Cross-Attractor Transformations: A Novel Machine Learning Framework to Minimize Forecast Error in the Presence of Model Bias
Imperfect models are often used for forecasting and state estimation of complex dynamical systems, typically by mapping a reference initial state into model phase space, making a forecast, and then mapping back to the reference space. In many cases these mappings are implicit, and forecast errors thus reflect a combination of model forecast errors and mapping errors. Techniques to infer parameterizations and parameters to reduce model bias have been the subject of intense scrutiny; however, we lack a general framework for discovering optimal mappings between system and model attractors. Here we propose a novel Machine Learning paradigm for inferring cross-attractor transformations (CATs) that minimize forecast error. CATs are pairs of transformations from the phase space of a reference system to the phase space of a model and vice versa that serve as a bridge between the attractors of a true system and an imperfect model. A computationally efficient analog approximation to tangent linear and adjoint models is developed to enable efficient stochastic gradient descent algorithms to train CAT parameters. Neural networks constructed with a custom analog-adjoint layer permit specification of affine transformations as well as more general nonlinear transformations.