Improving the predictive capability and computational cost of dynamical models is often at the heart of augmenting computational physics with machine learning (ML). However, most learning results are limited in interpretability and generalization over different computational grid resolutions, initial and boundary conditions, domain geometries, and physical or problem-specific parameters. In the present study, we simultaneously address all these challenges by developing the novel and versatile methodology of unified neural partial delay differential equations. We augment existing/low-fidelity dynamical models directly in their partial differential equation (PDE) forms with both Markovian and non-Markovian neural network (NN) closure parameterizations. The melding of the existing models with NNs in the continuous spatiotemporal space followed by numerical discretization automatically allows for the desired generalizability. The Markovian term is designed to enable extraction of its analytical form and thus provides interpretability. The non-Markovian terms allow accounting for inherently missing time delays needed to represent the real world. We obtain adjoint PDEs in the continuous form, thus enabling direct implementation across differentiable and non-differentiable computational physics codes, different ML frameworks, and treatment of nonuniformly-spaced spatiotemporal training data. We demonstrate the new generalized neural closure models (gnCMs) framework using four sets of experiments based on advecting nonlinear waves, shocks, and ocean acidification models. Our learned gnCMs discover missing physics, find leading numerical error terms, discriminate among candidate functional forms in an interpretable fashion, achieve generalization, and compensate for the lack of complexity in simpler models. Finally, we analyze the computational advantages of our new framework.

翻译：提升动力模型的预测能力和计算效率通常是计算物理与机器学习结合的核心目标。然而，大多数学习结果在可解释性以及在不同计算网格分辨率、初始和边界条件、域几何形状、物理或问题特定参数下的泛化能力方面存在局限。本研究通过开发新颖且通用的统一神经偏时滞微分方程组方法，同时应对上述所有挑战。我们直接在偏微分方程形式中，利用马尔可夫和非马尔可夫神经网络闭合参数化方法，增强现有/低保真动力模型。在连续时空域中将现有模型与神经网络融合，随后进行数值离散，可自动实现所需的泛化能力。马尔可夫项被设计为支持解析形式提取，从而提供可解释性。非马尔可夫项则允许计入表征真实世界所必需的固有缺失时滞。我们以连续形式推导伴随偏微分方程，使其能够直接应用于可微和不可微的计算物理代码、不同机器学习框架，以及非均匀时空训练数据的处理。基于四组涉及非线性平流波、激波和海洋酸化模型的实验，我们验证了新型广义神经闭合模型框架的有效性。训练后的gnCM能够发现缺失物理过程、识别主导数值误差项、以可解释方式区分候选函数形式、实现泛化能力，并弥补简单模型复杂度的不足。最后，我们分析了新框架的计算优势。