标题： 用于低级别脑肿瘤正向和反向建模的物理信息神经网络 显示英文标题

标题： Physics informed neural network for forward and inverse modeling of low grade brain tumors

摘要： 低级别肿瘤是一种生长较慢的肿瘤，与高级别肿瘤相比，其扩散的可能性较低。 使用偏微分方程（PDEs）的数学模型在描述肿瘤行为、生长和进展中起着至关重要的作用。 本研究采用 Burgess 和扩展的 Fisher Kolmogorov 方程来模拟低级别脑肿瘤。 我们利用基于物理信息神经网络（PINNs）的算法开发这些模型的自动数值求解器，并探索其在解决脑肿瘤建模中的正问题和反问题的应用。 本研究旨在证明基于 PINN 的算法通过将深度学习与物理信息原理相结合，作为建模脑肿瘤动力学的先进方法。 此外，我们建立了以训练误差和求积误差为基准的广义误差界限。 对于两种模型，推导了神经网络的收敛性和稳定性。 数值测试证实了算法在线性和非线性情况下的准确性和效率。 此外，还提供了数值结果的统计分析。 显示英文摘要

摘要： A low grade tumor is a slow growing tumor with a lower likelihood of spreading compared to high grade tumors. Mathematical modeling using partial differential equations (PDEs) plays a crucial role in describing tumor behavior, growth and progression. This study employs the Burgess and extended Fisher Kolmogorov equations to model low-grade brain tumors. We utilize Physics Informed Neural Networks (PINNs) based algorithm to develop an automated numerical solver for these models and explore their application in solving forward and inverse problems in brain tumor modeling. The study aims to demonstrate that the PINN based algorithms serve as advanced methodologies for modeling brain tumor dynamics by integrating deep learning with physics-informed principles. Additionally, we establish generalized error bounds in terms of training and quadrature errors. The convergence and stability of the neural network are derived for both models. Numerical tests confirm the accuracy and efficiency of the algorithms in both linear and nonlinear cases. Additionally, a statistical analysis of the numerical results is presented.