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Asymptotics-guided learning and symbolic regression for dispersive resonances

Konstantinos Alexopoulos, Josselin Garnier

arXiv:2608.16152v1math.NAcs.LGmath-phmath.APphysics.optics

TL;DR

Resonance prediction in dispersive media is limited by leading asymptotic models away from the strict subwavelength regime. This paper uses asymptotics-guided residual learning with physics-informed logarithmic features, substantially improving single-resonator and dimer predictions while producing compact symbolic correction formulas.

  • Problem

    Leading asymptotic and reduced resonance models lose accuracy away from the strict subwavelength regime and incompletely capture higher-order and coupled-system effects.

  • Method

    The framework uses asymptotic resonance predictions as baselines, learns residuals with subwavelength-informed features, and applies symbolic regression to obtain compact correction formulas.

  • Results

    Asymptotics-guided corrections substantially improve predictions for both single resonators and dimers, with logarithmic frequency-dependent features contributing to the improvement.

  • Takeaways & Limitations

    Asymptotic analysis can organize accurate, low-dimensional, and interpretable data-driven correction models for dispersive resonance prediction.

  • Takeaways & Limitations

    Symbolic regression formulas are not unique and should be interpreted as compact correction laws rather than rigorous derivations of the next asymptotic term.

Abstract

from arXiv · show

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.

1 Introduction

The paper uses asymptotic resonances as leading-order approximations and learns physics-informed residual corrections for nonlinear resonance problems in dispersive media. Symbolic regression then produces interpretable analytic correction formulas.

  • Problem setting: The approach targets nonlinear spectral problems for resonances in dispersive media formulated through volume integral operators.These problems arise in subwavelength resonators and highly dispersive systems, with asymptotic expansions providing leading-order resonance formulas.
  • Asymptotics-guided residual learning: The framework learns residual errors between asymptotic approximations and accurate reference resonances rather than rediscovering the full resonance map.Its purpose is to correct missing higher-order terms while retaining the asymptotic model as the leading-order approximation.
  • Physics-informed feature design: Physics-informed features are derived from the subwavelength expansion, including logarithmic next-order terms and frequency-dependent logarithmic scales.The introduction states that ablation studies show these features are essential for accurate residual learning.
  • Interpretable correction formulas: Symbolic regression searches for explicit analytic expressions that fit the learned residual data, supporting interpretable correction formulas.It is used as an interpretable scientific machine-learning method rather than a black-box replacement for the asymptotic model.

2 Mathematical setting

The paper formulates dispersive resonances as nonlinear spectral problems for frequency-dependent volume integral operators in two dimensions. The setting includes Lorentz-type damping, logarithmic Green-function behavior, subwavelength scaling, and dimer coupling that produces two resonance branches.

  • Material and wave setting: The model describes two-dimensional time-harmonic wave propagation in a homogeneous background containing bounded resonators with possibly complex, frequency-dependent permittivity.The resonator material follows a Lorentz-type dispersive law, while the damping parameter γ controls the imaginary response and resonance linewidth.
  • Integral-operator formulation: The two-dimensional free-space Green’s function has a logarithmic singularity, producing the characteristic logarithmic terms in the subwavelength asymptotic expansion.The Lippmann–Schwinger formulation leads to a frequency-dependent volume integral operator whose asymptotic approximations yield reduced resonance conditions.
  • Resonance problem: Resonances are complex frequencies at which the homogeneous outgoing problem has a non-trivial solution, equivalently when the frequency-dependent volume integral operator is not invertible.The problem is nonlinear because both the material contrast and the integral operator depend on frequency.
  • Dimer geometry: For a dimer, common scaling by δ and reference separation κ produce two resonance branches associated with coupling between the two resonators.The branches correspond to symmetric and antisymmetric modes, and their splitting depends on κ.
  • Dimer geometry: Dimer residual approximation is more delicate than the single-resonator case because residuals combine higher-order self-interaction and coupling-induced corrections.The reference and asymptotic resonances are defined separately for the two branches, with corresponding branchwise residuals.

3 Asymptotic modelling

This section develops asymptotic models for single resonators and dimers in the subwavelength regime, emphasizing logarithmic two-dimensional corrections and frequency-dependent higher-order terms. These structures motivate learning the residual between reference and asymptotic resonances with physics-informed features.

