An encoder–decoder neural network has been used to examine the possibility for acceleration of a partial integro-differential equation, the Fokker–Planck–Landau collision operator. This is part of the governing equation in the massively parallel particle-in-cell code XGC, which is used to study turbulence in fusion energy devices. The neural network emphasizes physics-inspired learning, where it is taught to respect physical conservation constraints of the collision operator by including them in the training loss, along with the $\ell _2$ loss. In particular, network architectures used for the computer vision task of semantic segmentation have been used for training. A penalization method is used to enforce the ‘soft’ constraints of the system and integrate error in the conservation properties into the loss function. During training, quantities representing the particle density, momentum and energy for all species of the system are calculated at each configuration vertex, mirroring the procedure in XGC. This simple training has produced a median relative loss, across configuration space, of the order of $10^{-4}$ , which is low enough if the error is of random nature, but not if it is of drift nature in time steps. The run time for the current Picard iterative solver of the operator is $O(n^2)$ , where $n$ is the number of plasma species. As the XGC1 code begins to attack problems including a larger number of species, the collision operator will become expensive computationally, making the neural network solver even more important, especially since its training only scales as $O(n)$ . A wide enough range of collisionality has been considered in the training data to ensure the full domain of collision physics is captured. An advanced technique to decrease the losses further will be subject of a subsequent report. Eventual work will include expansion of the network to include multiple plasma species.

中文翻译：

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用于求解 XGC 中非线性 Fokker-Planck-Landau 碰撞算子的编码器-解码器神经网络

编码器-解码器神经网络已被用于检查加速部分积分-微分方程的可能性，即 Fokker-Planck-Landau 碰撞算子。这是大规模平行粒子细胞代码 XGC 中控制方程的一部分，用于研究聚变能量装置中的湍流。神经网络强调受物理启发的学习，它通过将碰撞算子包括在训练损失中来尊重碰撞算子的物理守恒约束，以及 $\ell _2$ 失利。特别是，用于语义分割的计算机视觉任务的网络架构已被用于训练。惩罚方法用于强制执行系统的“软”约束，并将守恒属性中的误差整合到损失函数中。在训练期间，在每个配置顶点计算代表系统所有物种的粒子密度、动量和能量的数量，反映 XGC 中的过程。这种简单的训练已经产生了跨配置空间的中值相对损失，大约为 $10^{-4}$ ，如果误差是随机的，它就足够低，但如果它在时间步长上具有漂移性质，则不是。算子当前 Picard 迭代求解器的运行时间为 $O(n^2)$ ， 在哪里 $n$ 是等离子体种类的数量。随着 XGC1 代码开始攻击包括大量物种在内的问题，碰撞算子的计算量将变得昂贵，这使得神经网络求解器变得更加重要，特别是因为它的训练只缩放为 $O(n)$ . 在训练数据中考虑了足够广泛的碰撞性，以确保捕获碰撞物理的整个领域。进一步减少损失的先进技术将成为后续报告的主题。最终的工作将包括扩大网络以包括多种等离子体物种。