TY - JOUR T1 - pETNNs: Partial Evolutionary Tensor Neural Networks for Solving Time-Dependent Partial Differential Equations AU - Kao , Tunan AU - Zhang , He AU - Zhang , Lei AU - Zhao , Jin JO - CSIAM Transactions on Applied Mathematics VL - 4 SP - 862 EP - 891 PY - 2025 DA - 2025/09 SN - 6 DO - http://doi.org/10.4208/csiam-am.SO-2024-0048 UR - https://global-sci.org/intro/article_detail/csiam-am/24505.html KW - Time-dependent partial differential equations, tensor neural networks, evolutionary deep neural networks, high-dimensional problems. AB -

We present partial evolutionary tensor neural networks (pETNNs), a novel approach for solving time-dependent partial differential equations with high accuracy and capable of handling high-dimensional problems. Our architecture incorporates tensor neural networks and evolutionary parametric approximation. A posteriori error bound is proposed to support the extrapolation capabilities. In 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, pETNNs enhance the accuracy of evolutionary deep neural networks and empower computational abilities to address high-dimensional problems. Numerical experiments demonstrate the superior performance of the pETNNs in solving complex time-dependent equations, including the incompressible Navier-Stokes equations, high-dimensional heat equations, high-dimensional transport equations, and dispersive equations of higher-order derivatives.