Explainable prediction and simulation of complex system dynamics through networks of manifolds

Complex systems such as brains and other interacting biological and physical processes are difficult to represent because they evolve across many variables, scales, and nonlinear interactions. To capture these multivariate, multiscale interactions we have developed Generative Manifold Networks (GMNs) a machine learning framework consisting of a network of linked dynamical systems. The network is discovered by an interaction function which can focus on causality, shared information, nonlinearity or other metric. Network nodes are low-dimensional data-driven state-space manifolds with generator functions accommodating multiscale dynamics. In contrast to many machine learning approaches GMNs have no latent or randomly initialized variables providing transparent explainability. GMNs generate short term dynamics of chaos on par with echo state networks while outperforming them in long term generation of chaos and neural dynamics, but with a markedly reduced number of dimensions and without sensitive dependence on reservoir parameters or random states. As a result of their holistic, multiscale representation GMNs can learn the complete dynamics of a complex system. We further show that GMNs are universal approximators. GMNs are demonstrated on chaotic dynamics, neural and behavioral recordings of the fruit fly and domestic rat with comparisons to echo state networks and crossformer - a time series transformer. SignificanceA major challenge in machine learning is to model complex systems accurately without losing interpretability. Many methods that succeed in prediction rely on latent variables obscuring mechanistic insight and complicating experimental testing. Generative manifold networks (GMN) construct a network of low-dimensional functional manifolds directly from observed variables with no latent or randomly initialized variables: the model remains transparent and experimentally testable. We prove that GMN are universal approximators showing that high representational power can be achieved without sacrificing explainability. GMN therefore provides a general framework for prediction and simulation in neuroscience and complex systems where unraveling the links between variables in an experimentally testable manner is as important as forecasting their behavior.