• Abstract

Deep learning has achieved great success in solving partial differential equations (PDEs), where the loss is often defined as an integral. The accuracy and efficiency of these algorithms depend greatly on the quadrature method. We propose to apply quasi-Monte Carlo (QMC) methods to the Deep Ritz Method (DRM) for solving the Neumann problems for the Poisson equation and the static Schrödinger equation. For error estimation, we decompose the error of using the deep learning algorithm to solve PDEs into the generalization error, the approximation error and the training error. We establish the upper bounds and prove that QMC-based DRM achieves an asymptotically smaller error bound than DRM. Numerical experiments show that the proposed method converges faster in all cases and the variances of the gradient estimators of randomized QMC-based DRM are much smaller than those of DRM, which illustrates the superiority of QMC in deep learning over MC.

• General Summary

• Abstract

A three-level linearized difference scheme for solving the Fisher equation is firstly proposed in this work. It has the good property of discrete conservative energy. By the discrete energy analysis and mathematical induction method, it is proved to be uniquely solvable and unconditionally convergent with the second-order accuracy in both time and space. Then another three-level linearized compact difference scheme is derived along with its discrete energy conservation law, unique solvability and unconditional convergence of order two in time and four in space. The resultant schemes preserve the maximum bound principle. The analysis techniques for convergence used in this paper also work for the Euler scheme, the Crank-Nicolson scheme and others. Numerical experiments are carried out to verify the computational efficiency, conservative law and the maximum bound principle of the proposed difference schemes.

• General Summary

• Abstract

A novel dynamical model with fixed-time convergence is presented to solve the system of absolute value equations (AVEs). Under a mild condition, it is proved that the solution of the proposed dynamical system converges to the solution of the AVEs. Moreover, in contrast to the existing inversion-free dynamical system (C. Chen et al., Appl. Numer. Math. 168 (2021), 170–181), a conservative settling-time of the proposed method is given. Numerical simulations illustrate the effectiveness of the new method.

• General Summary

• Abstract

Character models have enormous applications in industry. Efficient creation of detailed character models is an important topic. This paper proposes a new and easy-to-use technique to quickly create detailed character models from sketches. The proposed technique consists of two main components: primitive deformer and shape generators. With this technique, 2D silhouette contours of a character model are drawn or extracted from an image or sketch. Then, proper geometric primitives are selected and aligned with the corresponding 2D silhouette contours. After that, a primitive deformer is used to create a base mesh and three shape generators are used to add 3D details to the base mesh. The primitive deformer and three shape generators are developed from ODE-driven deformations. The primitive deformer deforms the aligned geometric primitives to exactly match the 2D silhouette contours in one view plane and obtains a base mesh of a character model consisting of deformed primitives. The shape generators are used to add 3D details to the base mesh by creating local 3D models. The experimental results demonstrate that the new technique can quickly create detailed 3D character models from sketches with few manual operations. The new technique is physics-based and easy to learn and use.

• General Summary

• Abstract

The rigorous relations between the propagators of transient Schrödinger equations and stationary Green functions are established. Based on the generalized Fourier transform, non-singular transparent boundary condition for transient problem is proposed in a representation of Green functions. The unified framework of Green function method is presented for converting an open boundary problem into a bounded boundary problem. Numerical scheme for time-dependent Schrödinger equation with non-singular transparent boundary condition is designed to simulate the propagations of a free Gaussian wave packet and the resonant tunnelling through double barriers. Numerical results validate the effectiveness of non-singular transparent boundary condition.

• General Summary