Neural Basis Functions

• Neural basis functions are learned representations that form a compact basis for approximating functions and operators using adaptive, low-dimensional coefficients.
• They enable models to separate nonlinear basis learning from linear coefficient estimation, facilitating robust regression, operator learning, and scientific computing.
• Applications span functional data analysis, neural fields, and control, supported by theoretical guarantees from RKHS methods and PCA-based dimensionality reduction.

Searching arXiv for recent and foundational papers on neural basis functions, function encoders, radial basis layers, and basis-learning operator methods. Neural basis function denotes a family of representations in which a model is organized around learned basis elements whose linear or multilinear combinations approximate functions, operators, signals, or control laws. Across recent literature, the term covers several closely related constructions: function encoders that learn basis functions $\{\psi_j\}_{j=1}^n$ spanning a Hilbert-space subspace (Low et al., 24 Sep 2025); radial-basis-function layers that expand inputs through localized responses before linear recombination (Jost et al., 2023); adaptive basis layers for functional data in which each basis function is itself a micro neural network (Yao et al., 2021); operator-learning architectures that separate basis learning from coefficient mapping (Ingebrand et al., 2024, Zhang et al., 2022); and domain-specific basis models for PDEs, neural fields, communications, and control (Ma et al., 19 Jan 2026, Shaw et al., 17 Apr 2025, Soares et al., 2024, Li, 16 Jun 2025). The unifying idea is that the expressive nonlinear component is concentrated in the learned basis functions, while prediction, reconstruction, or operator application is carried out in a lower-dimensional coefficient space.

1. Conceptual definition and scope

In the most explicit function-space formulation, a function encoder learns basis functions

$\{\psi_j\}_{j=1}^n,$

which define the feature map

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$

and represent functions in the learned span as

$f(x)=\langle c,\phi(x)\rangle$

for coefficients $c\in\mathbb{R}^n$ (Low et al., 24 Sep 2025). With sampled data $\{(x_i,y_i)\}_{i=1}^m$, the coefficients are obtained by regularized least squares,

$$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$

with corresponding normal equations

$$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$

(Low et al., 24 Sep 2025). In this formulation, the neural network is not used as an end-to-end predictor in the usual sense; rather, it learns a basis for a function space, and prediction is a linear solve in that basis (Low et al., 24 Sep 2025).

A broader formulation appears in multivariate approximation. NeuApprox introduces the neural basis function by leveraging an untrained neural network as the basis function, using an MLP-based map

$f_\theta : [0,1] \to \mathbb{R}^{N},$

typically written as

$$f_\theta(x) = \boldsymbol{W}_L \big(\cdots \sigma(\boldsymbol{W}_0 x + \boldsymbol{b}_0) \cdots \big) + \boldsymbol{b}_L$$

with periodic activations such as $\{\psi_j\}_{j=1}^n,$0 to better capture high-frequency details (Wu et al., 4 Mar 2026). In that setting, multivariate functions are decomposed into sums of block terms built from products of univariate neural basis functions and coefficient tensors (Wu et al., 4 Mar 2026).

Other papers use the same underlying principle with a more localized inductive bias. The U-RBF layer treats each input dimension separately, assigns to each dimension a set of Gaussian RBF neurons, maps each scalar input coordinate into a higher-dimensional representation, and then feeds that expanded representation into a fully connected layer (Jost et al., 2023). NeuRBF writes the representation explicitly as a weighted sum of adaptive radial basis functions centered at learned locations with learned shapes,

$\{\psi_j\}_{j=1}^n,$1

thereby casting neural fields as basis aggregation followed by decoding (Chen et al., 2023).

Taken together, these formulations suggest a common definition: a neural basis function is a learnable basis element, usually parameterized by a neural network or embedded in a neural architecture, whose outputs serve as coordinates, modes, kernels, or localized responses for representing structured families of functions.

2. Algebraic and kernel-theoretic formulations

The algebraic form of neural basis-function models is usually a linear or multilinear expansion over learned basis responses. Function encoders use the primal form $\{\psi_j\}_{j=1}^n,$2 with coefficients computed analytically or by regularized least squares (Low et al., 24 Sep 2025). Basis-to-Basis (B2B) operator learning represents input and output functions as

$\{\psi_j\}_{j=1}^n,$3

then learns the map $\{\psi_j\}_{j=1}^n,$4 in coefficient space (Ingebrand et al., 2024). BelNet similarly reconstructs operators in a low-rank basis-enhanced form, starting from

$\{\psi_j\}_{j=1}^n,$5

so that

$\{\psi_j\}_{j=1}^n,$6

and its neuralized version learns both projection and construction functions directly (Zhang et al., 2022).

