论文标题
关于线性二阶椭圆和抛物线类型PDE的物理信息的收敛性神经网络
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs
论文作者
论文摘要
物理知情的神经网络（PINN）是解决计算科学和工程中包含的偏微分方程（PDE）的深度学习技术。在数据和物理定律的指导下，Pinn找到了一个近似于PDE系统的神经网络。通过最大程度地减少对PDE和数据的任何先验知识的损失函数来获得这种神经网络。尽管在一个或三维问题上取得了显着的经验成功，但针对Pinn的理论理由很少。 随着数据数量的增加，PINN会生成一系列最小化器，这些序列对应于一系列神经网络。我们想回答以下问题：最小化器的顺序是否将解决方案融合到PDE？我们考虑两类PDE：线性二阶椭圆和抛物线。通过调整Schauder方法和最大原则，我们表明，最小化器的顺序在$ C^0 $中强烈收敛到PDE解决方案。此外，我们表明，如果每个最小化器都满足初始/边界条件，则收敛模式将变为$ H^1 $。提供了计算示例来说明我们的理论发现。据我们所知，这是显示PINNS一致性的第一部理论工作。
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.
