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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.
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