Energy-Dissipative Evolutionary Deep Operator Neural Networks
Jiahao Zhang Shiheng Zhang Jie Shen Guang Lin
Energy-Dissipative Evolutionary Deep Operator Neural Network is an operator
learning neural network. It is designed to seed numerical solutions for a class
of partial differential equations instead of a single partial differential
equation, such as partial differential equations with different parameters or
different initial conditions. The network consists of two sub-networks, the
Branch net and the Trunk net. For an objective operator G, the Branch net
encodes different input functions u at the same number of sensors, and the
Trunk net evaluates the output function at any location. By minimizing the
error between the evaluated output q and the expected output G(u)(y), DeepONet
generates a good approximation of the operator G. In order to preserve
essential physical properties of PDEs, such as the Energy Dissipation Law, we
adopt a scalar auxiliary variable approach to generate the minimization
problem. It introduces a modified energy and enables unconditional energy
dissipation law at the discrete level. By taking the parameter as a function of
time t, this network can predict the accurate solution at any further time with
feeding data only at the initial state. The data needed can be generated by the
initial conditions, which are readily available. In order to validate the
accuracy and efficiency of our neural networks, we provide numerical
simulations of several partial differential equations, including heat
equations, parametric heat equations and Allen-Cahn equations.