APTNN 方法求解 3D 高场动理学方程，单 GPU 训练不到 30 分钟
Asymptotic-Preserving Tensor Neural Networks for 3D High-Field Kinetic Equations with Applications to Uncertainty Quantification
数值方法论文，用张量网络 CP 分解把高维动理学方程的成本从指数压到多项式，3D3V 例子单 GPU 半小时训完，做等离子体或半导体仿真模拟的可以看看。
论文提出渐近保持张量神经网络（APTNN），用于求解高场区间的 Vlasov–Poisson–Fokker–Planck（VPFP）和 Boltzmann–Poisson 等动理学方程。方法采用张量神经网络（TNN）的 canonical polyadic（CP）分解结构，将高维损失积分化为一维积分组合，使计算与存储成本随维度从指数级降至多项式级。在基准 AP 损失基础上加入一阶矩方程损失项，直接约束极限通量封闭，降低高场区间的预测误差。在含多个随机输入的 3D3V VPFP 问题上，单 GPU 训练用时不到 30 分钟。
Asymptotic-Preserving Tensor Neural Networks for 3D High-Field Kinetic Equations with Applications to Uncertainty Quantification
Kinetic equations in the high-field regime describe particle transport dominated by electric-field acceleration and collisional relaxation. These equations are widely used in simulations of plasmas and semiconductor devices. Typical models include the Vlasov--Poisson--Fokker--Planck (VPFP) and Boltzmann--Poisson systems. Their high dimensionality and multiscale nature pose two of the major challenges to accurate and efficient numerical solution. To address these challenges, we propose an asymptotic-preserving tensor neural network (APTNN) method. Specifically, to reduce high-dimensional computational costs and memory requirements, we adopt the canonical polyadic (CP) decomposition structure of TNNs to approximate their solutions. We show that the equation residuals retain this decomposition form, allowing high-dimensional loss integrals to be evaluated through combinations of one-dimensional integrals. The resulting computational and storage costs are reduced from exponential to polynomial growth with dimension. To improve the prediction accuracy in the high-field limit, we enhance the baseline AP loss by incorporating the first moment equation. The added loss term directly constrains the limiting flux closure to reduce its approximation error in the high-field regime. The enhanced AP loss can still be evaluated dimension-wise. Numerical results show that APTNN improves prediction accuracy in the high-field regime with significantly reduced computational costs and memory usage. For a 3D3V VPFP problem with multiple random inputs, training takes less than 30 minutes on a single GPU.