Adaptive trajectories sampling for solving PDEs with deep learning methods

In this paper, we propose a new adaptive technique, named adaptive trajectories sampling (ATS), which is used to select training points for the numerical solution of partial differential equations (PDEs) with deep learning methods. The key feature of the ATS is that all training points are adaptively selected from trajectories that are generated according to a PDE-related stochastic process. We incorporate the ATS into three known deep learning solvers for PDEs, namely the adaptive derivative-free-loss method (ATS-DFLM), the adaptive physics-informed neural network method (ATS-PINN), and the adaptive temporal-difference method for forward-backward stochastic differential equations (ATS-FBSTD). Our numerical experiments demonstrate that the ATS remarkably improves the computational accuracy and efficiency of the original deep learning solvers for the PDEs. In particular, for some specific high-dimensional PDEs, the ATS can even improve the accuracy of the PINN by two orders of magnitude.Comment: 18 pages, 12 figures, 42 reference

A C0C^0 Linear Finite Element Method for a Second Order Elliptic Equation in Non-Divergence Form with Cordes Coefficients

In this paper, we develop a gradient recovery based linear (GRBL) finite element method (FEM) and a Hessian recovery based linear (HRBL) FEM for second order elliptic equations in non-divergence form. The elliptic equation is casted into a symmetric non-divergence weak formulation, in which second order derivatives of the unknown function are involved. We use gradient and Hessian recovery operators to calculate the second order derivatives of linear finite element approximations. Although, thanks to low degrees of freedom (DOF) of linear elements, the implementation of the proposed schemes is easy and straightforward, the performances of the methods are competitive. The unique solvability and the $H^2$ seminorm error estimate of the GRBL scheme are rigorously proved. Optimal error estimates in both the $L^2$ norm and the $H^1$ seminorm have been proved when the coefficient is diagonal, which have been confirmed by numerical experiments. Superconvergence in errors has also been observed. Moreover, our methods can handle computational domains with curved boundaries without loss of accuracy from approximation of boundaries. Finally, the proposed numerical methods have been successfully applied to solve fully nonlinear Monge-Amp\`{e}re equations