Kernel-Based Operator Approach

• Kernel-Based Operator Approach is a computational framework employing operator-valued kernels and RKHS to model nonlinear operators between function spaces.
• It leverages the representer theorem and kernel ridge regression to ensure data fitting, regularization, and the enforcement of system-theoretic constraints such as IQCs.
• The method is applied across system identification, PDE discovery, and control tasks, offering theoretical error guarantees, interpretability, and scalability.

A kernel-based operator approach refers to a class of computational and theoretical frameworks that employ reproducing kernel Hilbert space (RKHS) constructions to model, identify, approximate, or infer nonlinear operators acting between function spaces or trajectories. Such operators—potentially nonlinear, multi-output, and nonparametric—arise in diverse contexts: input-output system identification, partial differential equation (PDE) operator learning, control, operator-valued regression, and probabilistic message-passing, among others. The unifying element is the utilization of operator-valued or function-valued kernels, which encode input-output structure and enable the transduction of fitting, regularization, and even system-theoretic constraints into explicit, tractable optimization problems.

1. Operator-Valued Kernels and RKHS Fundamentals

Let $\mathcal U$ and $\mathcal Y$ be Hilbert spaces of input and output trajectories, e.g., $\mathcal U=L_2(\mathbb R^+, \mathbb R^m)$, $\mathcal Y=L_2(\mathbb R^+, \mathbb R^p)$. A (possibly nonlinear) operator $f: \mathcal U \to \mathcal Y$ is embedded in a vector- or operator-valued RKHS $\mathcal H$ determined by a positive-definite kernel $K: \mathcal U \times \mathcal U \to B(\mathcal Y)$. The Moore–Aronszajn theorem guarantees a unique, complete $\mathcal H$ associated with $K$, satisfying the reproducing property: $$\langle f, K(\cdot, u) y \rangle_\mathcal H = \langle f(u), y \rangle_\mathcal Y$$ for all $f \in \mathcal H$, $u \in \mathcal U$, $y \in \mathcal Y$. The representer theorem ensures that any minimizer of a regularized empirical risk in $\mathcal H$ can be written as a finite kernel expansion in terms of the training data; for example, fitting $N$ input-output trajectory pairs $\{(u^{(i)}, y^{(i)})\}_{i=1}^N$ leads to

$$\hat f(\cdot) = \sum_{j=1}^N K(\cdot, u^{(j)}) c_j,\qquad c_j \in \mathcal Y.$$

The coefficients $\{c_j\}$ are found by solving a block-structured linear system associated with the kernel Gram operator.

Crucially, the form and properties of $K$—separable, entangled, diagonal, compact, etc.—control the function class and can encode structural output dependencies, input-output couplings, or nonexpansivity, as required for analytic or physical constraints.

2. Regularization, Integral Quadratic Constraints, and System Analysis

Regularization in the kernel-based operator framework takes the form $\lambda \|f\|^2_{\mathcal H}$, ensuring well-posedness and controlling model complexity. For systems-theoretic tasks, additional constraints are often imposed, such as incremental integral quadratic constraints (IQCs), which can guarantee robust stability, passivity, monotonicity, or finite gain. For instance, given a supply-rate matrix $\Phi \in \mathbb R^{(m+p)\times(m+p)}$, the incremental IQC is

$\int_0^T w(t)^\top \Phi w(t) dt \geq 0$

for $w(t) = [u(t) - u'(t); y(t) - y'(t)]$ and all input/output pairs. In the RKHS context, such constraints can be implemented by selecting "non-expansive" kernels (where a certain operator-induced semi-metric does not expand under the kernel) and regularization parameter $\lambda$ large enough to ensure contractivity in the RKHS norm (e.g., $\|f\|_{\mathcal H} \leq 1$).

This renders the operator estimation fully compatible with input-output small-gain/passivity analysis, contrasting with standard nonlinear state-space approaches that fit data well but cannot enforce dissipativity or robustness properties directly.

3. Kernel-Based Operator Learning Algorithms

The core algorithmic principles are a direct consequence of RKHS duality and the representer theorem:

• Kernel Ridge Regression (KRR) for Operator-Valued or Vector-Valued Outputs: Construct the block or operator-valued Gram matrix $G_{ij} = K(u^{(i)}, u^{(j)})$ and solve

$(G + \lambda I) c = y$

for the stacked coefficients $c$ corresponding to the output data $y$. For high-dimensional or structured outputs (e.g., functions, vectors, time series), separable kernel constructions or efficient iterative solvers (conjugate gradients, low-rank approximations) may be used to handle large-scale Gram matrices.

• Operator Constraints: To enforce IQC-type properties, choose $K$ such that

$$\| K(u,u) - K(u,v) - K(v,u) + K(v,v) \|_{B(\mathcal Y)}^{1/2} \leq \|u-v\|_{\mathcal U}$$

and calibrate $\lambda$ so that $\|f\|_{\mathcal H} \leq 1$.

• Solving Nonlinear Operator Learning Tasks: The above applies to integral operators (as in PDE solution mapping), spatio-temporal models (with operator-valued separable kernels reflecting space-time tensor structures), and even message operators in probabilistic inference, by constructing appropriate embeddings or feature maps and random feature approximations.

Algorithmic variations include multiple operator-valued kernel learning (block coordinate descent on kernel weights and regression coefficients (Kadri et al., 2012)), entangled kernel learning (alignment-based optimization beyond separable structures (Huusari et al., 2021)), and stochastic approximation with dimension-free rates and rich interpolation for misspecified targets (Yang et al., 14 Sep 2025).

