Evolution of Kernels: Concepts & Applications

• Evolution of kernels is a concept describing the adaptive change or optimization of kernel functions in both mathematical models and computational algorithms.
• It encompasses methods from quantum field theory, operating system tuning, machine learning, and graph analysis to enhance precision and performance.
• Applications include threshold resummation in QCD, adaptive system policies in OS, genetic programming in ML, and dynamic graph classification techniques.

The Evolution of Kernels (EoK) encompasses a spectrum of mathematical, computational, and physical concepts describing how kernel functions, kernel-based operators, or kernel-inspired model components change, adapt, or are optimized in response to domain-specific requirements. Across quantum physics, high-energy theory, signal processing, machine learning, graph dynamics, and operating system design, the term “evolution of kernels” captures the intricate interplay between structure, function, and optimality—often under nontrivial constraints such as symmetry, causality, statistical regularization, or algorithmic efficiency.

1. Kernel Evolution in Quantum Field Theory and QCD

In the context of perturbative quantum chromodynamics (QCD) and related field theories, kernels serve as generators of the scale dependence of physical observables such as parton densities and fragmentation functions. The evolution of these kernels, particularly at high orders of perturbation theory, encodes the resummation of large logarithms and the interplay between real and virtual corrections.

A central object is the evolution kernel $K_a$ that governs the $Q^2$ dependence of hard scattering observables (“structure functions”) $F_{a}(x, Q^2)$: $\frac{d}{d\ln Q^2} F_{a} = K_{a} F_{a}$ Here, $K_{a}$ incorporates both the standard splitting functions and scheme-dependent coefficient functions, via

$$K_{a} = P_{a} + \beta(a_{s}) \frac{d}{d a_{s}} \ln C_{a}$$

where $P_{a}$ is the splitting function, $C_{a}$ the coefficient function, and $\beta(a_s)$ the QCD beta function (Vogt et al., 2010).

A key insight is that physical evolution kernels exhibit only single-logarithmic enhancement at large $x$—in contrast to double-logarithmic enhancements seen in the coefficient functions. Explicitly, non-singlet splitting functions take the schematic form: $$P_{\mathrm{ns}}^{(\ell)}(x) = A_{\ell+1} \left[\frac{1}{1-x}\right]_+ + B_{\ell+1} \delta(1-x) + C_{\ell+1}\ln(1-x) + \cdots$$ while individual coefficient functions can display terms as singular as

$$c_{a,\mathrm{ns}}^{(\ell)}(x) \sim \left[\frac{1}{1-x}\right]_+ \ln^{2\ell-1}(1-x) + \cdots$$

The observed cancellation of double logarithms in $K_a$ is central for threshold resummation and enables precise predictions of the highest double logarithms in uncalculated higher-order corrections—most notably for four-loop singlet splitting functions (Vogt et al., 2010). The Mellin-N space analysis reveals the underlying exponentiation for both leading and subleading contributions.

Moreover, in the context of double parton distributions (DPDs), colour structure induces nontrivial modifications to the evolution kernels. At two-loop order, kernels project onto colour representations $R$ and differ from the standard DGLAP splitting functions through additional $c^{(n,R)}$ and $\gamma_J^{(n,R)}$ contributions, especially in the endpoint (delta-function) terms. For example: $$P^{(R)}_{ab}(x, \zeta/\mu^2) = \hat{P}^{(n)}_{ab}(R)(x) + \frac{1}{2}\delta_{RR} \delta_{ab} \delta(1-x) [d_a^{(n)} + c^{(n,R)} - \frac{1}{2}\gamma_J^{(n,R)}\ln(\zeta/\mu^2)]$$ with distinct mixing patterns and polarisation behaviors in various channels (Diehl et al., 2022). The non-singlet colour structure is particularly relevant for small-$x$ and endpoint dynamics in hadron–hadron collisions.

2. Monolithic Kernel Policy Evolution and Operating Systems

In the context of modern operating system (OS) kernels, “evolution of kernels” refers to the automated, in situ adaptation of system-level policies and parameters to optimize workload performance. EOS is representative: it enables the OS kernel to incrementally adapt to production workloads using an explicit API for parameter annotation, a policy cache for workload signature memorization, and a hierarchical search engine for bottleneck-driven optimization (Cui et al., 2015).

