AI 中文总结
该研究提出结合解析函数与PINN的迭代方法，用改进的DELTAsnake激活函数训练PINN，在Mu2e实验探测器螺线管磁场建模中，显著降低了约化卡方与残差均方根，减少了所需测量次数。
AI 中文摘要
我们展示了一种对大孔径螺线管中稀疏测量的磁场进行建模的迭代方法，该方法结合了传统技术与机器学习技术。传统技术是利用拉普拉斯方程的级数解进行线性最小二乘拟合，机器学习技术则涉及在最小二乘拟合残差上训练物理信息神经网络（PINN）。我们使用新定义的激活函数“DELTAsnake”，这是Ziyin等人提出的snake激活函数的改进版本，可实现更强的曲率和非单调性。组合模型近似满足麦克斯韦方程，其精度足以生成高质量的物理模拟与分析。我们将该方法应用于Mu2e实验探测器螺线管中预期磁场的高真实度计算，其中包含预期统计测量不确定度的简化模型。通过10次模拟测量，我们对比了该模型与单独最小二乘法的性能：单独最小二乘法的约化卡方统计量为${2.15 \boldsymbol{\times} 0.01}$，而我们的方法将约化卡方提升至${1.034 \boldsymbol{\times} 0.005}$。此外，对于平均模拟测量，我们发现三个磁场分量残差的均方根范围从0.07-0.37高斯缩小至0.05-0.07高斯。我们发现，该新方法对来自霍尔探头校准偏差的真实系统不确定度具有鲁棒性，且可用于显著减少构建精确模型所需的测量次数。
英文摘要
We demonstrate an iterative approach to modeling a sparsely measured magnetic field in a large-bore solenoid. This approach uses a hybrid of traditional and machine learning techniques. The traditional technique is a linear least-squares fit using a series solution to Laplace's equation, while the machine learning technique involves the training of a physics-informed neural network (PINN) on the least-squares fit residuals. We use a newly defined activation function "DELTAsnake," a modification to the snake activation function proposed by Ziyin et al. that allows for stronger curvature and non-monotonicity. The combined model approximately obeys Maxwell's equations to a level sufficient for producing high quality physics simulations and analysis. Our approach is applied to a highly realistic calculation of the expected magnetic field in the Mu2e experiment's Detector Solenoid which includes a simple model for the expected statistical measurement uncertainties. Using ten toy measurement simulations, we demonstrate the capabilities of our model in comparison to the least-squares method alone; the least-squares method alone results in a reduced chi-squared statistic of ${2.15 \pm 0.01}$, while our approach improves the reduced chi-square to ${1.034 \pm 0.005}$. Furthermore, for an average toy simulation, we show that the range of the RMS of the three field component residuals reduces from ${0.07-0.37}$ Gauss to ${0.05-0.07}$ Gauss. We find that this novel method is robust against a realistic systematic uncertainty deriving from Hall probe calibration bias and can be used to significantly reduce the number of measurements required to achieve an accurate model.
Comments54 pages, 18 figures, 6 tables; includes 19 pages of supplementary material. Submitted to APL Machine Learning
