The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.

翻译：在求解刚性非线性时间演化方程的数值近似解时，采用隐式时间步进方案具有众所周知的优势，通常包括更好的稳定性表现、相应支持更大的时间步长以及更好的结构保持特性。然而，其代价是在数值方案的每个时间步都必须求解一个非线性方程。在本工作中，我们提出了一种新颖的基于深度学习的混合牛顿法，以加速刚性时间演化非线性方程的非线性时间步系统的求解。我们提出了一种有针对性的学习策略，该策略便于在离线阶段进行稳健的无监督学习，并为牛顿迭代提供高效的初始化，从而实现牛顿法的一致加速。我们量化了通过改进初始化所实现的牛顿法改进速率，并分析了无监督学习策略泛化误差的上界。这些理论结果得到了大量数值结果的支持，证明了我们提出的神经混合求解器在一维和二维情况下的高效性。