The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy, classical discretization methods such as the finite element methods, remains a significant challenge. Deep learning methods usually struggle to reliably decrease the error in their approximate solution. A new methodology to better control the error for deep learning methods is presented here. The main idea consists in computing an initial approximation to the problem using a simple neural network and in estimating, in an iterative manner, a correction by solving the problem for the residual error with a new network of increasing complexity. This sequential reduction of the residual of the partial differential equation allows one to decrease the solution error, which, in some cases, can be reduced to machine precision. The underlying explanation is that the method is able to capture at each level smaller scales of the solution using a new network. Numerical examples in 1D and 2D are presented to demonstrate the effectiveness of the proposed approach. This approach applies not only to physics informed neural networks but to other neural network solvers based on weak or strong formulations of the residual.

翻译：使用深度学习方法求解偏微分方程在处理若干类初边值问题时已展现出良好前景。然而，其在精度上能否超越经典离散化方法（如有限元法）仍面临重大挑战。深度学习方法通常难以可靠地降低其近似解的误差。本文提出了一种新的方法论，用于更好地控制深度学习方法中的误差。其核心思想在于：首先利用简单神经网络计算问题的初始近似解，随后通过求解残差误差——采用复杂度递增的新网络——以迭代方式估算修正量。这种对偏微分方程残差的序贯约减使得解误差得以降低，在某些情况下甚至可降至机器精度。其内在机理在于，该方法能通过新网络逐级捕获解中的更小尺度特征。本文通过一维与二维数值算例验证了所提方法的有效性。该方法不仅适用于物理信息神经网络，亦可推广至基于残差弱形式或强形式的其他神经网络求解器。