Physics-Informed Neural Network (PINN) has proven itself a powerful tool to obtain the numerical solutions of nonlinear partial differential equations (PDEs) leveraging the expressivity of deep neural networks and the computing power of modern heterogeneous hardware. However, its training is still time-consuming, especially in the multi-query and real-time simulation settings, and its parameterization often overly excessive. In this paper, we propose the Generative Pre-Trained PINN (GPT-PINN) to mitigate both challenges in the setting of parametric PDEs. GPT-PINN represents a brand-new meta-learning paradigm for parametric systems. As a network of networks, its outer-/meta-network is hyper-reduced with only one hidden layer having significantly reduced number of neurons. Moreover, its activation function at each hidden neuron is a (full) PINN pre-trained at a judiciously selected system configuration. The meta-network adaptively ``learns'' the parametric dependence of the system and ``grows'' this hidden layer one neuron at a time. In the end, by encompassing a very small number of networks trained at this set of adaptively-selected parameter values, the meta-network is capable of generating surrogate solutions for the parametric system across the entire parameter domain accurately and efficiently.

翻译：物理信息神经网络（Physics-Informed Neural Network, PINN）已被证明是一种利用深度神经网络的表达能力和现代异构硬件的计算能力来获取非线性偏微分方程（PDEs）数值解的有力工具。然而，其训练仍然耗时，尤其是在多查询和实时仿真场景下，且参数化往往过于冗余。本文提出生成式预训练物理信息神经网络（Generative Pre-Trained PINN, GPT-PINN），以缓解参数化PDEs场景中的这两大挑战。GPT-PINN代表了一种针对参数化系统的新颖元学习范式。作为一种网络之网络，其外部/元网络采用超简化结构，仅包含一个隐藏层且神经元数量显著减少。此外，每个隐藏神经元的激活函数是一个（完整）PINN，该PINN在精心选择的系统配置上预训练而成。元网络自适应地“学习”系统的参数依赖性，并逐次“生长”该隐藏层的神经元。最终，通过包含极少数在该自适应选择的参数集上训练的网络，元网络能够高效且精确地生成跨整个参数域的参数化系统的替代解。