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 case
Asymptotic mean error
Corrected mean error
Improvement factor
Single, in-distribution
9.09×10−4
2.22×10−7
4.10×103
Single, large- δ OOD
2.29×10−3
7.89×10−7
2.90×103
Single, large- γ OOD
9.94×10−4
7.92×10−7
1.26×103
Dimer, in-distribution
1.22×10−4
3.47×10−6
35.2
Dimer, large- δ OOD
3.68×10−4
7.35×10−5
5.01
Dimer, small- κ OOD
1.47×10−4
3.83×10−5
3.82
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.
Configuration
Asymptotic
Ridge
Symbolic
Assisted symbolic
Single resonator
9.09×10−4
2.22×10−7
4.70×10−8
1.89×10−11
Dimer
1.22×10−4
3.47×10−6
8.35×10−7
9.30×10−11
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 . Right: comparison of the predicted and reference resonances in the complex plane. The asymptotic approximation deteriorates as δ 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 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 . 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 and qj .
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 , 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 , damping and branch coupling.
Configuration
Model
Features
Median
90th perc.
Single
Asymptotic baseline
0
5.85×10−4
2.28×10−3
Single
M0 : raw
2
1.26×10−4
2.12×10−4
Single
M1 : raw + logs
4
3.81×10−5
6.38×10−5
Single
M2 : + asymptotic resonance
6
6.95×10−6
1.49×10−5
Single
M3 : + algebraic scales
9
1.42×10−7
3.73×10−7
Single
M4 : + frequency-log scales
12
8.19×10−8
2.61×10−7
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.
Strategy
Feature or structure
Single
Dimer
Feature augmentation
proxy
14/20
15/40
δ2q2
14/20
21/40
δ2q2logδ
1/20
3/40
δ2q2logq
6/20
13/40
Preconditioning
proxy
20/20
40/40
δ2q2
20/20
40/40
Table 4: Stability of the assisted symbolic-regression expressions. The counts are computed over ten independent runs, corresponding to 20 symbolic expressions for the single resonator and 40 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 -dependent terms and splitting-dependent terms, respectively.
Configuration
Strategy
Mean
Median
Standard deviation
Single
next-order proxy
7.31×10−3
1.85×10−5
1.97×10−2
Single
asymptotic preconditioning
5.07×10−5
1.54×10−6
1.22×10−4
Single
feature augmentation
1.66×10−7
7.53×10−8
1.53×10−7
Dimer
next-order proxy
3.45×10−3
9.68×10−5
5.26×10−3
Dimer
asymptotic preconditioning
1.10×10−4
1.86×10−5
1.76×10−4
Dimer
feature augmentation
4.87×10−6
3.18×10−6
4.11×10−6
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
N
Mean
Median
90th percentile
25
3.56×10−4
8.29×10−5
1.18×10−3
50
1.71×10−4
3.88×10−5
5.75×10−4
80
9.68×10−5
2.26×10−5
3.21×10−4
120
3.44×10−5
6.97×10−6
1.14×10−4
160
1.69×10−5
3.53×10−6
5.53×10−5
200
0
0
0
Appendix
Table 6: Dimer discretization convergence. The errors are measured relative to the finest discretization Nref=200 , using ∣ωN−ωNref∣ , and are aggregated over both resonance branches.
Figure 11: Dimer discretization convergence as a function of the number N of quadrature points per resonator. The reference discretization is Nref=200 .
High-fidelity finite-element simulations can provide accurate numerical predictions for side-branch resonators, but large simulation datasets are expensive to generate and purely data-driven surrogates may become unreliable when simulation-labelled data are scarce. This study develops an analytical-prior learning framework that reuses a low-cost analytical model to improve data efficiency under limited high-fidelity simulation budgets. Two complementary routes are considered. When the analytical model remains available at inference, it is retained as an explicit baseline and the simulation data are used to learn only the analytical-to-simulation discrepancy. When a self-contained predictor is required, the analytical mapping is first distilled from abundant low-cost evaluations into a learned prior and then calibrated with the limited simulation data. The framework is evaluated on rectangular side-branch Helmholtz resonators using 86 simulation-labelled geometries and 8,998 non-overlapping analytical-only geometries. The analytical model achieved a mean absolute error (MAE) of 1.333 Hz. Direct support vector regression (SVR) achieved 3.375 Hz, while residual SVR reduced the MAE to 0.426 Hz. A direct multilayer perceptron (MLP) achieved 1.109 Hz, whereas analytical-prior pretraining reduced the error to 0.556 Hz with frozen-prior residual adaptation and 0.371 Hz with full-model fine-tuning. Across training budgets of 20 to 70 simulation-labelled cases, both analytical correction and analytical-prior pretraining consistently improved data efficiency relative to direct learning. These results show that analytical prior information can substantially improve high-fidelity prediction when simulation data are scarce, with explicit correction and prior distillation serving complementary deployment needs.
Jiaming Li
School of Informatics, Computing, and Cyber Systems, Northern Arizona University, Flagstaff, Arizona 86011, USA
Symbolic regression combines structural search with numerical fitting, but aggregate fit scores do not describe how the remaining error varies across inputs. We introduce RISR, a residual-informed method that uses these error patterns to guide formula discovery and learn which corrections are worth fitting. A residual encoder compresses aligned inputs, targets, current predictions, and residuals into continuous tokens that condition a language model to propose formulas. For subsequent refinement, a dual-view relational encoder uses additive and regularized multiplicative residuals to predict the post-fit utility of candidate corrections. We evaluate RISR on scientific tasks from the LLM-SRBench. RISR achieves 63.57% and 38.50% ID accuracy at the 1% and 0.1% pointwise relative-error tolerances, respectively. The corresponding OOD accuracies are 56.07% and 38.24%. RISR outperforms the reported baselines using the same backbone. The results show that our residual-informed approach can improve numerical equation recovery.
Haobo Li, Wenshuo Zhang, Wenxiao Zhao +6
Shanghai AI Laboratory · The Hong Kong University of Science and Technology · University of California, Los Angeles
Symbolic regression (SR) aims to discover explicit mathematical expressions that explain observed data and is widely used in domains where interpretability is essential. Because interpretability requires expressions to reflect meaningful regularities, SR is sensitive to observations that deviate from the dominant relationship. Such irregular observations, or outliers, are common in real-world data and can hinder SR from identifying underlying regularities. Robust regression mitigates this by downweighting observations with large residuals. However, deciding which observations should be treated as outliers is often ambiguous and depends on user interpretation and domain knowledge, a perspective largely overlooked in existing SR studies. This motivates approaches that present multiple candidate expressions, allowing users to examine different residual patterns and choose expressions consistent with their expertise. We propose diversified residual symbolic regression (DRSR), which achieves high predictive accuracy while promoting diversity with respect to residual patterns based on the Quality-Diversity paradigm. DRSR collects multiple expressions that fit the data well but differ in how residuals are distributed, enabling post-search selection aligned with domain knowledge. On a synthetic mixture dataset, DRSR produces more diverse expressions than conventional SR while capturing multiple underlying relationships. On a real-world astronomical dataset, DRSR discovers multiple expressions consistent with known physical relationships.
Koki Ikeda, Masahiro Nomura, Ryoki Hamano
CyberAgent Shibuya, Tokyo, Japan · Institute of Science Tokyo Meguro, Tokyo, Japan