Artificial intelligence and deep learning are currently reshaping numerical simulation frameworks by introducing new modeling capabilities. These frameworks are extensively investigated in the context of model correction and parameterization where they demonstrate great potential and often outperform traditional physical models. Most of these efforts in defining hybrid dynamical systems follow {offline} learning strategies in which the neural parameterization (called here sub-model) is trained to output an ideal correction. Yet, these hybrid models can face hard limitations when defining what should be a relevant sub-model response that would translate into a good forecasting performance. End-to-end learning schemes, also referred to as online learning, could address such a shortcoming by allowing the deep learning sub-models to train on historical data. However, defining end-to-end training schemes for the calibration of neural sub-models in hybrid systems requires working with an optimization problem that involves the solver of the physical equations. Online learning methodologies thus require the numerical model to be differentiable, which is not the case for most modeling systems. To overcome this difficulty and bypass the differentiability challenge of physical models, we present an efficient and practical online learning approach for hybrid systems. The method, called EGA for Euler Gradient Approximation, assumes an additive neural correction to the physical model, and an explicit Euler approximation of the gradients. We demonstrate that the EGA converges to the exact gradients in the limit of infinitely small time steps. Numerical experiments are performed on various case studies, including prototypical ocean-atmosphere dynamics. Results show significant improvements over offline learning, highlighting the potential of end-to-end online learning for hybrid modeling.

翻译：人工智能与深度学习正通过引入新型建模能力重塑数值模拟框架。这些框架在模型校正与参数化领域得到广泛研究，展现出巨大潜力，且常优于传统物理模型。当前构建混合动力系统的大多数尝试采用离线学习策略，即训练神经参数化（此处称为子模型）输出理想校正。然而，这类混合模型在定义能转化为良好预测性能的合适子模型响应时面临硬限制。端到端学习方案（亦称在线学习）可通过允许深度学习子模型基于历史数据训练来应对这一缺陷。但为混合系统中神经子模型的校准定义端到端训练方案，需要处理涉及物理方程求解器的优化问题。在线学习方法要求数值模型可微分，而多数建模系统不具备该特性。为攻克这一难题并绕过物理模型的可微分性挑战，我们提出一种高效实用的混合系统在线学习方法。该方法名为欧拉梯度近似（EGA），其假设对物理模型施加加性神经校正，并采用显式欧拉近似计算梯度。我们证明EGA在时间步长趋近于零的极限条件下可收敛至精确梯度。针对各类案例（包括典型的海洋-大气动力学）开展数值实验，结果表明该方法相较于离线学习有显著改进，凸显了端到端在线学习在混合建模中的潜力。