CoPhy-PGNN：学习物理引导的神经网络与竞争损失函数求解特征值问题
CoPhy-PGNN: Learning Physics-guided Neural Networks with Competing Loss Functions for Solving Eigenvalue Problems
摘要：
物理引导神经网络（PGNNs）代表了一类新兴的神经网络，它们使用物理引导（PG）损失函数（捕获具有已知物理的网络输出中的违规）以及数据中包含的监督进行训练。PGNNs中的现有工作已经证明了在神经网络目标中添加单个PG损失函数的有效性，使用恒定的权衡参数，以确保更好的泛化能力。然而，在存在具有竞争梯度方向的多个PG函数的情况下，需要在训练过程期间自适应地调整不同PG损失函数的贡献以达到可推广的解决方案。我们证明了存在竞争PG损失的通用神经网络的问题，解决最低（或最高）的特征向量的物理为基础的特征值方程，这是常见的许多科学问题。我们提出了一种新的方法来处理竞争PG损失，并证明其有效性，在两个激励应用量子力学和电磁传播的学习可推广的解决方案。本工作中使用的所有代码和数据都可以在https://github.com/jayroxis/Cophy-PGNN上获得。
英文摘要：
Physics-guided Neural Networks (PGNNs) represent an emerging class of neural networks that are trained using physics-guided (PG) loss functions (capturing violations in network outputs with known physics), along with the supervision contained in data. Existing work in PGNNs has demonstrated the efficacy of adding single PG loss functions in the neural network objectives, using constant trade-off parameters, to ensure better generalizability. However, in the presence of multiple PG functions with competing gradient directions, there is a need to adaptively tune the contribution of different PG loss functions during the course of training to arrive at generalizable solutions. We demonstrate the presence of competing PG losses in the generic neural network problem of solving for the lowest (or highest) eigenvector of a physics-based eigenvalue equation, which is commonly encountered in many scientific problems. We present a novel approach to handle competing PG losses and demonstrate its efficacy in learning generalizable solutions in two motivating applications of quantum mechanics and electromagnetic propagation. All the code and data used in this work is available at https://github.com/jayroxis/Cophy-PGNN.
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