Solving nonlinear partial differential equations (PDEs) with multiple solutions using neural networks has found widespread applications in various fields such as physics, biology, and engineering. However, classical neural network methods for solving nonlinear PDEs, such as Physics-Informed Neural Networks (PINN), Deep Ritz methods, and DeepONet, often encounter challenges when confronted with the presence of multiple solutions inherent in the nonlinear problem. These methods may encounter ill-posedness issues. In this paper, we propose a novel approach called the Newton Informed Neural Operator, which builds upon existing neural network techniques to tackle nonlinearities. Our method combines classical Newton methods, addressing well-posed problems, and efficiently learns multiple solutions in a single learning process while requiring fewer supervised data points compared to existing neural network methods.

翻译：利用神经网络求解具有多解的非线性偏微分方程（PDEs）在物理、生物和工程等多个领域已获得广泛应用。然而，在应对非线性问题固有的多解性时，经典神经网络方法如物理信息神经网络（PINN）、深度Ritz方法及DeepONet等常面临挑战，可能遭遇不适定性问题。本文提出一种称为牛顿信息神经算子的新方法，该方法在现有神经网络技术基础上构建，以处理非线性问题。我们的方法结合了经典牛顿法（适用于适定问题），能够在单次学习过程中高效获取多个解，且相较于现有神经网络方法需要更少的监督数据点。