  • Two-dimensional asymptotic structure: Two-dimensional asymptotics contain characteristic terms involving δ, k0, and log(δk0), which guide the construction of physics-informed learning features.The logarithmic dependence arises from the two-dimensional Green’s function and separates size effects from frequency-dependent corrections.
  • Spectral expansion: The leading spectral behavior is determined by a static logarithmic operator on the reference geometry, while higher-order eigenvalue corrections depend on frequency and background wavenumber.The higher-order operator terms include the two-dimensional frequency-dependent scale inherited from the Green’s-function expansion.
  • Residual scaling: The frequency residual inherits the scale of neglected eigenvalue and operator terms, multiplied by a smooth branch-dependent factor under the simple-resonance assumption.This follows from first-order implicit-function analysis of the nonlinear resonance condition.
  • Dimer asymptotics: For dimers, a finite-dimensional reduced model yields symmetric and antisymmetric branches through coupling-dependent off-diagonal interactions.The resulting residuals combine self-interaction corrections with coupling-induced, branch-dependent errors.
  • Motivation for residual learning: Because asymptotic accuracy deteriorates away from the strict subwavelength regime and coupled corrections are difficult to compute explicitly, the framework learns ωref −ωasymp while preserving asymptotic structure.The identified logarithmic and frequency-dependent scales are used to design input features and interpret symbolic-regression formulas.

4 Learning the asymptotic residual

The section formulates resonance prediction as residual learning: asymptotic or reduced models provide the baseline, while models learn the discrepancy to reference resonances. Features are designed from physical parameters, model outputs, and asymptotically motivated scales so that compact corrections can capture neglected higher-order and interaction effects.

  • Residual-learning formulation: The method learns the residual between an asymptotic or reduced resonance ωasymp and its reference value ωref, rather than learning resonances directly.The corrected resonance adds the learned discrepancy ΔωML to the baseline prediction.
  • Residual-learning formulation: The residual-learning formulation encodes leading-order behavior in ωasymp and targets a smaller correction expected to reflect neglected asymptotic terms.This structure enables simpler models when features are chosen consistently with the underlying asymptotics.
  • Asymptotics-guided features: Inputs combine raw physical and geometric parameters, asymptotic-model quantities, and physics-informed features motivated by next-order terms.The feature set includes δ, γ, and dimer separation κ; real and imaginary parts of ωasymp; approximate eigenvalues; branch-splitting quantities; and logarithmic or frequency-dependent scales.
  • Asymptotics-guided features: Feature design exposes dominant scales of the missing correction without replacing the asymptotic model, enabling ablation tests of physically meaningful structure.For dimers, relevant quantities and features are evaluated branchwise.
  • Ridge correction model: Ridge regression jointly learns the real and imaginary residual components as a low-dimensional linear combination of asymptotically informed features.The linear model is chosen because the baseline captures dominant behavior and the features encode expected higher-order scalings.
  • Interpretation and next step: The framework extends asymptotic modeling by learning higher-order terms and coupling effects, while motivating symbolic regression for more compact analytic residual formulas.Symbolic regression is introduced as the next step to improve interpretability and compare learned formulas with expected asymptotic structure.

5 Symbolic regression of the residual

Symbolic regression learns compact, interpretable formulas for asymptotic resonance residuals rather than rediscovering the full resonance. Physics-informed features and asymptotic guidance focus the search on higher-order, branch- and interaction-dependent corrections.

  • Method: Symbolic regression searches mathematical expressions for residuals, providing compact analytic corrections instead of coefficients within a fixed functional form.The target is the asymptotic residual, with real and imaginary parts fitted separately for single resonators and dimer branches.
  • Physics-informed features: The feature space uses asymptotic quantities such as δ, γ, ωasymp, branch eigenvalues, spectral splitting, and logarithmic two-dimensional scales.Terms involving δ2q2_asymp log(δq_asymp) are motivated by the next-order operator expansion, but selected formulas remain data-driven fitted corrections.
  • Single resonator: Single-resonator corrections are dominated in the real part by δ2 terms, while their imaginary part includes δ2q2_asymp log q_asymp and damping-linked frequency factors.These structures are consistent with the asymptotically motivated feature scales.
  • Asymptotic guidance: Asymptotic preconditioning and feature augmentation inject next-order information by changing the target or input representation, respectively.The assisted strategies are designed to make symbolic corrections simpler and more stable; numerical results are reported as improving accuracy for both systems.

6 Numerical experiments

Numerical experiments show that asymptotically designed corrections substantially improve resonance predictions, with symbolic regression producing compact, accurate formulas. Ablations and stability tests indicate that asymptotically meaningful scales, especially algebraic and logarithmic terms, organize the learned residual.

