In this paper, we propose a generic algorithm to train machine learning-based subgrid parametrizations online, i.e., with \textit{a posteriori} loss functions, but for non-differentiable numerical solvers. The proposed approach leverages a neural emulator to approximate the reduced state-space solver, which is then used to allow gradient propagation through temporal integration steps. We apply this methodology on a chaotic two-timescales Lorenz-96 system and a single layer quasi-geostrophic system with zonal dynamics, known to be highly unstable with offline strategies. Using our algorithm, we are able to train a parametrization that recovers most of the benefits of online strategies without having to compute the gradient of the original solver. We found that training the neural emulator and parametrization components separately with different loss quantities is necessary in order to minimize the propagation of approximation biases. Experiments on emulator architectures with different complexities also indicates that emulator performance is key in order to learn an accurate parametrization. This work is a step towards learning parametrization with online strategies for climate models.

翻译：本文提出了一种通用算法，用于在线训练基于机器学习的次网格参数化方案，即采用后验损失函数，但适用于不可微数值求解器。该方法利用神经仿真器近似约化状态空间求解器，从而允许梯度通过时间积分步骤传播。我们将此方法应用于混沌双时间尺度Lorenz-96系统以及具有纬向动力学的单层准地转系统——这两个系统已知在离线策略下具有高度不稳定性。通过所提算法，我们成功训练出能够恢复在线策略大部分优势的参数化方案，且无需计算原始求解器的梯度。研究发现，必须使用不同的损失量分别训练神经仿真器与参数化组件，以最小化近似偏差的传播。对不同复杂度仿真器架构的实验也表明，仿真器性能是学习精确参数化的关键。本工作为气候模型在线学习参数化策略迈出了重要一步。