Radial Basis Operator Networks

• Radial Basis Operator Networks (RBONs) are models that extend classical RBF architectures to learn nonlinear operators between infinite-dimensional function spaces.
• RBONs employ dual RBF layers—one applied to discretized inputs (branch) and one to query points (trunk)—to effectively approximate mappings in PDE and graph-based tasks.
• Training methods for RBONs, including K-means-based initialization, least squares optimization, and gradient descent, ensure robust and efficient operator regression performance.

Radial Basis Operator Networks (RBONs) generalize classical radial basis function (RBF) architectures to operator learning, enabling the approximation of nonlinear operators that map between infinite-dimensional function spaces. These models leverage the expressivity of RBF representations and are structurally adapted for tasks such as scientific computing, PDE solution mapping, and operator regression on both vectorial, graph-based, and functional domains. The RBON paradigm encompasses pure operator approximators ("Radial-Basis Operator Networks"), adaptively parametrized operator nets rooted in Kolmogorov–Arnold theory, and graph-aware RBF networks for relational data.

1. Mathematical Formulation and Core Architecture

Let $G:\mathcal{U}\to\mathcal{V}$ denote a continuous nonlinear operator acting between Banach spaces, frequently with $\mathcal{U}\subset C(K_1)$ and $\mathcal{V}\subset C(K_2)$. The modern RBON architecture approximates $G$ as $G^\dagger$ using a single hidden layer formed by the tensor product of two sets of RBFs: one ("branch") applied to discretized inputs $u^m=(u(x_1),\ldots,u(x_m))\in\mathbb{R}^m$, and one ("trunk") applied to the query point $y\in K_2$. The model's canonical form is

$$G^\dagger(u^m)(y) = \sum_{i=1}^M\sum_{k=1}^N \xi_{ik}\;\varphi(u^m;c_i^b,\sigma_i^b)\;\varphi(y;c_k^t,\sigma_k^t)$$

where $\varphi(z;c,\sigma)=\exp(-\|z-c\|^2/(2\sigma^2))$ is the Gaussian kernel, $(c_i^b,\sigma_i^b)$ and $\mathcal{U}\subset C(K_1)$0 are branch/trunk RBF centers and widths, and $\mathcal{U}\subset C(K_1)$1 are the linear output weights (Kurz et al., 2024).

In normalized RBONs (NRBON), the hidden layer activation is normalized by the $\mathcal{U}\subset C(K_1)$2 sum before output mapping. For operator learning on complex domains (e.g., frequency space), the same structure is applied with complex-valued inputs/centers and the complex-Euclidean norm.

2. Graph-based RBONs and Relational Data

RBONs can be adapted to graph-based settings where data is available only as a weighted adjacency matrix $\mathcal{U}\subset C(K_1)$3 encoding relations or metric-derived distances among $\mathcal{U}\subset C(K_1)$4 objects. The hidden layer constructs graph-RBF units as follows:

• Define prototype weights $\mathcal{U}\subset C(K_1)$5, normalized to $\mathcal{U}\subset C(K_1)$6.
• For node $\mathcal{U}\subset C(K_1)$7 and prototype $\mathcal{U}\subset C(K_1)$8, generalized "distance" is computed as

$\mathcal{U}\subset C(K_1)$9

• The unit activation is $\mathcal{V}\subset C(K_2)$0.

This construction ensures that the RBON output is equivalent to a standard vector-RBF network whenever the adjacency matrix $\mathcal{V}\subset C(K_2)$1 encodes squared Euclidean distances between (potentially unobserved) latent vectors $\mathcal{V}\subset C(K_2)$2 (Sledge et al., 2019).

3. Training Methods and Parameter Selection

The training of RBONs proceeds via closed-form or gradient-based methods, depending on context:

• For operator RBONs (Kurz et al., 2024):
  • The loss is the relative $\mathcal{V}\subset C(K_2)$3 error: $\mathcal{V}\subset C(K_2)$4.
  • The output weights $\mathcal{V}\subset C(K_2)$5 are solved for each trunk location via ordinary least squares or Moore–Penrose pseudoinverse. Centroids and widths are set using $\mathcal{V}\subset C(K_2)$6‑means clustering: branch RBF centers $\mathcal{V}\subset C(K_2)$7 are clusters of input samples; trunk centers $\mathcal{V}\subset C(K_2)$8 are clusters of query-points; widths are average intra-cluster radii.
  • Early stopping is applied to mitigate overfitting, especially for normalized variants.
• For graph-RBONs (Sledge et al., 2019):
  • Parameters $\mathcal{V}\subset C(K_2)$9 are updated via gradient descent:

  $G$0

  $G$1 - Prototype parameters and RBF widths are updated by gradients directly on $G$2. - All updates rely explicitly on entries of the adjacency matrix.

