Fourier Learning Machines Overview

• Fourier Learning Machines are architectures that use Fourier transforms, sinusoidal bases, and adaptive spectral filters to represent functions for improved computational and sample efficiency.
• They integrate methodologies like kernel approximation, spectral convolution, and tensor decompositions to model high-frequency and multi-scale phenomena in scientific and quantum computing.
• Practical applications include enhanced PDE solvers, signal processing systems, and quantum-classical hybrid models, with regularization strategies that mitigate spectral bias and boost convergence.

A Fourier Learning Machine (FLM) is a machine learning architecture—classical or quantum—where learning, feature representation, and/or function approximation are grounded in the explicit or implicit manipulation of Fourier series, Fourier transforms, or parametrized (possibly learned) bases of sinusoids or complex exponentials. FLMs span a continuum from analytic transforms (classical DFT, FFT) to architectures which embed learned, adaptive, or content-adaptive spectral representations within modern deep or quantum models. The core principle of FLMs is that by parameterizing, optimizing, or manipulating representations in the Fourier domain, both sample and computational efficiency, spectral bias, and expressiveness for high-frequency or multi-scale phenomena can be fundamentally improved and understood.

1. Fundamental Principles and Mathematical Foundations

The distinguishing characteristic of FLMs is the use of Fourier-based representations either for input features, model coefficients, or entire operator mappings. For any periodic or shift-invariant system, a function $f(x)$—either scalar- or vector-valued—can be decomposed as a (finite or truncated) sum:

$f(x) = \sum_{n=-N}^N c_n e^{i n x}$

or, equivalently for real data, as

$$f(x) = a_0 + \sum_{n=1}^N\left[ a_n \cos(n x) + b_n \sin(n x) \right].$$

In higher dimensions, the Fourier basis generalizes to sums over multi-indices in $\mathbb{Z}^d$.

The machinery of FLMs rests on three interrelated frameworks:

• Kernel approximation via random or deterministic Fourier features: By Bochner’s theorem, any shift-invariant positive-definite kernel $K(x-y)$ has a nonnegative spectrum $p(\omega)$; sampled Fourier features $$\phi(x) = \sqrt{2/m}\left[\cos(\omega_j^T x + b_j)\right]_{j=1}^m$$ approximate $K(x,y) \approx \phi(x)^T \phi(y)$ (Băzăvan et al., 2012).
• Operator learning by parameterized spectral convolution: In FLMs such as Fourier Neural Operators (FNO), the forward map is cast as

$$\mathcal{K}(z)(x) = \text{IFFT}\left[ R \cdot \text{FFT}(z) \right](x)$$

with learnable multipliers $R(\cdot)$ in the frequency domain (Tran et al., 2021, Lanthaler et al., 2024, Kalimuthu et al., 5 Apr 2025).

• Fourier-parameterized neural architectures and quantum circuits: Feedforward networks with trigonometric activations and trainable frequencies, amplitudes, and phase-shifts realize nonharmonic Fourier expansions, enabling function approximation or solution of scientific problems (Rubel et al., 10 Sep 2025, Ren et al., 14 Oct 2025); variational quantum circuits by construction yield outputs as truncated Fourier series in the data-encoding variables, modulated by circuit weights (Atchade-Adelomou et al., 2023, Casas et al., 2023).

2. Architectures and Model Construction

FLMs materialize in several canonical forms:

• Fourier Feature Maps as Inputs or Hidden-Layer Embeddings: Random or learned Fourier features encode input data, reducing spectral bias and enabling efficient kernel approximation. Selection of feature bandwidths and frequencies, as in GFF-PIELM, is adapted via output monitoring to span the relevant spectrum (Ren et al., 14 Oct 2025, Băzăvan et al., 2012, Li et al., 2021).
• Spectral Operator Networks: Fourier Neural Operators (FNO) and their variants (e.g., LOGLO-FNO, Factorized FNO) construct neural network layers whose transformations are performed in frequency space, using learnable, potentially axis-separable, spectral filters (Tran et al., 2021, Kalimuthu et al., 5 Apr 2025). LOGLO-FNO further introduces parallel local spectral convolutions and high-frequency propagation paths to address limitations in classical global FNO architectures (Kalimuthu et al., 5 Apr 2025).
• Nonharmonic Fourier Neural Networks: FLMs based on phase-shifted cosine activations with trainable frequencies and phases represent multidimensional nonharmonic series. The architecture guarantees a one-to-one mapping between phase-shifted and separable trigonometric representations, enabling interpretable and adaptive expansions for both periodic and nonperiodic problems (Rubel et al., 10 Sep 2025).
• Tensor-Product and CPD Models: In high dimensions, tensor-product Fourier features are parameterized via low-rank Canonical Polyadic Decomposition, with feature hyperparameters optimized jointly with model weights via alternating least squares (ALS) or block coordinate descent (Saiapin et al., 2 Dec 2025, Wesel et al., 2021).
• Quantum Fourier Learning Machines: Parameterized quantum circuits employing data re-uploading and Hamiltonian encoding implement finite multidimensional Fourier series in their measurement outcomes. The expressivity is bounded by the circuit structure; full universality for Fourier learning is realized by appropriate scaling of qubits and encoding layers (Atchade-Adelomou et al., 2023, Casas et al., 2023, Strobl et al., 28 Aug 2025).

