在热传导方程的研究中,物理信息神经网络(PINN)的应用已初显成效,其损失函数由多个损失项的加权和组成,这些损失项的加权组合对PINN的有效训练具有关键作用。为此,我们引入了一个基于高斯概率模型的损失项定义,通过噪声参数来描述每个损失项的权重,并提出了一种基于极大似然估计原理的自适应损失函数方法,该方法通过不断更新每个训练周期中的噪声参数,实现损失权重的自动分配。采用自适应物理信息神经网络(SalPINN)对一维瞬态热传导方程进行求解,并与传统PINN方法对比,结果显示SalPINN在模拟热传导方程方面表现出更高的精确性和有效性。In the field of research into heat transfer equations, the application of physical information neural network (PINN) has achieved some results. The loss function of PINN consists of a weighted sum of multiple loss terms, and the weighted combination of these loss terms plays an important role in PINN’s effective training. Therefore, we construct a loss term definition based on a Gaussian probability model, where the introduction of noise parameters is used to describe the weight of each loss term. We propose a self-adaptive loss function method based on the maximum likelihood estimation principle to automatically assign loss weights by constantly updating noise parameters in each training cycle. Then, we use self-adaptive loss physical information neural network (SalPINN) to solve the one-dimensional transient heat transfer equation, and compare it with the traditional PINN method, and the results show that SalPINN is more accurate and effective in simulating the heat transfer equation.
本文以热传导方程为背景,分别研究了不同位置上的常数型未知参数与函数型未知参数,并给出统计反演算法,完成数值反演试验。两类方法均能高效完成反演任务,且在面对高噪声观测信息时,具有很强的鲁棒性。Focusing on the heat conduction equation, this paper investigates constant-type unknown parameters and function-type unknown parameters at different positions, and provides statistical inversion algorithms to complete numerical inversion experiments. Both methods can efficiently accomplish the inversion tasks and exhibit strong robustness in the presence of high-noise observational information.
针对COB-LED(Chip on Board-Light Emitting Diode)散热问题,文中基于二维热传导方程建立了一个可快速计算COB-LED散热器表面热分布的数学模型。为了便于模型求解,采用有限差分法求解该数学模型并选择交替方向隐格式作为其差分格式。根据模型中的边界条件和初始条件设计COB-LED常温点亮实验,并基于ANSYS有限元分析软件进行仿真分析。通过比较求解结果、仿真结果和实验结果验证该数学模型的合理性。结果表明,求解结果与实验结果中最高温度相对误差约23.57%,且两者的温度变化趋势一致。求解结果与仿真结果中最高温度相对误差约34.84%,且温度分布较为接近,证明了该数学模型的合理性与正确性。