  • Prediction accuracy: Learned Ridge and symbolic corrections reduce errors across single-resonator and dimer predictions, while symbolic formulas retain explicit dependence on asymptotic quantities.The dimer symbolic corrections depend on asymptotic eigenvalues, branch splitting, and expected powers of δ and qj.
  • Dimer experiments: The dimer correction consistently improves both branches, although its errors show broader scatter because the residual depends on δ, separation κ, damping γ, and resonance branch.Dimer improvements are smaller than for a single resonator because self-interaction, interaction-block, eigenvalue-splitting, and coupling effects contribute simultaneously.
  • Feature ablation: 1.82 × 10−7 is the single-resonator mean error for M3, versus 8.23 × 10−6 for M2; the full M4 feature set reaches 1.25 × 10−7.For the dimer, adding algebraic asymptotic scales reduces mean error from 2.21 × 10−5 for M2 to 3.67 × 10−6 for M3.
  • Feature ablation: The sharp M2-to-M3 improvement, followed by further M3-to-M4 gains, shows that accuracy comes from asymptotically meaningful scales rather than simply supplying more variables.The ablation supports organizing the learned residual by the same scales appearing in the asymptotic expansion.
  • Symbolic-regression stability: The asymp log qasymp proxy appears in every preconditioned run, while the proxy is selected in 14/20 single-resonator and 15/40 dimer feature-augmentation expressions.The δ2-type contribution appears in every run for both configurations, whereas individual logarithmic variables are selected less often.
  • Proxy-guided symbolic regression: Symbolic regression reduces mean relative error to approximately 1.89 × 10−11 for the single resonator and approximately 6.04 × 10−10 for the dimer after asymptotic proxy corrections.The proxy alone reduces the baseline from approximately 9.09 × 10−4 to 1.83 × 10−5 for the single resonator and from 1.22 × 10−4 to 8.27 × 10−6 for the dimer.

7 Discussion and perspectives

The paper presents asymptotics-guided learning as a framework that predicts resonances by learning residuals organized by asymptotic variables and scales. Symbolic regression then extracts compact, interpretable correction laws, connecting asymptotic modelling with data-driven prediction.

  • Asymptotic analysis identifies variables and scales organizing approximation errors, while learning targets only the residual between asymptotic and numerical reference resonances.The asymptotic model serves as the baseline rather than the sole prediction.
  • Physics-informed features from the subwavelength expansion substantially improve residual prediction for both single resonators and dimers compared with raw-parameter learning.Two-dimensional logarithmic, frequency-dependent scales effectively correct missing higher-order terms.
  • Symbolic regression produces compact correction laws organized around size-, frequency-, damping-dependent, and branch-splitting terms.These formulas are data-driven laws valid in the sampled parameter regime, not rigorous asymptotic expansions.
  • The proposed methodology combines asymptotic structure identification, learned residual correction, and symbolic regression to obtain interpretable predictions for nonlinear spectral problems.This creates a bridge between asymptotic modelling and data-driven prediction in dispersive media.

A Numerical and learning details

The appendix specifies the numerical and software choices used to generate datasets and reproduce the learning experiments, with fixed random seeds throughout.

  • Numerical and learning details: All computations use Python, with NumPy and pandas for data handling, scikit-learn for Ridge regression and feature standardization, and PySR with SymbolicRegression.jl for symbolic regression.Random seeds are fixed throughout the experiments.

A.1 Parameter ranges and dataset construction

The datasets sample size and damping logarithmically, with dimer samples additionally varying separation. Retained data exclude failed solves and keep only outgoing damped resonances on the physical branch.

  • Single-resonator dataset: Single-resonator sampling uses δ ∼10U(−2,−1/2) and γ ∼10U(−3,−1), with samples outside the retained physical branch discarded.The nominal ranges are δ ∈[10−2, 10−1/2] and γ ∈[10−3, 10−1].
  • Dimer dataset: Dimer sampling uses δ ∼10U(−2,−1/2), γ ∼10U(−3,−1), and κ ∼10U(−2,0).The final realized ranges are approximately δ ∈[1.01 × 10−2, 3.16 × 10−1], γ ∈[1.00 × 10−3, 9.95 × 10−2], and κ ∈[1.00 × 10−2, 9.99 × 10−1].
  • Retention criteria: Samples with nonlinear resonance solver failures are discarded, retaining only outgoing damped resonances satisfying Re ω > 0 and Im ω < 0.These conditions define the retained resonance data.

A.2 Reference and reduced models

The study compares asymptotic and reference resonances for a single resonator, and reduced and reference resonances for a dimer using different operator approximations.