A major development is the kernel interpretation of learned neural bases. Once $\{\psi_j\}_{j=1}^n,$7 is learned, function encoders induce the kernel

$\{\psi_j\}_{j=1}^n,$8

which is automatically symmetric positive semidefinite and therefore a valid reproducing kernel (Low et al., 24 Sep 2025). In the dual, the same predictor can be written as

$\{\psi_j\}_{j=1}^n,$9

with coefficients satisfying

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$0

(Low et al., 24 Sep 2025). The vector-valued extension replaces the scalar kernel by the operator-valued kernel

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$1

(Low et al., 24 Sep 2025).

This kernel-theoretic reading is important because it connects learned neural bases to classical RKHS reasoning while preserving the computational advantage of a compact primal representation. In the function-encoder setting, test-time cost is naturally $\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$2 per test point rather than scaling with the number of training examples $\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$3 (Low et al., 24 Sep 2025). A plausible implication is that neural basis-function models occupy an intermediate position between fixed-basis approximation and data-dependent kernel learning: the basis is explicit and finite-dimensional, but the induced similarity structure is learned rather than prescribed.

3. Architectural families

The literature contains several distinct architectural realizations of neural basis functions.

Function encoders and related Hilbert-space models learn neural network basis functions and then solve for coefficients by least squares or ridge regression (Low et al., 24 Sep 2025). The central object is the learned span itself, not an end-to-end predictor. B2B operator learning extends this idea to paired input and output spaces, with separate learned bases and a coefficient-space operator, and derives algorithms directly analogous to eigendecomposition and singular value decomposition for linear operators (Ingebrand et al., 2024).

Functional-data models realize the basis directly inside a neural network layer. AdaFNN introduces a Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. For a functional input $\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$4, basis coefficients are computed as

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$5

where each $\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$6 is parameterized as

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$7

(Yao et al., 2021). The low-dimensional coefficient vector is then passed to a standard feedforward network (Yao et al., 2021).

Radial-basis architectures use localized functions as the basis family. The U-RBF layer maps each scalar input coordinate into Gaussian responses

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$8

then forms the next-layer output by weighted summation over these activations (Jost et al., 2023). Classical and adaptive RBF networks appear in several other forms: Beta Basis Function Neural Networks use beta transfer functions in the hidden layer and linear output combinations (Chouikhi et al., 2018); DeepLABNet replaces fixed activations with feature-wise scalar RBF interpolators of the form

$\phi(x)=\big[\psi_1(x),\ldots,\psi_n(x)\big]^\top,$9

(Hryniowski et al., 2019); NeuRBF uses adaptive centers $f(x)=\langle c,\phi(x)\rangle$0 and anisotropic shapes $f(x)=\langle c,\phi(x)\rangle$1 with inverse-quadratic kernels

$f(x)=\langle c,\phi(x)\rangle$2

and optional normalized weights (Chen et al., 2023).

Complex-valued basis-function networks form another family. Phase-transmittance radial basis function neural networks use complex-valued inputs, outputs, weights, biases, center vectors, and variances, with split-complex radial kernels that propagate real and imaginary parts separately (Soares et al., 2024). Deep complex-valued RBF networks extend the classical shallow C-RBF by stacking multiple Gaussian basis layers

$f(x)=\langle c,\phi(x)\rangle$3

(Soares et al., 2024).

A useful summary is that neural basis-function architectures differ chiefly in what is learned as the basis element: global Hilbert-space modes (Low et al., 24 Sep 2025), inner-product basis functions for functional data (Yao et al., 2021), localized radial responses (Jost et al., 2023, Chen et al., 2023), learned activation functions (Hryniowski et al., 2019), operator-domain bases (Ingebrand et al., 2024, Zhang et al., 2022), or reusable basis modules for initialization and transfer (Hu et al., 9 Oct 2025).

