MPINeuralODE: Multiple-Initial-Condition Physics-Informed Neural ODEs for Globally Consistent Dynamical System Learning

Neural ordinary differential equations (Neural ODEs) often fit training trajectories while generalizing poorly to unseen initial conditions and long horizons. We propose MPINeuralODE, which combines a soft physics-informed residual with a Multiple-Initial-Condition (MIC) multiple-shooting curriculum whose ingredients are structurally complementary: the physics term anchors the vector-field magnitude on the support that MIC enlarges. We evaluate along three axes — out-of-sample error, long-horizon stability, and Hamiltonian drift — that together expose whether the learned dynamics have recovered the true vector field. On Lotka–Volterra, MPINeuralODE achieves the lowest out-of-sample and long-horizon MSE among data-driven methods (a 26% reduction over the baseline Neural ODE) while essentially matching the PINN ablation on Hamiltonian drift.