Higher-order error estimates for physics-informed neural networks approximating the primitive equations

Error estimates for obstacle problems of higher order

For obstacle problems of higher order involving power growth functionals we prove a posteriori error estimates using methods from duality theory. These estimates can be seen as a reliable measure for the deviation of an approximation from the exact solution being independent of the concrete numerical scheme under consideration.

Blow-up Estimates for Higher-order Semilinear Parabolic Equations

Higher Order Neural Networks and Neural Networks for Stream Learning

The goal of this thesis is to explore some variations of neural networks. The thesis is mainly split into two parts: a variation of the shaping functions in neural networks and a variation of learning rules in neural networks. In the first part, we mainly investigate polynomial perceptrons a perceptron with a polynomial shaping function instead of a linear one. We prove the polynomial perceptro...

Higher Order Recurrent Neural Networks

In this paper, we study novel neural network structures to better model long term dependency in sequential data. We propose to use more memory units to keep track of more preceding states in recurrent neural networks (RNNs), which are all recurrently fed to the hidden layers as feedback through different weighted paths. By extending the popular recurrent structure in RNNs, we provide the models...

High order quadrature based iterative method for approximating the solution of nonlinear equations

In this paper, weight function and composition technique is utilized to speeds up the convergence order and increase the efficiency of an existing quadrature based iterative method. This results in the proposition of its improved form from a two-point quadrature based method of convergence order ρ = 3 with efficiency index EI = 1:3161 to a three-point method of convergence order ρ = 8 with EI =...