Tensorial neural networks (TNNs) combine the successes of multilinear algebra with those of deep learning to enable extremely efficient reduced-order models of high-dimensional problems. Here, I describe a deep neural network architecture that fuses multiple TNNs into a larger network, intended to solve a broader class of problems than a single TNN. I evaluate this architecture, referred to as a "stacked tensorial neural network" (STNN), on a parametric PDE with three independent variables and three parameters. The three parameters correspond to one PDE coefficient and two quantities describing the domain geometry. The STNN provides an accurate reduced-order description of the solution manifold over a wide range of parameters. There is also evidence of meaningful generalization to parameter values outside its training data. Finally, while the STNN architecture is relatively simple and problem agnostic, it can be regularized to incorporate problem-specific features like symmetries and physical modeling assumptions.

翻译：张量神经网络（TNNs）融合了多重线性代数与深度学习的优势，能够高效实现高维问题的降阶建模。在此，我描述了一种深度神经网络架构，该架构将多个TNN融合成一个更大的网络，旨在解决比单个TNN更广泛的问题类别。我评估了这种被称为"堆叠张量神经网络"（STNN）的架构，将其应用于一个具有三个独立变量和三个参数的参数化偏微分方程。这三个参数分别对应一个PDE系数和两个描述域几何的量。STNN能在广泛的参数范围内对解流形提供精确的降阶描述。同时，有证据表明它对训练数据范围之外的参数值也具备有意义的泛化能力。最后，尽管STNN架构相对简单且与具体问题无关，但它可以通过正则化来融入特定问题的特征，如对称性和物理建模假设。