporating the strong-form governing equations into the loss function. It reduces the solving of thin plate problem into an optimization problem. On the other hand, the deep energy method utilizes energy-driven neural networks based on the principle of minimum potential energy, indicating that of all displacements satisfying given boundary and equilibrium conditions, the actual displacement is the one that minimizes the total potential energy at stable equilibrium. Thus, we can build a loss function from the total potential energy. With the boundary conditions penalized to the loss form, the problem is reduced to an unconstrained optimization one. The physics-informed and energy-driven neural networks are based on the universal approximation theorem. Due to the introduction of physical and energy information, the neural networks become difficult to train. An improved two-step optimization algorithm is presented to train the neural network. From the numerical results, it is clearly seen that both methods are suitable for solving thin plate bending problems, easy to implement, and truly "meshfree".