The evolutionary workflow in EOS involves:

• Annotation of tunable parameters by developers (through C macros and structures for subsystems, parameters, and dependencies)
• Recording workload sensor measurements and matching them to previously optimized parameter sets (using signature similarity conditions such as $|s_1 - s_2| \leq \alpha s_2$)
• Adaptive tuning through a hierarchical, modified orthogonal search that respects parameter interdependencies
• Incremental and automated performance improvement via policy reuse and staged search

This approach not only automates the tedious task of kernel tuning but also embodies a paradigm for future OS kernel development where adaptability and workload awareness are built into the system (Cui et al., 2015).

3. Genetic and Programmatic Evolution of Machine Learning Kernels

The evolution of kernels in machine learning most concretely refers to evolutionary, typically genetic-programming-based, discovery and optimization of kernel functions for tasks such as classification, regression, object categorization, or time-series extrapolation.

• In multiple kernel learning (MKL), genetic programming evolves non-linear kernel combinations far beyond linear summations, encoding base kernels as expression trees with operators “$+$”, “$*$” (Korra, 2016).
• For Gaussian Processes, highly flexible kernel function evolution via expression trees allows principled search over a vast functional space (e.g., combinations of basic kernels, elementary arithmetic functions, and non-linearities), subject to strong type and positive-definiteness constraints (Roman et al., 2019). Fitness is evaluated via Bayesian Information Criterion (BIC) for regression, with hyperparameter optimization nested within the evolutionary loop.
• Multi-objective approaches (e.g., MOECov) augment kernel evolution to balance predictive accuracy (Pearson correlation, negative log predictive density) and computational cost, using Pareto-optimal selection (Roman et al., 2019). The evolutionary operators—tree mutation and crossover—handle both topology and parameter inheritance.

Empirically, genetically evolved kernels yield performance improvements and can capture structure or class-relevant features missed by conventional hand-designed or linearly composed kernels. A prominent implication is that evolved kernels display strong transferability across related tasks or domains, provided the underlying data representation is compatible (Roman et al., 2019).

4. Kernel Evolution in Nonlocal and Complex Systems

In mathematical physics and probability, nonlocal evolution equations are often governed by kernels encoding spatial interactions, stochastic transitions, or jump process dynamics. Recent work considers homogenization limits for jump processes on domains partitioned into microstructured subdomains, with distinct smooth kernels $J(x,y), G(x,y), R(x,y)$ for intra- and inter-domain jumps (Capanna et al., 2020):

• The nonlocal generator $L_n$ acts on densities $u_n(t, x)$ and the scale-dependent structure (characteristic functions converging weakly) leads to a limit system where kernels are weighted by local volume fractions—a coupling reminiscent of classical homogenization but intrinsically nonlocal and probabilistic.
• The convergence and limit theorems depend on precise boundary conditions (Neumann or Dirichlet) and the initial data (smooth or singular/delta functions).

The evolution kernel as a generator for the limiting Markov process directly encodes the macroscopic dynamics in terms of micro-scale jump rates, with implications for the scaling limits of stochastic processes and applications to composite materials, diffusion in heterogeneous media, and similar systems (Capanna et al., 2020).

5. Adaptive and Automated Kernel Optimization Using LLMs

Recent advances in LLM-driven code generation have introduced a new dimension to the evolution of computational kernels, particularly for domains lacking extensive optimization reference corpora (e.g., RISC-V). EoK (Chen et al., 14 Sep 2025) integrates:

• Automated mining of optimization “ideas” and granular “actionable thoughts” from mature kernel library development histories, using LLM-based embeddings and hierarchical clustering to formalize reusable strategies.
• An evolutionary search process (EoH) guided by softmax-weighted sampling from an Idea Pool and a database of prior kernels, with kernel generation steered by LLMs using retrieval-augmented context (e.g., hardware manuals, ISA docs, and hardware-specific examples).
• Empirical validation by demonstrating that, on a set of 80 diverse kernel design tasks, EoK consistently produces kernels outperforming human reference implementations (median speedup $\sim$1.27$\times$) and surpasses previous LLM-based approaches by 20%.

The framework operates iteratively: it selects a (weighted) optimization idea and a reference kernel, seeds an LLM evolution, evaluates variants on target hardware (e.g., Spacemit K1 RISC-V system), and updates the database of candidate kernels and the performance scores of the ideas. The process explicitly accommodates LLM prompt mutation, crossover, and multi-generation evolution cycles.

A concise mathematical representation of the Idea Pool $\mathcal{P}^I$ is: $$\mathcal{P}^I = \{ (\text{idea\_msg}_1, \{\theta_1^{(1)}, \theta_1^{(2)}, \ldots\}), \; \ldots, \; (\text{idea\_msg}_n, \{\theta_n^{(1)}, \theta_n^{(2)}, \ldots\}) \}$$ where each $\theta_k^{(j)}$ is an actionable thought with an estimated effectiveness score (Chen et al., 14 Sep 2025).

