AI 中文总结
本研究利用PINNs和DeepONets求解无穷及p-Laplace问题，处理p从2到1000的大值情形，在2D/3D域上优于传统网格求解器，并给出收敛性及通用逼近理论，数值实验验证了其有效性。
AI 中文摘要
我们研究了神经网络求解器在无穷和$p$-Laplace问题中的应用，这些问题在非线性分析中具有基础性地位，并有实际应用价值。我们的方法采用物理信息神经网络（PINNs）和深度算子网络（DeepONets）来解决与大的$p$值（范围从$2$到$1000$）相关的计算挑战，这些计算在各种二维和三维域上进行。我们的方法相比传统的基于物理的求解器具有优势，尤其是在三维情况下，对于这些问题，基于网格的求解器变得非常昂贵。我们还建立了针对这两个问题的PINN逼近的条件收敛结果，以及针对参数化$p$-Poisson问题的DeepONet的通用逼近结果。我们通过数值实验证明了这些神经网络求解器的有效性，并将其性能与传统方法进行了比较。
英文摘要
We investigate the use of neural network solvers for infinity and $p$-Laplace problems, which are fundamental in nonlinear analysis and have practical applications. Our approach employs Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) to address computational challenges associated with large $p$ values, ranging from $2$ to $1000$, on various 2D and 3D domains. Our method offers advantages over traditional physics-based solvers, especially in three dimensions where mesh-based solvers become very costly for these problems. We also establish conditional convergence results for PINN approximations of both problems and a universal approximation result for DeepONet on the parametric $p$-Poisson problem. We demonstrate the effectiveness of these neural network solvers through numerical experiments and compare their performance with conventional methods.