4. Applications Across System Identification, PDEs, and Operator Inference

The kernel-based operator paradigm is highly versatile and appears in multiple domains:

• Nonlinear Input-Output System Identification: Identification of operators $f: \mathcal U \to \mathcal Y$ from trajectory data, guaranteed to satisfy incremental dissipation/passivity via kernel design and regularization (Waarde et al., 2021). This blends nonparametric data-fitting with control-theoretic design constraints, supporting robust closed-loop control.
• Spatio-Temporal Dynamics and Koopman Analysis: Operator-valued RKHSs allow the identification and spectral approximation of dynamical generators (Koopman/Perron-Frobenius) in high-dimensional settings, with provable convergence and error estimates (Withanachchi, 23 Aug 2025, Williams et al., 2014, Klus et al., 2018, Klus et al., 2020). Applications include molecule dynamics, forecasting, reduced-order modeling, and stability analysis.
• PDE Discovery and Solution Operators: Multi-stage frameworks denoise empirical solutions, regress PDE operators in feature space, and solve the resulting equations via kernel-based operator inversion with rigorous convergence in suitable Sobolev norms (Long et al., 2022, Bao et al., 2022).
• Control and Optimization: Kernel operators lift nonlinear optimal control problems to convex programs in occupation kernel spaces, achieving linearity over trajectories and circumventing the intractability of dynamic programming in high dimension (Kamalapurkar et al., 2021).
• Probabilistic Graphical Models: Message-passing operators in Expectation Propagation (EP) are replaced by learned kernel regression mappings between distributional messages, accommodating principled uncertainty quantification and scalable online learning (Jitkrittum et al., 2015, Jitkrittum et al., 2015).

5. Theoretical Guarantees: Representer Theorems, Error Rates, and Convergence

The theoretical foundations rely on:

• Representer Theorem for Operator-Valued RKHS: Ensures finitely parameterized solutions under quadratic regularization, even for infinite-dimensional inputs/outputs.
• Regularity Theory: Underlying error bounds relate the fill distance of input/output measurement points, kernel smoothness (e.g., Matérn, Sobolev indices), and the target operator's regularity. For instance, spatial and temporal approximation errors in spatio-temporal learning decay like $O(N^{-\beta/d})$ for appropriate smoothness of the target (Withanachchi, 23 Aug 2025).
• Stochastic Approximation and Misspecification: Polynomial, dimension-free convergence rates are established for SGD-like learning in vector-valued RKHS, including precise interpolation space characterization for target operators outside the RKHS (Yang et al., 14 Sep 2025).
• Structural Advantages over State-Space and Neural Operator Methods: Kernel-based operator learning offers closed-form training, a priori error guarantees, and model interpretability through RKHS norms, often matching or surpassing the empirical performance of DeepONet, FNO, and similar neural architectures on various PDE and dynamical benchmarks (Batlle et al., 2023).

6. Advanced Topics: Entangled Kernels, Integral Operators, and Structure

Beyond classical separable (diagonal) kernel constructions $K(x,x')=k(x,x')T$, recent advances include:

• Entangled Operator-Valued Kernels: These are defined via quantum-inspired constructions (partial traces, block-wise Kronecker structure) and learn complex input-output dependencies not captured by Kronecker-separable designs. Learning is performed via generalized kernel alignment and low-rank Choi–Kraus decompositions, improving multi-output regression and dimensionality reduction for high $p$ or when $p \gg n$ (Huusari et al., 2021).
• Integral Operator Representation: Fredholm-type integral operators are directly learned in kernel RKHS, with rigorous Mercer conditions and explicit SGD updates in the function-space (Yang et al., 14 Sep 2025, Bao et al., 2022). Hybrid architectures with neural-parameterized kernels (branch-trunk separation) or encoder-decoder frameworks are used for high-dimensional or compositional tasks.

7. Empirical Performance and Deployment Considerations

Extensive numerical experiments testify to the flexibility, accuracy, and computational scalability of kernel-based operator methods:

• Accuracy: On PDE benchmarks (Burgers, Darcy, Helmholtz, Navier–Stokes, etc.), kernel-based operator learners match or outperform leading NN-based architectures, especially in data-limited regimes or for highly nonlinear operators (Batlle et al., 2023, Bao et al., 2022).
• Computational Cost: Per-iteration (training) complexity is typically $O(N^3)$ for dense Gram matrices, but large-scale deployment is enabled by exploiting separable/product structures, low-rank approximations, random features, and iterative solvers.
• Interpretability and Model Selection: The RKHS norm yields direct control over complexity, and kernel hyperparameters (bandwidth, regularization) can be calibrated by cross-validation or Bayesian marginal likelihood.

A summary comparison of some practical features appears below:

Feature	Kernel Operator Method	Deep Neural Operator	State-Space Model
Closed-form training	Yes	No	No
Theoretical error bounds	Yes	Limited	No
Structured constraints	Highly flexible	Limited	No
Nonlinear operator support	Yes	Yes	Limited
Model interpretability	High (RKHS norm, coeffs)	Low	Moderate
Scalability	Moderate–High*	High	High (linear only)
Mesh independence	Yes	Mixed	No

*Scalability enhanced via product/separable kernel structures, low-rank, stochastic approximation, and random feature expansions.

8. Outlook and Extensions

Current directions include leveraging operator-valued kernel learning for:

• Efficient online learning and control in large-scale interconnected systems, leveraging dimension-free convergence.
• Bayesian uncertainty quantification for PDE solution maps, control laws, and inference routines.
• Hybrid and entangled kernel architectures for high-dimensional multi-output problems and structured operator learning.
• Extensions to operator-valued kernels on manifolds, graphs, or irregular domains.
• Integration with physical priors (e.g., divergence-free, symmetry), active learning, and multi-fidelity settings.

As the kernel-based operator approach continues to mature, it provides a theoretically grounded and computationally transparent foundation for robust, structure-preserving, and interpretable operator learning across control, scientific computing, and probabilistic machine learning.