To understand and predict large, complex, and chaotic systems, Earth scientists build simulators from physical laws. Simulators generalize better to new scenarios, require fewer tunable parameters, and are more interpretable than nonphysical deep learning, but procedures for obtaining their derivatives with respect to their inputs are often unavailable. These missing derivatives limit the application of many important tools for forecasting, model tuning, sensitivity analysis, or subgrid-scale parametrization. Here, we propose to overcome this limitation with deep emulator networks that learn to calculate the missing derivatives. By training directly on simulation data without analyzing source code or equations, this approach supports simulators in any programming language on any hardware without specialized routines for each case. To demonstrate the effectiveness of our approach, we train emulators on complete or partial system states of the chaotic Lorenz-96 simulator and evaluate the accuracy of their dynamics and derivatives as a function of integration time and training data set size. We further demonstrate that emulator-derived derivatives enable accurate 4D-Var data assimilation and closed-loop training of parametrizations. These results provide a basis for further combining the parsimony and generality of physical models with the power and flexibility of machine learning.

中文翻译：

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用于地球科学模拟器中的微分、预测和参数化的深度模拟器

为了理解和预测大型、复杂和混乱的系统，地球科学家根据物理定律构建了模拟器。模拟器对新场景的泛化能力更好，需要的可调参数更少，并且比非物理深度学习更具可解释性，但获得其输入导数的程序通常不可用。这些缺失的导数限制了许多重要工具在预测、模型调整、敏感性分析或亚网格尺度参数化方面的应用。在这里，我们建议通过学习计算缺失导数的深度仿真器网络来克服这一限制。通过在不分析源代码或方程的情况下直接对仿真数据进行训练，这种方法支持在任何硬件上使用任何编程语言的模拟器，而无需针对每种情况设置专门的例程。为了证明我们方法的有效性，我们在混沌 Lorenz-96 模拟器的完整或部分系统状态上训练模拟器，并根据积分时间和训练数据集大小评估其动力学和导数的准确性。我们进一步证明了仿真器衍生的衍生物能够实现准确的 4D-Var 数据同化和参数化的闭环训练。这些结果为进一步将物理模型的简约性和通用性与机器学习的强大功能和灵活性相结合提供了基础。我们进一步证明了仿真器衍生的衍生物能够实现准确的 4D-Var 数据同化和参数化的闭环训练。这些结果为进一步将物理模型的简约性和通用性与机器学习的强大功能和灵活性相结合提供了基础。我们进一步证明了仿真器衍生的衍生物能够实现准确的 4D-Var 数据同化和参数化的闭环训练。这些结果为进一步将物理模型的简约性和通用性与机器学习的强大功能和灵活性相结合提供了基础。