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.

翻译：能量耗散演化型深度算子神经网络是一种算子学习神经网络，旨在为一类偏微分方程（而非单个方程）生成数值解，例如不同参数或不同初始条件下的偏微分方程。该网络由两个子网络组成：分支网络（Branch net）和主干网络（Trunk net）。对于目标算子G，分支网络在相同传感器数量上编码不同的输入函数u，而主干网络在任意位置评估输出函数q。通过最小化评估输出q与期望输出G(u)(y)之间的误差，DeepONet能够生成算子G的良好近似。为保留偏微分方程的关键物理特性（如能量耗散定律），我们采用标量辅助变量方法构建最小化问题。该方法引入修正能量，并在离散层面实现无条件能量耗散律。通过将参数视为时间t的函数，该网络仅需在初始状态输入数据，即可预测任意后续时刻的精确解。所需数据可由初始条件直接生成，易于获取。为验证神经网络的准确性与效率，我们提供了多个偏微分方程的数值模拟实例，包括热方程、参数化热方程和Allen-Cahn方程。