Interpretable AI-Assisted Early Reliability Prediction for a Two-Parameter Parallel Root-Finding Scheme
Authors: Bruno Carpentieri, Andrei Velichko, Mudassir Shams, Paola Lecca
Organizations: Faculty of Engineering, Free University of Bozen–Bolzano, 39100 Bolzano, Italy · Institute of Physics and Technology, Petrozavodsk State University, 185910 Petrozavodsk, Russia · Department of Mathematics, Faculty of Arts and Science, Balikesir University, 10145 Balıkesir, Turkey · Department of Mathematics and Statistics, Riphah International University, I-14, Islamabad 44000, Pakistan
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.
Figures & tables
Figure 1: Conceptual pipeline of the proposed reliability diagnostic. Solver ensembles are generated on a (α,β) grid, yielding kNN–LLE proxy profiles and interpretable profile metrics. Multi-horizon prefixes form an ML dataset used for early prediction of Smom(T) , followed by validation via metric curves and test-only heatmaps.
Figure 2: Ground-truth heatmaps of profile-based reliability metrics over the (α,β) grid: (a) Smin and (b) Smom .
Figure 3: Representative kNN–LLE proxy profiles (raw vs. smoothed) spanning increasing Smom levels (a)–(f). Higher Smom typically corresponds to earlier and deeper contractive dips (more negative values) in the smoothed profile, whereas low Smom cases tend to exhibit weak or delayed contractivity.
Figure 4: Timing statistics for the smoothed proxy profile λ~1(t) , comparing the good subset (top 20% by Smom ) against the rest: (a) distribution of the minimum location tmin ; (b) distribution of the first negative entry time tenter_neg .
Figure 5: Center split: prediction performance for Smom as a function of prefix length T , reported up to T≤3Tmin=36 . The dashed line indicates Tmin=12 (median minimum-location index of the good region), and the dotted line indicates 3Tmin .
T
best model
R2
fit time (s)
test time (s)
test per-sample (s)
1
grad_boost
0.476
2.91e-01
3.06e-03
2.12e-06
3
grad_boost
0.670
6.30e-01
3.05e-03
2.12e-06
5
random_forest
0.791
3.12e-01
5.00e-02
3.47e-05
7
random_forest
0.836
3.46e-01
9.87e-02
6.85e-05
9
random_forest
0.873
3.67e-01
4.93e-02
3.42e-05
11
random_forest
0.893
4.48e-01
4.90e-02
3.40e-05
Table 1: Center split: best model by R2 at each scanned horizon T . R2 is rounded to three decimals; timing values are reported in seconds (scientific notation, two decimals).
Figure 6: Center split: heatmap-level comparison between the theoretical Smom landscape (a) and test-only predictions at increasing horizons (b)–(d). White region indicates training points (not plotted), while colored pixels correspond to test-region predictions on a common scale.
Figure 7: Random split: prediction performance for Smom as a function of prefix length T , reported up to T≤3Tmin=36 . The dashed line indicates Tmin=12 , and the dotted line indicates 3Tmin .
T
best model
R2
fit (s)
test (s)
test/sample (s)
1
grad_boost
0.486
3.00e-01
3.41e-03
2.37e-06
3
grad_boost
0.673
4.57e-01
3.44e-03
2.39e-06
5
random_forest
0.791
3.00e-01
5.00e-02
3.47e-05
7
knn
0.852
2.50e-03
2.61e-03
1.81e-06
9
knn
0.879
2.54e-03
2.54e-03
1.76e-06
11
knn
0.905
2.53e-03
2.50e-03
1.74e-06
Table 2: Random split: best model by R2 at each scanned horizon T . R2 is rounded to three decimals; timing values are reported in seconds (scientific notation, two decimals).
Figure 8: Random split: heatmap-level comparison between the theoretical Smom landscape (a) and test-only predictions at increasing horizons (b)–(d). White cells indicate training points (not plotted), while colored cells correspond to test-region predictions on a common scale.
T
approx. solver iterations
speedup vs. full K=200
center split R2
random split R2
1
10
20.0×
0.476
0.486
3
12
16.7×
0.670
0.673
11
20
10.0×
0.893
0.905
35
44
4.5×
0.956
0.958
Table 3: Approximate diagnostic cost versus predictive quality at representative horizons. The solver-iteration requirement is estimated as kreq(T)≈T+9 for the present proxy construction ( L=5 , hmax=5 ), and the full diagnostic uses K=200 iterations per trajectory.
Appendix figures & tables4 assets
Supplementary material from the paper’s appendix.
Appendix
Iteration k
Near-root init.
Moderate init.
Random init.
Order (approx.)
1
1.26×10−2
3.35×10−1
8.44×10−1
–
2
2.86×10−7
5.71×10−5
3.47×10−4
3.92
3
4.66×10−21
7.44×10−9
9.78×10−10
4.15
4
2.16×10−59
2.43×10−17
4.72×10−27
4.97
5
6.38×10−75
8.04×10−49
7.01×10−25
5.67
6
9.17×10−191
0.21×10−160
3.47×10−90
5.97
Appendix
Table 4: Early iteration error behavior of the proposed scheme under different initialization scenarios.
Figure 9: Early-iteration convergence of the proposed parallel scheme under different initialization scenarios, showing log10(E(k)) versus iteration index.
Method
Iterations
CPU time (s)
Memory (MB)
Maximum error
Proposed Scheme [ 21 ]
5
0.012
123.65
2.05×10−189
Existing Scheme [ 31 ]
11
0.018
256.76
5.14×10−45
Existing Scheme [ 32 ]
10
0.015
254.89
0.77×10−37
Appendix
Table 5: Computational performance of the proposed solver.
Method
Order (theoretical)
Order (observed)
Proposed Scheme [ 21 ]
6
5.97
Existing Scheme [ 31 ]
6
4.56
Existing Scheme [ 32 ]
6
3.19
Appendix
Table 6: Estimated convergence order of the methods.
Algorithmic stability measures how sensitive a learning algorithm’s outputs are to small changes in its training data, linking robustness to generalization.