Support Vector Machine Kernels as Quantum Propagators

TL;DR

This work establishes a formal correspondence between SVM kernels and quantum Green's functions, arguing that kernel performance is governed by spectral alignment with the system's propagator. It introduces both analytically derived kernel-selection rules for analytically tractable systems (Regime I) and a Kernel Polynomial Method-based framework (Regime II) to construct custom, physics-aligned kernels, ensuring positive semi-definiteness via spectral clipping. Through extensive regression studies on copper conductivity, graphene band structure, anharmonic oscillators, photonic crystals, and Fibonacci quasicrystals, the authors demonstrate that physics-informed kernels outperform standard choices, especially in nonlocal or fractal spectral regimes. The approach offers a path toward automated, physics-driven kernel design with potential applications to topological materials and quantum devices, advancing the integration of quantum theory and machine learning for predictive modeling. All mathematical notation is presented with appropriate delimiters, emphasizing the role of spectral content, operator inverses, and Green's-function decompositions in kernel construction.

Abstract

Selecting optimal kernels for regression in physical systems remains a challenge, often relying on trial-and-error with standard functions. In this work, we establish a mathematical correspondence between support vector machine kernels and quantum propagators, demonstrating that kernel efficacy is determined by its spectral alignment with the system's Green's function. Based on this isomorphism, we propose a unified, physics-informed framework for kernel selection and design. For systems with known propagator forms, we derive analytical selection rules that map standard kernels to physical operators. For complex systems where the Green's function is analytically intractable, we introduce a constructive numerical method using the Kernel Polynomial Method with Jackson smoothing to generate custom, physics-aligned kernels. Numerical experiments spanning electrical conductivity, electronic band structure, anharmonic oscillators, and photonic crystals demonstrate that this framework consistently performs well as long as there is an alignment with a Green's function.

Figures (7)

• Figure 1: Schematic illustration of the operator inversion equivalence between SVM and quantum mechanics.
• Figure 2: Comparison of four SVM kernels in predicting copper conductivity. (a) Mean Squared Error (MSE) for each kernel. The RBF kernel achieves the lowest error, indicating strong ability to model nonlinear relationships; the sigmoid kernel yields a very high MSE, reflecting poor compatibility. We also compare a polynomial kernel of $\mathrm{degree}=3$, and a simpler linear kernel. (b) Coefficient of determination $R^2$ across the same four kernels, again highlighting the superior performance of the RBF kernel.
• Figure 3: Performance comparison of SVM with different kernels in predicting the band structure of graphene. The linear and RBF kernels outperformed others, indicating better alignment with the physical process.
• Figure 4: Comparison of polynomial kernels at different degrees ($2,3,4,5,6$) for the anharmonic oscillator. Degree 3 and degree 2 consistently yield the best predictive accuracy.
• Figure 5: Performance comparison of four different kernel types (RBF, linear, sigmoid, and a polynomial of degree 3) for predicting the anharmonic oscillator energy levels. The degree 3 polynomial kernel outperforms the others on average.