Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically, to converge to the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in solving the highly non-convex optimization problems resulting from the neural network discretization, which are difficult to treat both theoretically and practically. It is our goal in this work to take a step toward remedying this. For this purpose, we develop a novel greedy training algorithm for shallow neural networks. Our method is applicable to both the variational formulation of the PDE and also to the residual minimization formulation pioneered by physics informed neural networks (PINNs). We analyze the method and obtain a priori error bounds when solving PDEs from the function class defined by shallow networks, which rigorously establishes the convergence of the method as the network size increases. Finally, we test the algorithm on several benchmark examples, including high dimensional PDEs, to confirm the theoretical convergence rate. Although the method is expensive relative to traditional approaches such as finite element methods, we view this work as a proof of concept for neural network-based methods, which shows that numerical methods based upon neural networks can be shown to rigorously converge.

翻译：近年来，神经网络被广泛应用于求解偏微分方程（PDEs）。尽管这类方法在实际工程问题中取得了显著成功，但理论上或经验上均未证明其能任意高精度地收敛至PDE的精确解。主要困难在于神经网络离散化所产生的高度非凸优化问题，这类问题在理论和实践层面均难以处理。本工作的目标正是为解决这一难题迈出关键一步。为此，我们提出了一种适用于浅层神经网络的新型贪婪训练算法。该方法既可应用于PDE的变分形式，也可应用于物理信息神经网络（PINNs）开创的残差最小化框架。我们对该方法进行了分析，并在浅层网络定义的函数类中求解PDE时，得到了先验误差界，严格证明了方法随网络规模增大时的收敛性。最后，我们在多个基准算例（包括高维PDE）上测试了算法，验证了理论收敛速率。虽然该方法相较于有限元等传统方法计算成本较高，但我们将其视为神经网络方法的一个概念验证，证明基于神经网络的数值方法可以实现严格的收敛。