Physics-informed neural networks (PINNs) have garnered significant interest for their potential in solving partial differential equations (PDEs) that govern a wide range of physical phenomena. By incorporating physical laws into the learning process, PINN models have demonstrated the ability to learn physical outcomes reasonably well. However, current PINN approaches struggle to predict or solve new PDEs effectively when there is a lack of training examples, indicating they do not generalize well to unseen problem instances. In this paper, we present a transferable learning approach for PINNs premised on a fast Pseudoinverse PINN framework (Pi-PINN). Pi-PINN learns a transferable physics-informed representation in a shared embedding space and enables rapid solving of both known and unknown PDE instances via closed-form head adaptation using a least-squares-optimal pseudoinverse under PDE constraints. We further investigate the synergies between data-driven multi-task learning loss and physics-informed loss, providing insights into the design of more performant PINNs. We demonstrate the effectiveness of Pi-PINN on various PDE problems, including Poisson's equation, Helmholtz equation, and Burgers' equation, achieving fast and accurate physics-informed solutions without requiring any data for unseen instances. Pi-PINN can produce predictions 100-1000 times faster than a typical PINN, while producing predictions with 10-100 times lower relative error than a typical data-driven model even with only two training samples. Overall, our findings highlight the potential of transferable representations with closed-form head adaptation to enhance the efficiency and generalization of PINNs across PDE families and scientific and engineering applications.

翻译：物理信息神经网络因其在求解描述广泛物理现象的偏微分方程方面的潜力而备受关注。通过将物理定律融入学习过程，PINN模型已展现出合理学习物理结果的能力。然而，当前PINN方法在训练样本不足时难以有效预测或求解新的PDE，表明其对未见问题实例的泛化能力不足。本文提出一种基于快速伪逆PINN框架的可迁移学习方法。该框架在共享嵌入空间中学习可迁移的物理信息表示，并通过基于PDE约束的最小二乘最优伪逆实现闭式头部自适应，从而快速求解已知和未知PDE实例。我们进一步研究了数据驱动多任务学习损失与物理信息损失之间的协同效应，为设计性能更优的PINN提供了 insights。我们在泊松方程、亥姆霍兹方程和伯格斯方程等多个PDE问题上验证了Pi-PINN的有效性，实现了无需任何数据即可快速准确地求解未知实例。Pi-PINN的预测速度比典型PINN快100-1000倍，即使在仅有两个训练样本的情况下，其预测相对误差也比典型数据驱动模型低10-100倍。总体而言，我们的研究结果凸显了基于闭式头部自适应的可迁移表示在提升PINN跨PDE族及科学与工程应用中的效率和泛化能力方面的潜力。