4. Basis learning, coefficient estimation, and model compression

A recurring feature of neural basis-function methods is the separation between basis learning and coefficient estimation. In B2B, sampled data $f(x)=\langle c,\phi(x)\rangle$4 are projected onto the learned basis by solving

$f(x)=\langle c,\phi(x)\rangle$5

with closed-form solution

$f(x)=\langle c,\phi(x)\rangle$6

(Ingebrand et al., 2024). Function encoders use the analogous regularized least-squares solve in the learned feature space (Low et al., 24 Sep 2025). BelNet computes latent coefficients from arbitrary input samples and reconstructs output functions through learned basis functions on the output domain (Zhang et al., 2022). In the reusable initialization framework, pretrained neural basis networks approximate polynomial basis functions on a reference domain and are then assembled for target functions as

$f(x)=\langle c,\phi(x)\rangle$7

with coefficients usually computed by least squares (Hu et al., 9 Oct 2025).

Compactness and intrinsic dimension are major concerns. Function encoders propose two PCA-guided schemes for learning compact bases (Low et al., 24 Sep 2025). In progressive training, basis functions are learned sequentially, coefficients are computed across training tasks, and the covariance

$f(x)=\langle c,\phi(x)\rangle$8

is used to compute explained variance ratios $f(x)=\langle c,\phi(x)\rangle$9; training stops when cumulative explained variance exceeds a threshold $c\in\mathbb{R}^n$0 such as $c\in\mathbb{R}^n$1 (Low et al., 24 Sep 2025). In train-then-prune, an overparameterized basis set of size $c\in\mathbb{R}^n$2 is trained jointly, PCA is applied to the learned coefficient covariance $c\in\mathbb{R}^n$3, the effective rank is inferred as

$c\in\mathbb{R}^n$4

and the bases are scored by

$c\in\mathbb{R}^n$5

before pruning and fine-tuning (Low et al., 24 Sep 2025).

The polynomial benchmark in the same paper provides a particularly clean example of intrinsic-dimension recovery. Degree-$c\in\mathbb{R}^n$6 polynomials have intrinsic dimension $c\in\mathbb{R}^n$7, and the eigenvalue scree plots show elbows exactly at that dimension; the progressive method stops after $c\in\mathbb{R}^n$8, $c\in\mathbb{R}^n$9, and $\{(x_i,y_i)\}_{i=1}^m$0 bases for degrees $\{(x_i,y_i)\}_{i=1}^m$1, $\{(x_i,y_i)\}_{i=1}^m$2, and $\{(x_i,y_i)\}_{i=1}^m$3, respectively, while train-then-prune identifies the same cutoffs (Low et al., 24 Sep 2025).

This use of PCA is not merely a visualization device. It functions as a basis-selection criterion revealing how many learned modes are actually required to span the dominant variability of a function family. This suggests a broader interpretation of neural basis-function learning as supervised or task-aware dimension reduction in function space.

5. Theoretical properties

Several papers place neural basis functions on a formal approximation-theoretic footing. NeuApprox states a Neural Approximation Theorem: for any continuous multivariate function $\{(x_i,y_i)\}_{i=1}^m$4 and any desired accuracy $\{(x_i,y_i)\}_{i=1}^m$5, there exists a finite set of neural basis functions and corresponding coefficient tensors such that

$\{(x_i,y_i)\}_{i=1}^m$6

for

$\{(x_i,y_i)\}_{i=1}^m$7

(Wu et al., 4 Mar 2026). The proof combines Stone–Weierstrass density of separable products with universal approximation for the univariate neural factors (Wu et al., 4 Mar 2026).

AdaFNN provides a consistency result for functional mappings $\{(x_i,y_i)\}_{i=1}^m$8, where $\{(x_i,y_i)\}_{i=1}^m$9 is a finite-dimensional linear continuous map and $$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$0 is continuous nonlinear. Under accurate numerical integration and sufficient network capacity, there exists a network $$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$1 such that

$$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$2

for any $$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$3 (Yao et al., 2021). The same paper also derives a small-generalization-error result, with expected generalization gap bounded as

$$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$4

under compactness and Lipschitz assumptions with SGD step sizes $$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$5 (Yao et al., 2021).

Function encoders develop finite-sample guarantees for the fixed-basis stage. If each basis is bounded by $$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$6 and outputs satisfy $$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$7, the learned predictor obeys a Rademacher-style risk bound of the form

$$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$8

and asymptotically

$$\min_{c\in\mathbb{R}^n}\ \frac{1}{m}\sum_{i=1}^m\big(y_i-\langle c,\phi(x_i)\rangle\big)^2+\lambda\|c\|^2,$$9

(Low et al., 24 Sep 2025). The same analysis yields

$$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$0

making explicit the tradeoff between basis count, data size, and regularization (Low et al., 24 Sep 2025). A PAC-Bayes bound is also derived using truncated Gaussian priors and posteriors over coefficients, with a key KL estimate $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$1 (Low et al., 24 Sep 2025).

