基于热传导方程的自适应损失物理信息神经网络算法研究

Study on Self-Adaptive Loss Physical Information Neural Network Algorithm Based on Heat Transfer Equation

摘要: 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。

Abstract: 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.