A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

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 operator 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.

Energy stable neural network for gradient flow equations

In this paper, we propose an energy stable network (EStable-Net) for solving gradient flow equations. The solution update scheme in our neural network EStable-Net is inspired by a proposed auxiliary variable based equivalent form of the gradient flow equation. EStable-Net enables decreasing of a discrete energy along the neural network, which is consistent with the property in the evolution process of the gradient flow equation. The architecture of the neural network EStable-Net consists of a few energy decay blocks, and the output of each block can be interpreted as an intermediate state of the evolution process of the gradient flow equation. This design provides a stable, efficient and interpretable network structure. Numerical experimental results demonstrate that our network is able to generate high accuracy and stable predictions.