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Physics informed neural network (PINN) is a new deep learning paradigm, which embeds the physical information delineated by PDEs in the loss function and optimizes the weights in the neural network. Based on PINN, an extended PINN(E-PINN) is proposed, which is a mixture of the polynomial function approximation method and PINN's learning framework. A preprocess layer is added before the classical PINN, using Legendre polynomials as the polynomial basis function. Therefore, E-PINN not only has the excellent approximation ability of the polynomial basis function, but also inherits the learning framework of the neural network method. In numerical experiments, the proposed E-PINN algorithms have high accuracy in solving 1D, 2D high-order nonlinear Fredholm equations and equations system, including the forward and inverse problems. [ABSTRACT FROM AUTHOR]

In molecular solvation, the size-modified electrostatic free energy is investigated. With the uniform ionic and solvent molecular sizes, the generalized Poisson–Boltzmann (PB) equation is considered. The first variation of the size-modified electrostatic free energy with respect to the location variation of the interface is derived. The concept of shape derivative is used to define such variations. The explicit formula of the dielectric boundary force is derived. [ABSTRACT FROM AUTHOR]