math.NAAug 17, 2026

Asymptotics-guided learning and symbolic regression for dispersive resonances

Authors: Konstantinos Alexopoulos, Josselin Garnier

Organizations: CMAP, CNRS, Ecole polytechnique, Institut Polytechnique de Paris, 91120 Palaiseau, France

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Abstract

We study resonance prediction in dispersive media, formulated as nonlinear spectral problems for volume integral operators. The main idea is to use asymptotic analysis not only as a baseline approximation, but also as a guide for constructing predictive correction models. We learn the residual between asymptotic and reference resonances using features suggested by the subwavelength expansion, including the logarithmic scales specific to two dimensions. The resulting corrections substantially improve single-resonator and dimer predictions, and symbolic regression produces compact formulas for the learned residual. The results show that asymptotic analysis can be used not only to approximate resonances, but also to design the feature space in which data-driven corrections become accurate, low-dimensional, and interpretable.

Figures & tables

Validation caseAsymptotic mean errorCorrected mean errorImprovement factor
Single, in-distribution9.09×10−49.09\times 10^{-4}2.22×10−72.22\times 10^{-7}4.10×1034.10\times 10^{3}
Single, large- δ\delta OOD2.29×10−32.29\times 10^{-3}7.89×10−77.89\times 10^{-7}2.90×1032.90\times 10^{3}
Single, large- γ\gamma OOD9.94×10−49.94\times 10^{-4}7.92×10−77.92\times 10^{-7}1.26×1031.26\times 10^{3}
Dimer, in-distribution1.22×10−41.22\times 10^{-4}3.47×10−63.47\times 10^{-6}35.235.2
Dimer, large- δ\delta OOD3.68×10−43.68\times 10^{-4}7.35×10−57.35\times 10^{-5}5.015.01
Dimer, small- κ\kappa OOD1.47×10−41.47\times 10^{-4}3.83×10−53.83\times 10^{-5}3.823.82
Table 1: In-distribution and out-of-distribution validation of the physics-informed residual correction. OOD denotes out-of-distribution. The corrected error corresponds to the Ridge residual model using physics-informed features.
ConfigurationAsymptoticRidgeSymbolicAssisted symbolic
Single resonator9.09×10−49.09\times 10^{-4}2.22×10−72.22\times 10^{-7}4.70×10−84.70\times 10^{-8}1.89×10−111.89\times 10^{-11}
Dimer1.22×10−41.22\times 10^{-4}3.47×10−63.47\times 10^{-6}8.35×10−78.35\times 10^{-7}9.30×10−119.30\times 10^{-11}
Table 2: Mean relative errors on the in-distribution held-out test sets for the asymptotic approximation, the full physics-informed Ridge correction, symbolic regression, and assisted symbolic regression.
Figure 1: Single-resonator prediction results. Left: relative error of the asymptotic approximation and of the corrected models as a function of the asymptotic parameter δ\delta . Right: comparison of the predicted and reference resonances in the complex plane. The asymptotic approximation deteriorates as δ\delta increases, while the learned corrections remain close to the reference resonances over the sampled regime.
Figure 2: Dimer prediction errors as functions of the separation distance κ\kappa for the two resonance branches. The reduced asymptotic model captures the leading coupling effect, but the Ridge correction significantly improves the approximation for both branches. The second branch exhibits a larger variability, reflecting its stronger sensitivity to interaction effects.
Figure 3: Symbolic-regression correction for the two dimer branches as functions of the separation distance κ\kappa . The symbolic formulas preserve the accuracy of the learned residual correction while providing explicit expressions involving the asymptotic eigenvalues, branch splitting, and the expected powers of δ\delta and qjq_{j} .
Figure 4: Dimer prediction results. Left: relative error of the asymptotic approximation and of the corrected models as a function of the asymptotic parameter δ\delta , with both resonance branches included. Right: symbolic-regression reconstruction of the two resonance branches in the complex plane. The corrected models reduce the baseline error across the sampled range, while the broader scatter compared with the single-resonator case reflects the additional dependence on the separation distance κ\kappa , damping and branch coupling.
ConfigurationModelFeaturesMedian90th perc.
SingleAsymptotic baseline05.85×10−45.85\times 10^{-4}2.28×10−32.28\times 10^{-3}
SingleM0M0 : raw21.26×10−41.26\times 10^{-4}2.12×10−42.12\times 10^{-4}
SingleM1M1 : raw + logs43.81×10−53.81\times 10^{-5}6.38×10−56.38\times 10^{-5}
SingleM2M2 : + asymptotic resonance66.95×10−66.95\times 10^{-6}1.49×10−51.49\times 10^{-5}