Operator-learning theory supplies another layer of structure. For linear operators, B2B proves that if $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$2 and $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$3 is linear, then the learned operator satisfies

$$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$4

because the coefficient map is linear and least-squares projection is linear in the data (Ingebrand et al., 2024). The same framework gives explicit SVD-style and eigendecomposition-style operator approximations in learned bases (Ingebrand et al., 2024).

These results do not imply identifiability of individual learned basis functions. AdaFNN states explicitly that the consistency theorem is for the network map $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$5 and does not mean the individual basis functions are identifiable, since many parameterizations can represent the same map (Yao et al., 2021). That point addresses a common misconception: neural basis functions are often meaningful as spans or coordinate systems even when particular basis elements are non-unique.

6. Applications and empirical behavior

Neural basis-function methods have been applied to regression, functional data analysis, operator learning, PDE solution, control, communications, and scientific computing.

In low-dimensional regression, U-RBF is evaluated on low-dimensional function regression, image regression, and PMLB datasets with at most $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$6 input dimensions. It is often the best overall among deep learning methods on the real-world low-dimensional regressions, performs very well on isotropic low-pass filtered white noise, and is competitive but not always superior on coordinate-to-RGB image regression, where Fourier-feature methods often perform best (Jost et al., 2023). The same paper emphasizes that no dedicated reinforcement-learning experiment is reported, despite control being mentioned as a general motivation (Jost et al., 2023).

In function-space representation, function encoders validate compact-basis learning on polynomial spaces and nonlinear dynamical systems. On the Van der Pol oscillator, prior work used $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$7 neural ODE bases, while the PCA-guided methods find that only $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$8 bases are sufficient to capture nearly all variance and attain comparable predictive accuracy; for the planar two-body problem, about $$\left(\frac{1}{m}\sum_{i=1}^m \phi(x_i)\phi(x_i)^\top+\lambda I\right)c = \frac{1}{m}\sum_{i=1}^m y_i\phi(x_i)$$9–$f_\theta : [0,1] \to \mathbb{R}^{N},$0 bases capture over $f_\theta : [0,1] \to \mathbb{R}^{N},$1 of the variance and reproduce the orbital dynamics accurately (Low et al., 24 Sep 2025).

In operator learning, B2B reports a two-orders-of-magnitude improvement in accuracy over existing approaches on several benchmark tasks and is especially strong on linear operators such as derivative and anti-derivative, while remaining robust to variable sensor locations that prevent DeepONet variants from converging (Ingebrand et al., 2024). BelNet targets mesh-free operator learning for parametric PDEs and reports lower mean relative error than DON on several multiscale and high-contrast problems, including multiscale elliptic equations and Richard’s equation, with the free-BelNet variant benefiting from varying input sensors (Zhang et al., 2022).

In functional data analysis, AdaFNN compares learned basis layers against raw discretized inputs, B-spline bases, and FPCA-based pipelines. It is reported as the best method on all nine real-data prediction tasks in the reported table, and it often achieves better accuracy with fewer basis functions than B-spline or FPCA pipelines (Yao et al., 2021). Orthogonality regularization often improves performance, while sparsity regularization mainly helps interpretability and sometimes predictive accuracy (Yao et al., 2021).

In neural fields and coordinate-based representations, NeuRBF demonstrates higher accuracy and compactness than prior methods on 2D image fitting, 3D signed distance fields, and neural radiance field reconstruction. Reported examples include about $f_\theta : [0,1] \to \mathbb{R}^{N},$2 dB PSNR on DIV2K, average IoU $f_\theta : [0,1] \to \mathbb{R}^{N},$3 and normal angular error $f_\theta : [0,1] \to \mathbb{R}^{N},$4 on 3D SDF tasks, and average PSNR $f_\theta : [0,1] \to \mathbb{R}^{N},$5, SSIM $f_\theta : [0,1] \to \mathbb{R}^{N},$6, and LPIPS$f_\theta : [0,1] \to \mathbb{R}^{N},$7 $f_\theta : [0,1] \to \mathbb{R}^{N},$8 on synthetic NeRF scenes (Chen et al., 2023).

