Learning Interpretable Parametric Neural ODEs for Bifurcating Dynamical Systems

Modelling parametric dynamical systems remains a huge challenge in physics and engineering, especially where the parametric envelope contains bifurcation points that alter the stability landscape. Many data-driven methods reproduce trajectories accurately at a fixed operating condition, yet the problem becomes considerably harder when a single model must remain valid across a parameter range. In that setting the stability profile is often captured poorly unless an explicit mechanism recovers it, and bifurcation points may be misplaced or omitted entirely. We introduce an interpretable parametric Neural ODE that separates these responsibilities. A compact parameter-dependent linear operator is identified jointly from trajectories and then frozen, carrying the equilibrium spectrum and the stability profile across the whole parameter range in a form that can be inspected directly. A state-dependent neural coefficient-matrix residual then learns what the operator cannot express, namely the nonlinear corrections that produce saturation, waveform and limit-cycle geometry. Because the operator is a component of the deployed model rather than a summary fitted to its outputs, the stability at any operating condition is read off the model directly and can be audited, rather than inferred from simulations after the fact. We demonstrate the method across fluid-structure interaction, bluff-body wake flow and thermoacoustics, where it reproduces stable decay, instability growth and limit-cycle amplitudes while remaining bounded in autonomous rollouts at every operating condition. Given the same states, parameterisation and training data, our method strongly outperforms both symbolic and neural-network-based model identification methods in trajectory and stability prediction alike.