文章基于物理信息神经网络(Physics-informed Neural Networks，PINN)求解非线性瞬态热传导问题并识别随温度变化的导热系数。首先，基于热传导问题的控制方程，利用初始条件和边界条件，构建损失函数。然后，应用自动微分算法求解控制方程中温度的偏导数。使用梯度下降算法，更新网络参数，最小化损失函数，实现热传导正问题的求解，并讨论了不同隐藏层数、神经元数量和域内数据点数量对计算结果的影响。最后，采用PINN识别随温度变化的导热系数，利用控制方程、测量温度和计算温度的残差构建损失函数，通过梯度下降算法，更新网络参数和导热系数，使其逼近于精确解，并比较了不同的测点数量和测量误差对计算结果的影响。结果表明，PINN能够有效求解非线性瞬态热传导问题并识别与温度相关的导热系数。

A method based on the physics-informed neural networks (PINN) is proposed to solve the transient nonlinear heat conduction problems and estimate the temperature-dependent thermal conductivity. Firstly, a loss function is formulated with the residual error of partial differential equation, initial conditions and boundary conditions for the heat conduction problems. Then, the automatic differentiation is applied to acquire the partial derivatives of temperature in the partial differential equation. The heat conduction problems are solved by applying gradient descent algorithm to update the network parameters and to minimize the loss function. The influences of different numbers of hidden layers, neurons and interior collection points on the results are discussed. Finally, the PINN is applied to identify the temperature-dependent thermal conductivities. The loss function is formulated with the residual error of governing equation, measured temperature and computed temperature. The network parameters and thermal conductivity are updated by the gradient descent algorithm to approximate the exact solution. The influences of different measurement points and errors on results are also compared. The results show that the proposed method is an effective approach to solve the transient heat conduction problem and estimate the temperature-dependent thermal conductivity.

国家自然科学基金(12002181); 中央高校基本科研业务费(JZ2022HGQA0165, JZ2022HGTB0243)