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This research introduces a physics-informed kernel strategy that leverages Green's function analogies and Jackson-damped Chebyshev polynomials to improve regression of physical observables with limited data. By integrating spectral structures directly into the machine learning architecture, the method provides an inspectable prior for complex systems ranging from photonic crystals to Dirac-like band dispersions without requiring exact identification between kernels and propagators. The approach represents a significant advancement in merging mathematical physics principles with statistical learning for finite-data regimes in quantum and condensed matter contexts.
Physics-Informed Support Vector Kernels via Green-Function Analogies and Jackson-Chebyshev Spectral Design
arXiv:2502.11153v4 Announce Type: replace Abstract: Kernel selection for regression of physical observables is often heuristic. We investigate a physics-informed strategy in which functional forms and spectral structures associated with Green's functions motivate kernel selection without requiring an exact identification between a machine-learning kernel and a physical propagator. The principal construction is a Jackson-damped Chebyshev kernel inspired by the kernel polynomial method (KPM); its explicit feature map yields a positive-semidefinite Gram matrix by construction and provides an inspectable spectral prior for structured observables. We evaluate standard and custom SVR models on copper-conductivity proxies, local Dirac-like band dispersion, quartic-oscillator energy levels, photonic-crystal transmission, and Fibonacci-chain transmission using repeated nested validation, learning curves, random-forest and multilayer-perceptron baselines, and low-rank Nystr\"om tests where relevant. The framework is intended for finite-data regression of precomputed observables while boundary conditions remain part of the physical model that generates those observables.
Abstract:Kernel selection for regression of physical observables is often heuristic. We investigate a physics-informed strategy in which functional forms and spectral structures associated with Green's functions motivate kernel selection without requiring an exact identification between a machine-learning kernel and a physical propagator. The principal construction is a Jackson-damped Chebyshev kernel inspired by the kernel polynomial method (KPM); its explicit feature map yields a positive-semidefinite Gram matrix by construction and provides an inspectable spectral prior for structured observables. We evaluate standard and custom SVR models on copper-conductivity proxies, local Dirac-like band dispersion, quartic-oscillator energy levels, photonic-crystal transmission, and Fibonacci-chain transmission using repeated nested validation, learning curves, random-forest and multilayer-perceptron baselines, and low-rank Nyström tests where relevant. The framework is intended for finite-data regression of precomputed observables while boundary conditions remain part of the physical model that generates those observables.
Quantum Physics (quant-ph); Statistical Mechanics (cond-mat.stat-mech); Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Mathematical Physics (math-ph)
arXiv:2502.11153 [quant-ph]
(or arXiv:2502.11153v4 [quant-ph] for this version)
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