Data-driven approximations of ordinary differential equations offer a promising alternative to classical methods of discovering a dynamical system model, particularly in complex systems lacking explicit first principles. This paper focuses on a complex system whose dynamics is described with a system of such equations, coupled through a complex network. Numerous real-world systems, including financial, social, and neural systems, belong to this class of dynamical models. We propose essential elements for approximating these dynamical systems using neural networks, including necessary biases and an appropriate neural architecture. Emphasizing the differences from static supervised learning, we advocate for evaluating generalization beyond classical assumptions of statistical learning theory. To estimate confidence in prediction during inference time, we introduce a dedicated null model. By studying various complex network dynamics, we demonstrate that the neural approximations of dynamics generalize across complex network structures, sizes, and statistical properties of inputs. Our comprehensive framework enables accurate and reliable deep learning approximations of high-dimensional, nonlinear dynamical systems.

翻译：数据驱动的常微分方程近似方法为发现动力学系统模型提供了经典方法的替代方案，尤其适用于缺乏明确第一性原理的复杂系统。本文聚焦于一类特殊复杂系统——其动力学由通过复杂网络耦合的此类方程组描述。众多真实系统（包括金融、社会及神经系统）均属于此类动力学模型范畴。我们提出了利用神经网络近似这些动力学系统的关键要素，包括必要的偏置项与合适的网络架构。区别于静态监督学习，我们主张在统计学习理论经典假设之外评估泛化能力。为在推理阶段预测置信度，我们引入了专用零模型。通过研究多种复杂网络动力学，我们证明：基于神经网络的动力学近似可跨复杂网络结构、规模及输入统计特性实现泛化。本研究所构建的综合框架能够实现高维非线性动力学系统的精确可靠深度学习近似。