Learning to Forecast on Convolution-Induced Geometry

Delay-coordinate methods provide a principled way to expose the geometry of dynamical systems from observed time series. However, classical constructions typically choose coordinates to recover or characterize the dynamics of observed trajectories, rather than to optimize a downstream forecasting objective. We ask whether such coordinates can instead be learned directly for prediction. Building on the connection between convolutional coordinates and finite-dimensional Koopman representations, we show that a causal convolution provides a learnable projection of a delay embedding, with Hankel-SVD convolutional coordinates arising from a particular choice of filters. This motivates a neural forecaster that learns delay coordinates end-to-end for prediction. We further use a Koopman regularizer to encourage approximately linear evolution in the learned coordinate space while retaining efficient direct multi-horizon prediction. Experiments show that the model achieves strong performance on standard forecasting benchmarks. More importantly, ablations and representation analyses reveal that forecasting supervision gives the learned coordinates predictive structure, while Koopman regularization helps organize their temporal evolution. Overall, our results show that strong neural forecasting and an explicit, analyzable dynamical representation can be achieved within the same learned coordinate space, connecting predictive accuracy with the study of what the forecaster learns.