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arXiv:2301.04900v4 Announce Type: replace-cross
Abstract: Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of differential equations present a promising alternative to traditional methods for uncovering a model of dynamical systems, especially in complex systems that lack explicit first principles. A recently employed machine learning tool for studying dynamics is neural networks, which can be used for data-driven solution finding or discovery of differential equations. Specifically for the latter task, however, deploying deep learning models in unfamiliar settings - such as predicting dynamics in unobserved state space regions or on novel graphs - can lead to spurious results. Focusing on complex systems whose dynamics are described with a system of first-order differential equations coupled through a graph, we show that extending the model's generalizability beyond traditional statistical learning theory limits is feasible. However, achieving this advanced level of generalization requires neural network models to conform to fundamental assumptions about the dynamical model. Additionally, we propose a statistical significance test to assess prediction quality during inference, enabling the identification of a neural network's confidence level in its predictions.