This work aims to solve large-scale variational inequalities (VIs), which are equivalent to high-dimensional systems of ordinary differential equations (ODEs). The existing physics-informed neural network (PINN) approach (Wu and Lisser, 2023) shows superior performance for VIs with less than 1000 variables, but fails for VIs of larger size, due to the increasing number of equations and the requirement of an extensive time interval. To overcome this limitation, we present two algorithms that dynamically adjust the initial condition for the PINN. The first algorithm uses multiple PINNs sequentially to decompose the task, where the best prediction from the current PINN serves as the initial condition for the next PINN. The second algorithm uses a single PINN throughout the solution process, immediately taking any improved prediction as an initial condition and refining the PINN to achieve a better prediction. Finally, we demonstrate the effectiveness of the proposed algorithms on a number of large-scale VI problems with up to 100,000 variables.

中文翻译：

---

通过动态调整物理信息神经网络中的初始条件来解决大规模变分不等式


这项工作旨在解决大规模变分不等式（VI），其相当于高维常微分方程（ODE）系统。现有的物理信息神经网络 (PINN) 方法（Wu 和 Lisser，2023）对于变量少于 1000 个的 VI 表现出优异的性能，但由于方程数量不断增加以及需要大量时间间隔。为了克服这个限制，我们提出了两种动态调整 PINN 初始条件的算法。第一个算法依次使用多个 PINN 来分解任务，其中当前 PINN 的最佳预测作为下一个 PINN 的初始条件。第二种算法在整个求解过程中使用单个 PINN，立即将任何改进的预测作为初始条件并改进 PINN 以实现更好的预测。最后，我们证明了所提出的算法在许多具有多达 100,000 个变量的大规模 VI 问题上的有效性。