We introduce the Transformed Generative Pre-Trained Physics-Informed Neural Networks (TGPT-PINN) for accomplishing nonlinear model order reduction (MOR) of transport-dominated partial differential equations in an MOR-integrating PINNs framework. Building on the recent development of the GPT-PINN that is a network-of-networks design achieving snapshot-based model reduction, we design and test a novel paradigm for nonlinear model reduction that can effectively tackle problems with parameter-dependent discontinuities. Through incorporation of a shock-capturing loss function component as well as a parameter-dependent transform layer, the TGPT-PINN overcomes the limitations of linear model reduction in the transport-dominated regime. We demonstrate this new capability for nonlinear model reduction in the PINNs framework by several nontrivial parametric partial differential equations.

翻译：我们提出变换生成式预训练物理信息神经网络（TGPT-PINN），在融合模型降阶的PINN框架中实现输运主导型偏微分方程的非线性模型降阶。基于近期发展的GPT-PINN（一种实现基于快照模型降阶的网络嵌套网络设计），我们设计并验证了一种能够有效处理含参数依赖不连续性问题的新型非线性模型降阶范式。通过引入激波捕捉损失函数分量及参数依赖变换层，TGPT-PINN克服了线性模型降阶在输运主导问题中的局限性。我们通过多个非平凡参数化偏微分方程实例，展示了PINN框架下非线性模型降阶的这一新能力。