Interpretable AI-Assisted Early Reliability Prediction for a Two-Parameter Parallel Root-Finding Scheme

We propose an interpretable AI-assisted reliability diagnostic framework for parameterized root-finding schemes based on kNN-LLE proxy stability profiling and multi-horizon early prediction. The approach augments a numerical solver with a lightweight predictive layer that estimates solver reliability from short prefixes of iteration dynamics, enabling early identification of stable and unstable parameter regimes. For each configuration in the parameter space, raw and smoothed proxy profiles of a largest Lyapunov exponent (LLE) estimator are constructed, from which contractivity-based reliability scores summarizing finite-time convergence are derived. Machine learning models predict the reliability score from early segments of the proxy profile, allowing the framework to determine when solver dynamics become diagnostically informative. Experiments on a two-parameter parallel root-finding scheme show reliable prediction after only a few iterations: the best models achieve R^2=0.48 at horizon T=1, improve to R^2=0.67 by T=3, and exceed R^2=0.89 before the characteristic minimum-location scale of the stability profile. Prediction accuracy increases to R^2=0.96 at larger horizons, with mean absolute errors around 0.03, while inference costs remain negligible (microseconds per sample). The framework provides interpretable stability indicators and supports early decisions during solver execution, such as continuing, restarting, or adjusting parameters.

Optimizing Parallel Schemes with Lyapunov Exponents and kNN-LLE Estimation

Inverse parallel schemes remain indispensable tools for computing the roots of nonlinear systems, yet their dynamical behavior can be unexpectedly rich, ranging from strong contraction to oscillatory or chaotic transients depending on the choice of algorithmic parameters and initial states. A unified analytical-data-driven methodology for identifying, measuring, and reducing such instabilities in a family of uni-parametric inverse parallel solvers is presented in this study. On the theoretical side, we derive stability and bifurcation characterizations of the underlying iterative maps, identifying parameter regions associated with periodic or chaotic behavior. On the computational side, we introduce a micro-series pipeline based on kNN-driven estimation of the local largest Lyapunov exponent (LLE), applied to scalar time series derived from solver trajectories. The resulting sliding-window Lyapunov profiles provide fine-grained, real-time diagnostics of contractive or unstable phases and reveal transient behaviors not captured by coarse linearized analysis. Leveraging this correspondence, we introduce a Lyapunov-informed parameter selection strategy that identifies solver settings associated with stable behavior, particularly when the estimated LLE indicates persistent instability. Comprehensive experiments on ensembles of perturbed initial guesses demonstrate close agreement between the theoretical stability diagrams and empirical Lyapunov profiles, and show that the proposed adaptive mechanism significantly improves robustness. The study establishes micro-series Lyapunov analysis as a practical, interpretable tool for constructing self-stabilizing root-finding schemes and opens avenues for extending such diagnostics to higher-dimensional or noise-contaminated problems.