KernelEvolve: Dynamic Kernel Methods

• KernelEvolve is a comprehensive framework that models evolving phenomena through dynamic kernel methods across control, machine learning, and physics applications.
• It leverages state-space formulations, fast kernel algebra, and evolutionary search to achieve scalable state estimation, control synthesis, and kernel optimization.
• The approach integrates robust simulation, theoretical insights, and hardware-accelerated strategies to deliver accurate, efficient, and adaptable models across various domains.

KernelEvolve encompasses a collection of algorithms, theoretical frameworks, and systems for modeling, inference, optimization, and control of evolving phenomena where kernel methods are central. Across application domains—machine learning, scientific computing, signal processing, systems theory, and computational physics—KernelEvolve denotes approaches that capture temporal change, nonstationarity, and evolution by explicitly modeling the dynamics of kernels, kernel-induced features, or kernel-based representations. This article reviews key formulations, methodologies, and representative applications, focusing on systems theory, scientific modeling, scalable computation, and foundational aspects.

1. Dynamical Modeling of Spatiotemporally Evolving Fields

KernelEvolve, as formalized in the systems-theoretic literature, models a time-dependent spatial field $f(x,t)$ via an RKHS expansion with $M$ dictionary centers $C = \{c_1,...,c_M\}$ and learned weights $w_j(t)$ (Kingravi et al., 2015):

$f(x,t) \approx \sum_{j=1}^M w_j(t) k(c_j, x)$

Temporal evolution is encoded by a linear dynamical systems prior in the weight space:

$$w_{k+1} = A w_k + B u_k + \eta_k,\quad y_k = K w_k + \zeta_k$$

where $A$ is the system matrix (to be identified), $B$ encodes control inputs, $K$ captures sensor geometry, and $\eta_k$, $\zeta_k$ model process/measurement noise. Observability and controllability are determined by algebraic conditions on the kernel matrices and measurement schedules, with shadedness and cyclic index of $A$ dictating sensor/actuator count for guaranteed state recovery and actuation.

System identification (Algorithm 1), Kalman-style state estimation (Algorithm 2), and LQR controller synthesis (Algorithm 3) enable practical deployment for large-scale data-driven control, e.g., sea-surface temperature estimation (37M points, 300–2000 sensors, sub-2% RMSE) and PDE control (diffusion, heat equation). Scalability is achieved by leveraging fast kernel algebra and exploiting sparsity and structure in sensor placement (Kingravi et al., 2015).

2. Kernel Evolution in High-Dimensional Machine Learning

The evolution of kernel objects under training dynamics is central in wide neural networks (Bordelon et al., 2022). The infinite-width limit admits a dynamical description via self-consistent field theory for inner-product feature and gradient kernels:

• Feature kernel: $$\Sigma^\ell_{\mu\alpha}(t,s) = \frac{1}{N} \phi(h^\ell_\mu(t))\cdot\phi(h^\ell_\alpha(s))$$
• Gradient kernel: $$\Gamma^\ell_{\mu\alpha}(t,s) = \frac{1}{N} g^\ell_\mu(t)\cdot g^\ell_\alpha(s)$$

The time evolution of $\Sigma^\ell, \Gamma^\ell$ is governed by nonlinear, stochastic recursive equations, closed by implicit functional relations reflecting the full training process. For linear networks, explicit algebraic solutions are available. Nonlinear dynamics require alternating Monte Carlo schemes for empirical kernel computation.

KernelEvolve provides a non-perturbative, width-invariant characterization of feature learning, connecting Neural Tangent Kernel limits, mean-field perturbations, and generalization theory. Empirical validation via CIFAR CNNs shows loss and kernel alignment dynamics collapse across widths at fixed feature-learning strength $\gamma_0$, substantiating theoretical predictions (Bordelon et al., 2022).

3. Kernel Evolution and Optimization for Heterogeneous Accelerators

In large-scale AI systems, KernelEvolve describes an agentic, retrieval-augmented kernel synthesis and optimization service for high-performance operator development across heterogeneous hardware (NVIDIA, AMD, custom accelerators) (Liao et al., 29 Dec 2025). This framework formulates kernel optimization as graph-based search, where each candidate kernel implementation is a node, and search is guided by selection policies (greedy, MCTS/UCT, evolutionary), universal LLM-driven operators, fitness evaluation (speedup over PyTorch baseline), and dynamic prompt synthesis informed by real-time profiling and knowledge base retrieval.

Retrieval-augmented prompts inject hardware-specific guidance, constraint reminders, and error traces for efficient context adaptation. KernelEvolve achieves full correctness and coverage on the KernelBench suite (250 problems, 100% pass), supports 160 PyTorch ATen operators across all hardware targets, and demonstrates speedups up to 17× for critical production workloads. Development times are reduced from weeks to hours; rigorous metadata, parallel FaaS evaluation, and fault tolerance ensure scalability (Liao et al., 29 Dec 2025).

