Data-driven approximations of ordinary differential equations offer a promising alternative to classical methods in 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 ordinary differential equations, coupled via a network adjacency matrix. Numerous real-world systems, including financial, social, and neural systems, belong to this class of dynamical models. We propose essential elements for approximating such 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 the neural network's ability to approximate various dynamics, generalize across complex network structures, sizes, and statistical properties of inputs. Our comprehensive framework enables deep learning approximations of high-dimensional, non-linearly coupled complex dynamical systems.

翻译：数据驱动的常微分方程逼近为发现动力系统模型（尤其是在缺乏显式第一性原理的复杂系统中）提供了一种有前景的替代方案。本文聚焦于一类由网络邻接矩阵耦合的常微分方程组描述其动态行为的复杂系统。包括金融、社会和神经动力学系统在内的众多现实世界系统均属于此类动力学模型。我们提出了使用神经网络逼近此类动力系统的关键要素，包括必要的偏置项和合适的神经架构。通过强调其与静态监督学习的差异，我们主张在经典统计学习理论假设之外评估泛化性能。为在推理阶段估计预测置信度，我们引入了专属零模型。通过研究多种复杂网络动力学，我们展示了神经网络逼近不同动力学行为、跨复杂网络结构及规模进行泛化，以及处理输入统计特性的能力。本文提出的综合框架使深度学习能够逼近高维非线性耦合的复杂动力系统。