Support Vector Machine Kernels as Quantum Propagators

TL;DR

This paper establishes a mathematical link between SVM kernels and quantum propagators, enabling physics-informed kernel design and selection for improved regression in physical systems.

Contribution

It introduces a novel isomorphism between kernels and quantum propagators, providing analytical and numerical methods for physics-aligned kernel construction.

Findings

Framework performs well with Green's function alignment

Analytical rules for kernel-system mapping

Numerical method for complex systems

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.