pETNNs: Partial Evolutionary Tensor Neural Networks for Solving Time-dependent Partial Differential Equations

TL;DR

pETNNs address the challenge of solving time-dependent PDEs in high dimensions by marrying tensor neural networks with evolutionary parametric approximation. A Dirac-Frenkel variational principle yields ODEs that evolve the time-dependent network parameters, while a partial update strategy and a posterior error bound enable efficient, extrapolative predictions. The method demonstrates superior accuracy over EDNNs and robust extrapolation on problems including incompressible Navier–Stokes, high-dimensional heat and transport equations, and high-order dispersive PDEs. These results suggest a scalable, mesh-free approach for high-dimensional time-dependent simulations with potential cross-disciplinary impact.

Abstract

We present the 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 evolutionary 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, 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 time-dependent complex equations, including the incompressible Navier-Stokes equations, high-dimensional heat equations, high-dimensional transport equations, and dispersive equations of higher-order derivatives.

Figures (12)

• Figure 1: Schematic diagram of the tensor neural networks (TNNs). If the hidden layers of each sub-network is of width $M$ and depth $N$, then the scale of the total free parameters is $O(M^2Nd + Mpd)$, which grows linearly in terms of the problem dimension $d$.
• Figure 2: Schematic diagram of the partial evolutionary tensor neural networks (pETNNs). The thick red line represents the updated parameters and other parameters remain unchanged. The embedding layer is used to ensure the boundary conditions.
• Figure 3: The averaged absolute errors of 3D transport equations over independent runs of Algorithm \ref{['algorithm2']}. We updated 1/3 of the parameters randomly per iteration. The four error curves correspond to four different initial $\theta_0$ configurations.
• Figure 4: Averaged absolute errors of updating 1/3 of the parameters randomly per iteration: 50 (left) and 500 (right) time units. $"vanilla"$ represents random update in every time step; $"w/~first~layer"$ represents that the inclusion of the update parameters in the first layer is mandatory; $"modified~ Euler"$ represents predictor-corrector (modified-Euler) method, and $"RK4"$ represents explicit fourth-order Runge-Kutta method.
• Figure 5: Averaged absolute errors of different update strategies. $"w/~first~layer"$ represents that the inclusion of the update parameters in the first layer is mandatory, and $1/3(1/2)$ represents that only $1/3(1/2)$ of parameters are updated at each time step.

Theorems & Definitions (3)

• Remark 2.1
• Proposition 2.1
• proof