Networks For Solving Research Articles

PHYSICS-INFORMED ENCODER-DECODER GATED RECURRENT NEURAL NETWORK FOR SOLVING TIME-DEPENDENT PDES

The ability of deep-learning approaches to approximate complex functions offers a promising alternative for solving partial differential equations (PDEs). The methodology of incorporating the physics prior to the deep neural network can significantly reduce the requirement for labeled data. In this study, a novel physics-informed encoder-decoder gated recurrent units neural network is proposed to solve the time-dependent PDEs without using any observed data. The encoder is utilized to approximate the underlying patterns and structures of solutions over the entire spatial-temporal domain. The approximated solution is processed by the decoder, which is the gated recurrent units layer. We utilize the initial condition as the initial state of the gated recurrent units to retain critical information in the hidden states. The boundary conditions are enforced in the final prediction to enhance the model's performance. Then, we incorporate physical laws into the neural network during the training process. The effectiveness of this algorithm is demonstrated by solving Burgers-Fisher and coupled two-dimensional burgers' equations. The ability to identify unknown parameters is demonstrated through the solution of inverse problems.

PHYSICS-INFORMED NEURAL NETWORK FOR SOLVING HAUSDORFF DERIVATIVE POISSON EQUATIONS

This paper proposed a new physics-informed neural network (PINN) for solving the Hausdorff derivative Poisson equations (HDPEs) on irregular domains by using the concept of Hausdorff fractal derivative. The present scheme transforms the numerical solution of partial differential equation into an optimization problem including governing equation and boundary conditions. Like the meshless method, the developed PINN does not require grid generation and numerical integration. Moreover, it can freely address irregular domains and non-uniformly distributed nodes. The present study investigated different activation functions, and given an optimal choice in solving the HDPEs. Compared to other existing approaches, the PINN is simple, straightforward, and easy-to-program. Numerical experiments indicate that the new methodology is accurate and effective in solving the HDPEs on arbitrary domains, which provides a new idea for solving fractal differential equations.

A NOVEL RECURRENT NEURAL NETWORK FOR SOLVING MLCPs AND ITS APPLICATION TO LINEAR AND QUADRATIC PROGRAMMING PROBLEMS

In this paper, we present a recurrent neural network for solving mixed linear complementarity problems (MLCPs) with positive semi-definite matrices. The proposed neural network is derived based on an NCP function and has a low complexity respect to the other existing models. In theoretical and numerical aspects, global convergence of the proposed neural network is proved. As an application, we show that the proposed neural network can be used to solve linear and convex quadratic programming problems. The validity and transient behavior of the proposed neural network are demonstrated by using five numerical examples.