基于物理信息神经网络的薄板反问题研究

Research on the Inverse Problem of Thin Plates Based on Physics-informed Neural Networks

为适应薄板反问题求解的智能化发展趋势,基于数据和模型混合驱动思想,建立了基于物理信息神经网络的薄板反问题求解新框架.针对神经网络计算高阶项时间长的问题,受小样本学习中基于微调的学习方式启发,提出了一种冻结参数分步训练法.针对监督学习薄板挠度时边缘处误差较大现象,提出了一种边缘切割方法以优化采样区域.在此基础上采用基于Kirchhoff薄板理论的挠度偏微分方程结合解析解公式计算并叠加随机噪声来模拟薄板挠度实测数据,从而搭建薄板反问题求解的物理信息神经网络模型.同时,进行消融实验以分析验证所提出的改进措施的有效性.数值实验结果显示:受均布载荷的四边简支与四边固支矩形薄板的均布载荷与抗弯刚度比、均布载荷和弹性模量等参数在NVIDIA RTX 3060显卡加速下反演耗时小于30秒,低噪声环境下相对误差小于3%;受静水压力作用的四边固支矩形薄板线性载荷反演相对误差小于11%.研究表明:基于物理信息神经网络的薄板反问题求解方法可行有效,能精准反演力学模型各种参数;相应的切割边缘-冻结参数分步训练反演算法具有速度快、精度高、鲁棒性强和参数冗余度小等特点.该研究为薄板反问题自适应高效准确求解创造了条件,为薄板结构智能健康监控提供了有益参考.

In order to adapt to the intelligent development trend of solving the inverse problem of thin plates,this paper establishes a novel framework for solving the inverse problem of thin plates using physics-informed neural networks based on the hybrid driving idea of data and modeling.Inspired by the finetuning method for few-shot learning,a step-by-step training method for freezing parameters is proposed to solve the time-consuming problem for neural networks to calculate higher-order terms.To optimize the sampling area in view of the large error at the edge for supervised learning of the deflection of a thin plate,an edge cutting method is proposed.The measured data of thin-plate deflections are obtained by calculating the partial differential equation for deflection based on the Kirchhoff thin-plate theory in combination with the analytical solution formula and superimposing random noise,and a model of physics-informed neural networks is thus constructed for solving the inverse problem of thin plates.In addition,ablation experiments are carried out to analyze and verify the effectiveness of the proposed improvement measures.The numerical results show that under the acceleration of NVIDIA RTX 3060 video card,the inversion time for the ratios of uniformly-distributed load to flexural rigidity,the uniformly-distributed loads and elastic moduli of simply-supported and clamped rectangular thin plates is less than 30 seconds and the relative error is less than 3%in a low-noise environment.For rectangular thin plates with four edges clamped,the relative error of linear load inversion clamped under hydrostatic pressure is less than 11%.The results show that the inversion method for thin-plate problems based on physics-informed neural networks is correct and effective,which can accurately invert the parameters of various mechanical models.The corresponding step-by-step training inversion algorithm of cutting edge-freezing parameters has the advantages of fast speed,high precision,strong robustness and small parameter redundancy.This research creates conditions for the adaptive,efficient and accurate solution of thin-plate inverse problems,and provides a useful reference for intelligent health monitoring of thin-plate structures.