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

TL;DR

MPINeuralODE enhances neural ODEs by integrating physics-informed residuals with a multiple-shooting curriculum, significantly improving out-of-sample accuracy and stability in dynamical system learning.

Contribution

It introduces MPINeuralODE, combining physics-informed residuals with multiple-initial-condition training to improve generalization and stability in neural ODE models.

Findings

Achieves 26% lower out-of-sample MSE than baseline Neural ODE.

Outperforms other data-driven methods on long-horizon stability.

Matches PINN on Hamiltonian drift, demonstrating effective physics integration.

Abstract

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, which together expose whether the learned dynamics recover the underlying vector field. On Lotka-Volterra, MPINeuralODE achieves the lowest out-of-sample and long-horizon MSE among data-driven methods, with a 26% reduction over the baseline Neural ODE, while essentially matching the PINN ablation on Hamiltonian drift.