Physics-informed neural networks (PINNs) are a newly emerging research frontier in machine learning, which incorporate certain physical laws that govern a given data set, e.g., those described by partial differential equations (PDEs), into the training of the neural network (NN) based on such a data set. In PINNs, the NN acts as the solution approximator for the PDE while the PDE acts as the prior knowledge to guide the NN training, leading to the desired generalization performance of the NN when facing the limited availability of training data. However, training PINNs is a non-trivial task largely due to the complexity of the loss composed of both NN and physical law parts. In this work, we propose a new PINN training framework based on the multi-task optimization (MTO) paradigm. Under this framework, multiple auxiliary tasks are created and solved together with the given (main) task, where the useful knowledge from solving one task is transferred in an adaptive mode to assist in solving some other tasks, aiming to uplift the performance of solving the main task. We implement the proposed framework and apply it to train the PINN for addressing the traffic density prediction problem. Experimental results demonstrate that our proposed training framework leads to significant performance improvement in comparison to the traditional way of training the PINN.

翻译：物理信息神经网络（PINNs）是机器学习领域新兴的研究前沿，其将支配特定数据集的物理定律（例如由偏微分方程描述的规律）融入基于该数据集训练神经网络的进程中。在PINNs中，神经网络充当偏微分方程的解近似器，而偏微分方程则为引导神经网络训练的先验知识，从而使神经网络在训练数据有限的情况下仍具有理想的泛化性能。然而，由于损失函数同时包含神经网络项与物理定律项所带来的复杂性，训练PINNs是一项具有挑战性的任务。本文提出一种基于多任务优化（MTO）范式的新型PINN训练框架。在该框架下，创建多个辅助任务并与给定（主）任务协同求解，通过自适应模式迁移求解某一任务所获得的有用知识来辅助求解其他任务，旨在提升主任务的求解性能。我们实现了所提出的框架，并将其应用于训练用于交通密度预测问题的PINN。实验结果表明，相较于传统PINN训练方式，本训练框架能显著提升性能。