  • Single resonator: For the single resonator, the asymptotic resonance uses the leading asymptotic eigenvalue, while the reference resonance uses a more accurate numerical value of the same eigenvalue.These provide the asymptotic and reference models for comparison.
  • Dimer: For the dimer, the reduced model uses a coarser discretization of the two-resonator integral operator, whereas the reference model uses a finer discretization.The two resulting resonances are treated as the two dimer branches.
  • Dimer: 25 sample points per resonator are used for the coarse dimer operator, compared with 120 sample points per resonator for the reference operator.The two discretization levels define the reduced and reference dimer eigenvalues.

A.3 Dimer discretization convergence

The dimer reference resonances are checked for discretization stability by varying the quadrature points per resonator and comparing errors with the finest discretization, Nref = 200. Errors decrease systematically, supporting the stability of the reference used to train learned corrections.

  • Dimer discretization convergence: The convergence study varies N, the number of quadrature points per resonator, over six values and measures |ωN −ωNref| against Nref = 200 across both resonance branches.The configurations are sampled from the same parameter ranges as the main experiments.
  • Dimer discretization convergence: The mean error drops from approximately 3.56×10−4 at N = 25 to approximately 1.69×10−5 at N = 160.This systematic decrease indicates that the learning reference is stable enough that learned corrections are not merely artifacts of coarse discretization.

A.4 Train-test splits and validation protocols

Models use fixed-seed 80–20 train-test splits for in-distribution evaluation and parameter-space holdouts for out-of-distribution extrapolation. The main datasets contain 10 000 single-resonator samples and 3 000 dimer configurations, with corresponding 8 000/2 000 and 2 400/600 train-test splits.

  • Train-test splits and validation protocols: Models use an 80–20 train-test split with fixed random seed for in-distribution tests.Out-of-distribution validation instead holds out a challenging parameter-space region.
  • Train-test splits and validation protocols: Out-of-distribution tests extrapolate to large δ and large γ for the single resonator, and to large δ and small κ for the dimer.The holdout threshold is generally set at the corresponding 80% quantile, with an exception for the small-κ dimer case in the supplied passage.
  • Train-test splits and validation protocols: The main prediction figures use 10 000 single-resonator samples split into 8 000 training and 2 000 test samples.These dataset sizes and splits are specified for the main prediction figures.

A.5 Ridge regression … A.9 Noise experiments

The paper combines standardized Ridge regression, asymptotic feature engineering, and symbolic regression to learn compact residual corrections for resonance prediction. Assisted symbolic regression incorporates a calibrated next-order proxy, while noise experiments assess robustness under controlled perturbations and clean or noisy evaluation.

  • A.5 Ridge regression: Ridge regression learns the real and imaginary parts of the resonance residual simultaneously rather than the full resonance.Inputs are standardized using training-set means and standard deviations, with αR = 10^-8 and no cross-validation tuning.
  • A.6 Feature sets: Feature sets include logarithmic parameter transforms and physics-informed combinations motivated by asymptotic scaling.Examples include log δ, log γ, log κ, δ2, δ2 log δ, and γδ2; frequency-dependent terms use δ2q2.
  • A.6 Feature sets: Dimer frequency-dependent features are evaluated branchwise, with coupling features including asymptotic eigenvalues and branch splitting.The branchwise evaluation uses the corresponding asymptotic resonance.
  • A.7 Symbolic regression: Symbolic regression fits separate compact formulas for the real and imaginary residual components and, for dimers, for each branch.Clean-data PySR uses +, −, and ×, maximum expression size 20, 20 populations, 500 iterations, and fixed random seeds.
  • A.7 Symbolic regression: Noisy symbolic rediscovery reruns PySR with division allowed and uses modified population, execution, and iteration settings only for robustness diagnostics.These runs use +, −, ×, /, maximum expression size 20, population size 40, deterministic serial execution, and 100 iterations unless otherwise specified.
  • A.8 Assisted symbolic regression: Assisted symbolic regression first fits a calibrated next-order proxy, then either subtracts it from the residual or supplies it as an additional input feature.The experiments test whether explicit next-order asymptotic information improves the symbolic search.
  • A.9 Noise experiments: Noise experiments perturb training resonances with independent standard Gaussian variables and evaluate models on a clean held-out test set.Noise levels are σ ∈ {0, 10^-4, 10^-3, 10^-2}; Ridge and proxy-based results average 20 independent realizations per level.
  • A.9 Noise experiments: A clean-versus-noise-trained symbolic experiment compares formulas discovered from clean and noisy training data against noisy test resonances.Both formulas are evaluated against noisy test resonances.
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