Residual-Guided Active Learning of Multi-Component Soliton Dynamics with Physics-Informed Neural Networks

NeurIPS 2025 Workshop NeurReps Submission43 Authors

Keywords: Physics-Informed Neural Networks (PINNs), Residual-driven collocation, Active learning, Manakov system, Nonlinear Schrödinger equation, Sample efficiency, Dispersive wave dynamics

Abstract: Physics-Informed Neural Networks (PINNs) provide a flexible framework for solving nonlinear partial differential equations (PDEs), but their efficiency is often limited by static collocation strategies that distribute points uniformly across the domain. This work develops and evaluates a residual-driven active learning approach for the three-component Manakov system, a coupled nonlinear Schrödinger-type model characterized by strongly inhomogeneous solution geometry. The method incrementally augments the training set by identifying regions of high PDE residual and concentrating new collocation points in these areas, thereby adapting the sampling distribution to the evolving dynamics of the solution. Numerical experiments show that this targeted placement reduces global loss and component-wise error relative to uniform baselines, achieving comparable accuracy with substantially fewer collocation points. The approach requires no modification to the network architecture or optimization procedure, ensuring that the observed improvements arise solely from the sampling policy. These results demonstrate that residual-driven collocation can improve sample efficiency and solution fidelity in multi-component dispersive systems, suggesting a scalable path for applying PINNs to more complex and stiff nonlinear wave equations. Limitations regarding computational overhead, reproducibility, and the lack of a formal error analysis are also discussed, highlighting directions for future investigation.

Submission Number: 43