4. Evolution Kernels in Computational Physics

KernelEvolve encompasses numerical schemes for evolving PDE solutions, most notably in advection–diffusion and conservation law problems. High-order explicit, unconditionally stable schemes leverage kernel-convolution representations for spatial derivatives, resolvent series truncation, SSP-RK temporal integration, WENO reconstruction for non-smooth regions, and $O(N)$ fast recursive convolution evaluation (Christlieb et al., 2017). Stability (A-stability) is proven via von Neumann analysis; the approach admits CFL-free timestepping and robust handling of degeneracies.

Meshfree kernel discretizations for first-order evolution PDEs combine SPH-type reconstructions with Eulerian ODE frameworks. Convergence is guaranteed under kernel moment conditions and solution smoothness; order is tunable via kernel construction (Vandermonde scaling, zero moment enforcement) (Ramming et al., 2016).

For surface PDEs, kernel-based collocation provides high-order meshless solvers, admitting both parametric and point-cloud-based surface evolution, spectral accuracy, and robust handling of high-curvature/discontinuous data. Intrinsic surface derivatives, tangent frame projection, and regularization ensure numerical stability and convergence (Chen et al., 2019).

The Stable Physics-Informed Kernel Evolution (SPIKE) paradigm applies reproducing-kernel representations and regularized parameter evolution to nonlinear conservation laws. SPIKE resolves the paradox of strong-form residual minimization for weak solutions, automatically enforces conservation and Rankine-Hugoniot conditions, and efficiently traverses dynamic shock formation without explicit detectors or viscosity. ODE-based parameter evolution and efficient block-tridiagonal solvers yield $O(N)$ complexity and robust solution profiles for Burgers’, Buckley–Leverett, and Euler systems (Su et al., 21 Oct 2025).

5. Evolution Kernels in QCD and Particle Physics

KernelEvolve also refers to evolution kernels in high-energy physics, notably DGLAP kernels in parton shower Monte Carlo frameworks and GPD evolution equations (Kusina et al., 2015, Bertone et al., 2022). Evolution kernel recalculation in MC-friendly schemes enables direct NLO usage within transverse-momentum-ordered parton showers, reconciles with $\overline{\mathrm{MS}}$ results at the inclusive level, and achieves precise Sudakov form factor construction for no-emission probabilities. Differences in exclusive distributions affect subleading observables without perturbing leading PDFs.

GPD evolution kernels are formulated for direct convolution implementation, ensuring compliance with sum rules, DGLAP and ERBL limits, and polynomiality. Public implementations (APFEL++, PARTONS) allow evolution across heavy-quark thresholds, confirm analytic limits, and benchmark against legacy codes (Bertone et al., 2022).

6. Evolutionary Kernel Search for Gaussian Processes

KernelEvolve, in the GP literature, can denote evolutionary algorithms for kernel selection. Expression trees constructed from elementary mathematical primitives define the kernel search space; strongly-typed GP recombination and mutation enforce PSD and parsimony constraints, while BIC penalization controls complexity. Explicit grammar rules allow construction of novel kernels that blend smooth and periodic components, outperforming fixed-basis compositional searches in time-series prediction benchmarks (Roman et al., 2019).

Fitness optimization is framed as marginal likelihood maximization with cross-validation variants; multi-start Powell search and parental hyperparameter inheritance ensure efficient hyperparameter tuning. Empirical results show lower RMSE and model complexity compared to state-of-the-art baselines, with robustness to bloat and non-PSD candidate rejection (Roman et al., 2019).

---

Summary Table: Major KernelEvolve Paradigms and Domains

Domain	Core KernelEvolve Concept	Representative Publication
Control & Estimation	RKHS field evolution + system theory	Kernel Controllers (Kingravi et al., 2015)
Wide Neural Networks	DMFT for feature/gradient kernel dynamics	Bordelon et al (Bordelon et al., 2022)
AI Systems–Accelerators	Agentic LLM–guided kernel search/optimization	Meta (Liao et al., 29 Dec 2025)
Scientific Computing	Explicit convolutional kernel PDE solvers	Li et al (Christlieb et al., 2017), Ramming & Wendland (Ramming et al., 2016)
Surface PDEs	Intrinsic meshless kernel collocation	Chen et al (Chen et al., 2019)
Conservation Laws	SPIKE: regularized kernel evolution	SPIKE (Su et al., 21 Oct 2025)
Particle Physics	Parton shower MC/GPD kernel evolution	Kusina et al (Kusina et al., 2015), Bertone et al (Bertone et al., 2022)
GP Kernel Learning	Evolutionary search in kernel grammar space	Garcıa et al (Roman et al., 2019)

---

KernelEvolve, in all its technical incarnations, reflects a convergence of kernel methods, dynamical systems, computational optimization, and learning-theoretic principles, enabling scalable, interpretable, and high-performance modeling of evolving processes across data-rich scientific, engineering, and AI domains.