🤖 AI Summary
This work addresses the challenge of efficiently and reliably tracking local solution manifolds in parametrized dynamical systems, particularly in regions exhibiting folds or degeneracies. To this end, we propose a residual-certified adaptive tracking method that dynamically assesses the validity of local reduced-order models via a residual-based threshold and employs distinct update strategies in regular versus degenerate parameter regimes. The approach integrates localized proper orthogonal decomposition (POD) reduction, full-physics residual verification, state-distance–driven snapshot forgetting, high-fidelity resampling, and lightweight physics-informed neural correction. Numerical experiments demonstrate that the method concentrates high-fidelity computations precisely in challenging parameter regions, achieving substantial gains in computational efficiency while preserving physical consistency and predictive accuracy of the solution.
📝 Abstract
This paper presents a residual-certified adaptive method for tracking local solution manifolds in parametric dynamical systems. The method combines local POD reduction, full physical residual checks, state-distance snapshot forgetting, high-fidelity resampling, and a lightweight physics-informed neural correction. Instead of learning one global parameter-to-state map, the algorithm maintains the currently active local branch and updates it when the residual indicates loss of validity. The analysis explains why residual thresholds are meaningful on regular branches through local residual-error control, and why stricter local updates are needed near folds or other degenerate neighborhoods. Numerical studies on Ostwald ripening, a particle population-balance model, and the Bratu equation test the approach across low-dimensional dynamics, nonlinear nonlocal residual compensation, and near-fold model failure. The results show that residual-certified local model management can concentrate high-fidelity computation in difficult parameter regions while preserving an interpretable link between surrogate prediction, physical consistency, and active-branch tracking.