• In adaptive RBF-based Kolmogorov–Arnold networks (Free-RBF-KAN), centroids and widths are fully trainable per layer; parameters are updated by backpropagation with constraints $G$3 and $G$4 restricted to the data domain for numerical stability (Chiu et al., 12 Jan 2026).

4. Universality and Approximation Guarantees

RBONs inherit the universal approximation property of RBF networks and extend it to operators:

• If $G$5 is a non-polynomial, rapidly-decaying RBF (e.g., Gaussian), then, for every continuous nonlinear operator $G$6 and any $G$7, there exist $G$8 and network parameters such that

$G$9

for the RBON mapping form above (Kurz et al., 2024). The proof extends Chen & Chen’s theorem to operator settings.

• For Free-RBF-KANs that utilize the Kolmogorov–Arnold superposition theorem, expressive power is established by showing that networks composed of sums of univariate RBF subnets can uniformly approximate any continuous mapping $G^\dagger$0 within arbitrary precision (Chiu et al., 12 Jan 2026).

A plausible implication is that RBONs offer rigorous universality in both function and operator spaces, although explicit error rates in Sobolev or Barron-type norms for operator settings remain an open problem.

5. Empirical Benchmarks and Comparative Analysis

Empirical results across canonical PDE operator learning tasks and standard regression benchmarks demonstrate that RBONs, including variants such as NRBON and frequency-domain RBON, offer strong generalization and compact representational efficiency, often outperforming paradigms such as DeepONet, LNO, and FNO (Kurz et al., 2024):

Network	Wave (ID/OOD)	Burgers (ID/OOD)	Beam (ID/OOD)
RBON	$G^\dagger$1	$G^\dagger$2	$G^\dagger$3
NRBON	$G^\dagger$4	$G^\dagger$5	$G^\dagger$6

RBONs with a single hidden layer (≤225 units) exhibit low in- and out-of-distribution (OOD) errors, with OOD generalization robust even across function classes. In scientific data tasks (e.g., learning CO₂→temperature operators), RBONs match or improve on LSTM, FNO, DeepONet, and LNO baselines, especially in long-term forecasts (Kurz et al., 2024).

In the context of structured function learning, Free-RBF-KAN matches or surpasses classical B-spline KAN and standard RBF-KAN, providing reduced training/inference times and adaptive expressivity due to freely learnable centroids and widths (Chiu et al., 12 Jan 2026).

6. Architectural Variants: Kolmogorov–Arnold Free-RBF-KANs

Free-RBF-KAN combines Kolmogorov–Arnold superposition with univariate RBF expansions and adaptively learned centroids and widths, yielding a multilayer operator network architecture:

• Layerwise, activations are

$G^\dagger$7

with all parameters updated using joint backpropagation.

• The architecture is embedded within operator learning pipelines, for example as trunk networks in DeepONet setups, consistently yielding lower relative $G^\dagger$8 errors for comparable or reduced parameter counts.
• Universality is guaranteed for continuous, non-polynomial RBF kernels (Chiu et al., 12 Jan 2026).

7. Limitations and Future Directions

• RBON performance depends on the initialization of RBF centers and widths; $G^\dagger$9-means clustering can be sensitive to local minima.
• The size of the hidden layer grows quadratically with the number of branch/trunk RBFs ($u^m=(u(x_1),\ldots,u(x_m))\in\mathbb{R}^m$0), impacting scalability for high-dimensional domains.
• Current theoretical results provide only qualitative universality, with quantitative error rates a subject of ongoing research.
• Exploration of alternative kernels beyond Gaussians (e.g., inverse-multiquadric) and sparse/low-rank tensor decompositions are open directions (Kurz et al., 2024).
• Enhanced automation of RBF parameter initialization (e.g., density-based or ensemble clustering) and extension to multimodal and convolutional tasks for scientific and structured data are proposed avenues for future work.

RBONs establish a rigorous and flexible framework for operator learning, bridging RBF-based regression, functional approximation, and modern neural operator methods while maintaining computational tractability and compactness (Kurz et al., 2024, Sledge et al., 2019, Chiu et al., 12 Jan 2026).