6. Kernel Evolution in Graphs and Complex Systems

The “Evolution Kernel Method” for graph classification introduces a transformation that constructs a temporal sequence (“evolution”) for each static graph via heat kernel diffusion: $H_t = e^{-t\mathcal{L}}$ where $\mathcal{L}$ is the normalized Laplacian matrix (Liu et al., 2023). The heat vector $u_t = H_t u_0$ (typically initialized as uniform) determines the node-level importance as a function of time.

A DropNode augmentation stochastically drops nodes at each time point according to the Boltzmann-modeled (normalized) heat probability, effectively generating a temporal episode $S(G) = \{G^{t_0}, ..., G^{t_N}\}$ per graph.

To compare the “evolution” of two graphs, the Graph Dynamic Time Warping (GDTW) distance is computed by aligning the temporal episodes using a cost matrix: $$\mathbf{M}(i, j) = \delta\left(G_m^{t_i}, G_n^{t_j}\right)$$ with recursive computation of the optimal path cost $\gamma(i, j)$. The GDTW-inducing kernel is then employed in supervised classification (e.g., via SVM), consistently improving accuracy across molecular and social network datasets (Liu et al., 2023). The approach highlights the informational richness encoded in the dynamic evolution of graph structure as opposed to static descriptors.

7. Theoretical Symmetry, Conformal Invariance, and Kernel Construction

Kernel evolution also arises in the context of conformal field theory and the renormalization-group analysis of operator mixing. Twist-two operator evolution in QCD is governed by evolution kernels invariant under the $\mathrm{SL}(2, \mathbb{R})$ collinear subgroup. Beyond one loop, quantum corrections deform canonical symmetry generators, requiring a similarity transformation that restores canonicity: $\mathscr{H}(a) = e^{-X(a)} \mathbb{H}(a) e^{X(a)}$ A compact recurrence procedure relates anomalous dimensions (the kernel’s spectrum) to integral kernels (explicit operator forms), with the weight function $h(\tau)$ reconstructed from the parity-respecting anomalous dimensions using a Mellin transform (Ji et al., 2023): $$h(\tau) = \int_C \frac{dN}{2\pi i} (2N+1) \Delta\hat{\gamma}(N) P_N\left( \frac{1+\tau}{1-\tau} \right)$$ where $P_N$ is the Legendre polynomial. This approach yields transparent analytic control of multi-loop evolution kernels, with applications extending to precision calculations in QCD and maximally supersymmetric Yang–Mills theory.

Summary Table: Kernel Evolution Contexts and Techniques

Domain	Kernel Evolution Mechanism	Key Feature/Technique
QCD / High-Energy Theory	Perturbative expansion, threshold exponentiation, color projection; Mellin-N analysis	Logarithmic enhancement structure, resummation, color-dependent corrections (Vogt et al., 2010, Diehl et al., 2022, Ji et al., 2023)
OS Kernel Policies	In vivo policy evolution via API, caching, hierarchical search	Automatic, adaptive system tuning (Cui et al., 2015)
ML / GP / MKL	Genetic programming of expressions, multi-objective evolution	Flexible, interpretable kernel discovery (Korra, 2016, Roman et al., 2019, Roman et al., 2019)
RISC-V Kernel Optimization	LLM-guided evolution using human-mined Idea Pools + RAG	Data-driven, hardware-aware code search (Chen et al., 14 Sep 2025)
Nonlocal/Probabilistic Systems	Jump operators, spatial kernel homogenization	Macroscale dynamics from microscale inhomogeneity (Capanna et al., 2020)
Graphs / Complex Networks	Time-augmented dynamics via heat kernels, DropNode, GDTW	Classification via temporal trajectory comparison (Liu et al., 2023)
Conformal / RG Dynamics	SL(2,R) invariance, similarity transformation, Mellin inversion	Restored symmetry, analytic control (Ji et al., 2023)

Conclusion

Kernel evolution is a unifying theme spanning many mathematical and computational disciplines, describing either the analytic or algorithmic change of kernel functions or the data-driven, automated adaptation of kernels for optimal system performance. Whether in physical evolution equations, computational architectures, machine learning model selection, or adaptive operating systems, the evolution of kernels serves as the principal mechanism by which structure and function respond to underlying constraints, new data, and domain-specific objectives.