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.

张量神经网络 (TNNs) 结合了多线性代数和深度学习的成果，能够实现高维问题的高效降维模型。本文描述了一种将多个 TNNs 融合成一个更大网络的深度神经网络架构，旨在解决比单个 TNN 更广泛的问题类别。作者在具有三个独立变量和三个参数的参数型 PDE 上评估了该架构，其中三个参数分别对应一个 PDE 系数和描述域几何的两个量。STNN 在参数范围内提供了准确的降维解描述，并显示出对训练数据之外的参数值有意义的泛化证据。最后，尽管 STNN 架构相对简单且不依赖具体问题，但它可通过正则化来融合特定问题的特征，如对称性和物理建模假设。