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

Bruno Carpentieri, Andrei Velichko, Mudassir Shams, Paola Lecca

Abstract

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.

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

Abstract

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.
Paper Structure (44 sections, 17 equations, 9 figures, 6 tables)

This paper contains 44 sections, 17 equations, 9 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Methodology
  3. Problem Setting and Iterative Scheme (Brief)
  4. kNN--LLE Proxy Profile Construction
  5. Profile-Based Reliability Metrics
  6. Dataset Construction for Multi-Horizon Learning
  7. Train/Test Splits: Random vs. Center-to-Periphery Generalization
  8. Random split.
  9. Center-to-periphery split.
  10. Learning Models and Evaluation (Brief)
  11. Experimental Setup
  12. Parameter Grid and Computational Budget
  13. Micro-Series Definition and Stabilization Settings
  14. kNN--LLE Proxy Hyperparameters
  15. Metric Window and Reliability Targets
  16. ...and 29 more sections

Figures (9)

  • Figure 1: Conceptual pipeline of the proposed reliability diagnostic. Solver ensembles are generated on a $(\alpha,\beta)$ grid, yielding kNN--LLE proxy profiles and interpretable profile metrics. Multi-horizon prefixes form an ML dataset used for early prediction of $S_{\mathrm{mom}}(T)$, followed by validation via metric curves and test-only heatmaps.
  • Figure 2: Ground-truth heatmaps of profile-based reliability metrics over the $(\alpha,\beta)$ grid: (a) $S_{\min}$ and (b) $S_{\mathrm{mom}}$.
  • Figure 3: Representative kNN--LLE proxy profiles (raw vs. smoothed) spanning increasing $S_{\mathrm{mom}}$ levels (a)--(f). Higher $S_{\mathrm{mom}}$ typically corresponds to earlier and deeper contractive dips (more negative values) in the smoothed profile, whereas low $S_{\mathrm{mom}}$ cases tend to exhibit weak or delayed contractivity.
  • Figure 4: Timing statistics for the smoothed proxy profile $\tilde{\lambda}_1(t)$, comparing the good subset (top $20\%$ by $S_{\mathrm{mom}}$) against the rest: (a) distribution of the minimum location $t_{\min}$; (b) distribution of the first negative entry time $t_{\mathrm{enter\_neg}}$.
  • Figure 5: Center split: prediction performance for $S_{\mathrm{mom}}$ as a function of prefix length $T$, reported up to $T\le 3T_{\min}=36$. The dashed line indicates $T_{\min}=12$ (median minimum-location index of the good region), and the dotted line indicates $3T_{\min}$.
  • ...and 4 more figures