In this work, we propose a novel layerwise adaptive construction method for neural network architectures. Our approach is based on a goal--oriented dual-weighted residual technique for the optimal control of neural differential equations. This leads to an ordinary differential equation constrained optimization problem with controls acting as coefficients and a specific loss function. We implement our approach on the basis of a DG(0) Galerkin discretization of the neural ODE, leading to an explicit Euler time marching scheme. For the optimization we use steepest descent. Finally, we apply our method to the construction of neural networks for the classification of data sets, where we present results for a selection of well known examples from the literature.

翻译：本文提出一种新颖的逐层自适应神经网络架构构建方法。该方法基于面向目标的双加权残差技术，用于神经微分方程的最优控制。由此导出一个常微分方程约束优化问题，其中控制变量作为系数，并采用特定损失函数。我们在神经ODE的DG(0)伽辽金离散化基础上实现该方法，得到显式欧拉时间推进格式。优化过程采用最速下降法。最后，我们将所提方法应用于数据分类的神经网络构建，并以文献中若干经典算例展示计算结果。