SingleM3M3 : + algebraic scales91.42×10−71.42\times 10^{-7}3.73×10−73.73\times 10^{-7}
SingleM4M4 : + frequency-log scales128.19×10−88.19\times 10^{-8}2.61×10−72.61\times 10^{-7}
Table 3: Ablation study for the Ridge residual correction. The table reports the distribution of relative errors on the held-out test sets. The nested feature hierarchy shows that the main improvement is obtained when asymptotic and physics-informed scales are added, rather than merely by increasing the number of input variables.
StrategyFeature or structureSingleDimer
Feature augmentationproxy14/2014/2015/4015/40
δ2q2\delta^{2}q^{2}14/2014/2021/4021/40
δ2q2log⁡δ\delta^{2}q^{2}\log\delta1/201/203/403/40
δ2q2log⁡q\delta^{2}q^{2}\log q6/206/2013/4013/40
Preconditioningproxy20/2020/2040/4040/40
δ2q2\delta^{2}q^{2}20/2020/2040/4040/40
Table 4: Stability of the assisted symbolic-regression expressions. The counts are computed over ten independent runs, corresponding to 2020 symbolic expressions for the single resonator and 4040 expressions for the dimer. The table reports how often each feature family or symbolic structure appears in the selected expressions. In the dimer case, the spectral/splitting row reports the frequencies of λasymp\lambda_{\mathrm{asymp}} -dependent terms and splitting-dependent terms, respectively.
ConfigurationStrategyMeanMedianStandard deviation
Singlenext-order proxy7.31×10−37.31\times 10^{-3}1.85×10−51.85\times 10^{-5}1.97×10−21.97\times 10^{-2}
Singleasymptotic preconditioning5.07×10−55.07\times 10^{-5}1.54×10−61.54\times 10^{-6}1.22×10−41.22\times 10^{-4}
Singlefeature augmentation1.66×10−71.66\times 10^{-7}7.53×10−87.53\times 10^{-8}1.53×10−71.53\times 10^{-7}
Dimernext-order proxy3.45×10−33.45\times 10^{-3}9.68×10−59.68\times 10^{-5}5.26×10−35.26\times 10^{-3}
Dimerasymptotic preconditioning1.10×10−41.10\times 10^{-4}1.86×10−51.86\times 10^{-5}1.76×10−41.76\times 10^{-4}
Dimerfeature augmentation4.87×10−64.87\times 10^{-6}3.18×10−63.18\times 10^{-6}4.11×10−64.11\times 10^{-6}
Table 5: Prediction errors for the assisted symbolic-regression stability study over ten independent runs. The reported quantities are computed from the mean relative error of each run.
Figure 5: Assisted symbolic regression using a next-order asymptotic proxy. The proxy already improves the asymptotic or reduced approximation, and symbolic regression further reduces the remaining residual by several orders of magnitude. The dimer plot is more dispersed because the correction also depends on the separation distance and on branch splitting.
Figure 6: Robustness with respect to noisy training resonances. The corrections remain stable for moderate noise levels and deteriorate when the noise becomes comparable to the residual.
Figure 7: Effect of the number of training measurements on the Ridge residual correction. The error decreases as the amount of training data increases, especially for noisy measurements.
Figure 8: Effect of the number of training measurements on the next-order asymptotic proxy. Increasing the training size improves the stability of the calibrated proxy under noisy measurements.
Figure 9: Effect of the number of training measurements on the Ridge correction assisted by the next-order proxy. The proxy feature improves the stability of the learned correction, with a particularly visible gain in the dimer case.
Figure 10: Effect of the number of symbolic-training measurements on PySR rediscovery under noisy data. For each training size and noise level, symbolic regression is rerun from the noisy residuals. Larger training sets improve the robustness of the rediscovered symbolic correction, especially at higher noise levels.
Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

NNMeanMedian90th percentile
25253.56×10−43.56\times 10^{-4}8.29×10−58.29\times 10^{-5}1.18×10−31.18\times 10^{-3}
50501.71×10−41.71\times 10^{-4}3.88×10−53.88\times 10^{-5}5.75×10−45.75\times 10^{-4}
80809.68×10−59.68\times 10^{-5}2.26×10−52.26\times 10^{-5}3.21×10−43.21\times 10^{-4}
1201203.44×10−53.44\times 10^{-5}6.97×10−66.97\times 10^{-6}1.14×10−41.14\times 10^{-4}
1601601.69×10−51.69\times 10^{-5}3.53×10−63.53\times 10^{-6}5.53×10−55.53\times 10^{-5}
200200000000
Appendix
Table 6: Dimer discretization convergence. The errors are measured relative to the finest discretization Nref=200N_{\mathrm{ref}}=200 , using ∣ωN−ωNref∣|\omega_{N}-\omega_{N_{\mathrm{ref}}}| , and are aggregated over both resonance branches.
Figure 11: Dimer discretization convergence as a function of the number NN of quadrature points per resonator. The reference discretization is Nref=200N_{\mathrm{ref}}=200 .

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