We present partial evolutionary tensor neural networks (pETNNs), a novel framework for solving time-dependent partial differential equations with high accuracy and capable of handling high-dimensional problems. Our architecture incorporates tensor neural networks and evolutional parametric approximation. A posterior error bounded is proposed to support the extrapolation capabilities. In the numerical implementations, we adopt a partial update strategy to achieve a significant reduction in computational cost while maintaining precision and robustness. Notably, as a low-rank approximation method of complex dynamical systems, the pETNNs enhance the accuracy of evolutional deep neural networks and empowers computational abilities to address high-dimensional problems. Numerical experiments demonstrate the superior performance of the pETNNs in solving time-dependent complex equations, including the incompressible Navier-Stokes equations, high-dimensional heat equations, highdimensional transport equations, and dispersive equations of higher-order derivatives.

翻译：本文提出部分演化张量神经网络（pETNNs），这是一种用于求解时间依赖偏微分方程的新型框架，具有高精度且能处理高维问题。该架构融合了张量神经网络与演化参数逼近方法。我们提出了后验误差界以支持其外推能力。在数值实现中，我们采用部分更新策略，在保持精度与鲁棒性的同时显著降低计算成本。值得注意的是，作为复杂动力系统的低秩逼近方法，pETNNs提升了演化深度神经网络的精度，并增强了处理高维问题的计算能力。数值实验表明，pETNNs在求解时间依赖复杂方程方面具有优越性能，包括不可压缩Navier-Stokes方程、高维热方程、高维输运方程以及高阶导数色散方程。