Following the success in machine learning tasks, neural network (NN) based methods are starting to show promise results in classical problems of applied mathematics. In this talk, we introduce two important recent approaches for solving partial differential equations: physics-informed neural networks (PINNs) and neural operators. PINNs employ NNs as approximators for the solution function to a PDE. Alternatively, neural operators aim at use NNs as the ansatz of the solution operator for a family of PDEs. Specifically, the neural operator generalizes the conventional NNs to the operator between infinite-dimensional function spaces. We contrast their ability and suitability for complex PDEs, and then introduce various applications.
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