Solving DCIS problem with free boundary conditions based on physical information neural network(基于物理信息神经网络求解带自由边界条件的DCIS问题)

Solving the ductal carcinoma in situ (DCIS) problem with free boundary conditions can better reveal the changing law of Ductal Carcinoma in situ, mean while reducing the cost of experiments and improving the accuracy of numerical results. This paper exploits physics-informed neural networks (PINNs) to solve the problem of ductal carcinoma in situ with free boundary conditions. We construct two independent deep neural networks to respectively approximate the nutrient concentration and the free boundary, the loss function is represented by the residuals of the partial differential equations, initial and boundary conditions, and solved using the Adam optimization algorithm. In order to verify the accuracy and computational efficiency of the solution of the free boundary problem by PINNs, numerical simulation was performed. The results not only show the effectiveness of PINNs in solving the problem, but also verify that the numerical solution agrees well with the exact solution.(求解带自由边界条件的原位管癌（ductal carcinoma in situ，DCIS）问题有助于深入揭示DCIS的变化规律，降低试验成本，提高数值结果的精度。本文基于物理信息神经网络（physics-informed neural networks，PINNs）求解带自由边界条件的DCIS正问题，通过构建2个独立的深度神经网络分别近似模拟营养物质浓度和自由边界，损失函数由偏微分方程、初始条件和边界条件的残差组成，用Adam优化算法进行求解，并通过数值模拟验证求解精度和计算效率，结果表明，数值解与精确解吻合较好，PINNs能有效求解DCIS正问题，且在处理复杂边界条件时具有优势。)