Bifurcations from Weight Space: JEPAWG on Parametric Dynamical Systems

Abstract

Trained network weights are increasingly treated as data, but there is rarely ground truth about what a population of models should encode, which hampers systematic validation of weight-space methods. This makes parametric dynamical systems a natural test-bed: the dynamics change qualitatively at submanifolds of parameter space that are, in well-studied families, known analytically or numerically. Fine-tuning a forecasting network at each point of a parameter grid yields a model zoo indexed by a physical parameter, against which weight-space diagnostics can be checked. We build such model zoos for four dynamical systems and analyse them with JEPAWG, a recently proposed joint-embedding hypernetwork built around a learned parameter--weight representation. It was first introduced for neural samplers for lattice field theories, where its latent geometry located the phase transition and estimated the 2D Ising critical exponent. Applied to dynamical systems, JEPAWG is competitive on forecasting with a hypernetwork trained directly for that task, despite being supervised on the parameter--weight correspondence rather than on forecasts. Beyond forecasting, the norm of the parameter-derivative of the decoded weights locates Hopf points and curves, tracks a period-doubling cascade closely enough to give a qualitative estimate of the Feigenbaum constant, and detects a global bifurcation, which has no eigenvalue signature and is therefore invisible to linearisation. As such, JEPAWG provides interpretable insight into the dynamics that is not available from other hypernetworks. Together, these results indicate that JEPAWG is a workable instrument for reading physical structure off a population of trained networks.