In scientific computing, reusable neural basis pretraining on polynomial families yields strong 1D and 2D approximation results and improved extrapolation through a domain mapping mechanism (Hu et al., 9 Oct 2025). RRNN solves multiscale elliptic equations by randomized Gaussian RBF bases on domain-decomposed subproblems, reducing training to a least-squares solve for output-layer weights (Wu et al., 2024). PIRBFNN solves Black–Scholes PDEs for single-asset, double-asset, and four-asset options by jointly optimizing RBF centers, shape parameters, weights, and network width under a physics-informed loss, with residual-based adaptive neuron insertion (Ma et al., 19 Jan 2026). For neural field equations on curved cortical surfaces, local RBF interpolatory projection combined with RBF quadrature provides a mesh-flexible, high-order framework capable of simulating labyrinthine patterns and traveling spots on deformed spheres, tori, and realistic cortical meshes (Shaw et al., 17 Apr 2025).

In control and communications, TGRBF combines an RBFNN branch with a GRU branch through a dynamic gate,

$f_\theta : [0,1] \to \mathbb{R}^{N},$9

and is used for system identification and Jacobian-driven adaptive control with Lyapunov-based guarantees of uniform ultimate boundedness (Li, 16 Jun 2025). In 5G-style communication tasks, deep PT-RBF and deep C-RBF networks require carefully derived initialization schemes; random and constellation-based initialization fail to converge in deep settings, whereas the proposed statistically informed initialization is reported as the only one to achieve successful convergence for the tested deep architectures (Soares et al., 2024, Soares et al., 2024).

7. Related concepts, misconceptions, and open directions

A common misconception is that neural basis functions are merely classical RBF networks under a new name. The literature shows a broader picture. Some models do use Gaussian, inverse-quadratic, or beta basis functions (Jost et al., 2023, Chen et al., 2023, Chouikhi et al., 2018), but others learn basis functions as general neural networks in Hilbert spaces (Low et al., 24 Sep 2025), as micro networks inside a Basis Layer (Yao et al., 2021), or as reusable modules pretrained on polynomial families (Hu et al., 9 Oct 2025). The term therefore encompasses both localized radial bases and nonlocal learned modes.

A second misconception is that the basis must be fixed once learned. Several papers emphasize adaptability after pretraining. The reusable initialization framework transfers pretrained polynomial basis networks to new domains using a domain transform and least-squares coefficient fitting (Hu et al., 9 Oct 2025). NeuApprox compares freezing, fine-tuning with LoRA (rank $$f_\theta(x) = \boldsymbol{W}_L \big(\cdots \sigma(\boldsymbol{W}_0 x + \boldsymbol{b}_0) \cdots \big) + \boldsymbol{b}_L$$0), and training from scratch, reporting that fine-tuning provides a better speed/accuracy trade-off than training from scratch on out-of-distribution data (Wu et al., 4 Mar 2026). TGRBF performs event-triggered online optimization with an experience buffer and momentum-explicit gradient descent to refine parameters during control execution (Li, 16 Jun 2025).

There are also objective limitations. U-RBF imposes an axial or univariate bias because each dimension is processed separately, which is helpful only when the problem aligns with that structure (Jost et al., 2023). DeepLABNet reports slower training, increased overfitting with model size, higher performance variance than ReLU, and greater sensitivity to learning-rate scheduling (Hryniowski et al., 2019). Function encoders make the expressivity–complexity tradeoff explicit through the dependence of generalization bounds on basis count $$f_\theta(x) = \boldsymbol{W}_L \big(\cdots \sigma(\boldsymbol{W}_0 x + \boldsymbol{b}_0) \cdots \big) + \boldsymbol{b}_L$$1 (Low et al., 24 Sep 2025). Deep complex-valued RBF models and PT-RBFs highlight that poor initialization can prevent convergence entirely in multilayer settings (Soares et al., 2024, Soares et al., 2024).

A plausible synthesis is that neural basis functions are best understood not as a single architecture but as a design principle: learn a compact, adaptive set of nonlinear basis elements whose coefficients can be solved, regressed, optimized, or controlled in a lower-dimensional space. Recent work pushes this principle in three directions. One is toward functional analysis and statistical guarantees, especially through function encoders and B2B (Low et al., 24 Sep 2025, Ingebrand et al., 2024). Another is toward task-structured and geometry-aware bases for scientific computing, operator learning, and neural fields (Zhang et al., 2022, Ma et al., 19 Jan 2026, Shaw et al., 17 Apr 2025). A third is toward modularity, transfer, and fine-tuning, where pretrained basis libraries become reusable building blocks rather than task-specific endpoints (Hu et al., 9 Oct 2025, Wu et al., 4 Mar 2026).