In this paper, we propose a generic algorithm to train machine learning-based subgrid parametrizations online, i.e., with $\textit{a posteriori}$ loss functions for non-differentiable numerical solvers. The proposed approach leverage neural emulators to train an approximation of the reduced state-space solver, which is then used to allows gradient propagation through temporal integration steps. The algorithm is able to recover most of the benefit of online strategies without having to compute the gradient of the original solver. It is demonstrated that training the neural emulator and parametrization components separately with respective loss quantities is necessary in order to minimize the propagation of some approximation bias.

翻译：本文提出一种通用算法，用于在线训练基于机器学习的亚网格参数化方案，即针对不可微数值求解器采用 $\textit{后验}$ 损失函数。所提方法利用神经仿真器训练简化状态空间求解器的近似模型，进而实现梯度在时间积分步骤间的传播。该算法无需计算原始求解器的梯度，即可获得在线策略的大部分优势。研究表明，为最小化近似偏差的传播，有必要使用各自对应的损失量分别训练神经仿真器与参数化组件。