A summary of representative architectures is shown below:

FLM Type	Key Representation	Learnable Elements
Random Fourier Features	$\phi(x)=\sqrt{\frac{2}{m}}[\cos(\omega_j^T x+b_j)]$	Kernel spectrum, frequency parameters
Fourier Neural Operator	Layer: IFFT of learnable spectral filter $\cdot$ FFT	Spectral filter $R$
Nonharmonic FLM (NN)	$f(x)=\sum_k A_k \cos(w_k\cdot x+ \phi_k)$	$A_k$, $w_k$, $\phi_k$
Tensor CPD Model	$w=\sum_r w_r^{(1)}\otimes\dots\otimes w_r^{(D)}$	Factor matrices, CPD weights
Quantum VQA FLM	$$f(x)=\langle0|U^\dagger(x;\theta)OU(x;\theta)|0\rangle$$	$U(x;\theta)$, circuit parameters

3. Applications and Empirical Performance

FLMs have demonstrated superior or competitive results in several domains requiring accurate modeling of spectral content:

• Large-Scale Kernel Machine Learning: RFF-based FLMs outperform or match kernel ridge regression and multiple kernel learning on large datasets, e.g., VOC2011 object recognition (RFF-SKL and RFF-GL matched nonlinear KRR and GMKL with order-of-magnitude reduced complexity and timing) (Băzăvan et al., 2012).
• Operator Learning and Scientific Computing: FNO and its derivatives (LOGLO-FNO, F-FNO) provide fast and accurate surrogates for PDE solution operators. LOGLO-FNO achieves up to 35% reduction in high-frequency RMSE on turbulent flow and advanced operator tasks, especially in turbulent or multi-scale regimes (Kalimuthu et al., 5 Apr 2025, Tran et al., 2021).
• Quantum and Classical Fourier Machine Learning: Expressivity studies confirm that quantum circuits with properly chosen structure can realize arbitrary Fourier series up to the bound imposed by the circuit Hilbert space (Casas et al., 2023). Quantum-classical hybrid FLMs benchmarked on trigonometric interpolation and signal processing tasks achieve competitive accuracy to classical Fourier analysis (Atchade-Adelomou et al., 2023).
• Time-Series, Audio, and Signal Processing: Neural frontends trained end-to-end to learn Fourier-like transforms reveal filter banks combining sinusoids, windowing, comb and onset detectors—outperforming fixed FFT spectrograms in downstream audio tasks (Verma, 2023). Neural Fourier Modelling (NFM) provides highly compact architectures for forecasting, anomaly detection, and classification, matching or surpassing models with 100x more parameters, and exhibiting remarkable sampling-rate invariance (Kim et al., 2024).
• Physics-Informed Learning and PDEs: GFF-PIELM and phase-shifted NEU-FLMs deliver low error and rapid convergence on high-frequency and multi-scale PDE benchmarks (advection, Klein-Gordon, Helmholtz, etc.), demonstrating that carefully initialized and tuned Fourier features alleviate the spectral bias of classical DNNs (Ren et al., 14 Oct 2025, Rubel et al., 10 Sep 2025).

4. Optimization, Regularization, and Spectral Bias

FLMs introduce regularization and inductive bias both via their explicit spectral structure and through controlled parametrization of frequency components:

• Spectral Regularization via Band-Limiting and Filtering: Iterative band-pass filtering in Fourier regression and clustering offers a principled, hyperparameter-free strategy for balancing under- and overfitting, and for selecting model capacity in a data-driven way (Mehrabkhani, 2019, Mehrabkhani, 2019).
• Learned Spectral Bases and Hyperparameter Optimization: Optimization strategies prioritize relevant frequency bands; e.g., adaptive initialization of Fourier feature frequencies in GFF-PIELM by output weight monitoring improves high-frequency accuracy (Ren et al., 14 Oct 2025). Group-Lasso penalties in the RFF-GL model encourage kernel sparsity via frequency-group selection (Băzăvan et al., 2012).
• Mitigation of Spectral Bias in Deep Neural Networks: Learned Fourier feature embeddings in RL and regression serve as functional regularization tools that enable adjustment of the learned frequency spectrum; properly tuned Fourier bases prevent overfitting to high-frequency noise and accelerate learning of desired low-frequency solutions (Li et al., 2021).
• Discretization and Approximation Error: For FLMs using FFT on grids (FNO and variants), the discretization error of each layer is rigorously shown to decay algebraically as $N^{-s}$, where $s$ is the Sobolev regularity of the input, independent of network depth, provided activations and encodings are sufficiently smooth (Lanthaler et al., 2024).

5. Theoretical Guarantees, Limitations, and Inductive Bias

FLMs provide explicit theoretical control over expressivity, approximation error, and inductive bias:

• Universality and Expressivity: Quantum and classical FLMs can achieve universal approximation of multidimensional Fourier polynomials up to the limit set by circuit parameters or feature rank. The universality of parameterized quantum circuits to fully fit a general M-dimensional Fourier polynomial requires a number of parameters at least $(2D+1)^M$ for degree $D$ in each coordinate (Casas et al., 2023, Strobl et al., 28 Aug 2025).
• Statistical Correlations and Model Selection: In quantum FLMs, unavoidable correlations between Fourier coefficients (Fourier-coefficient correlations, or FCCs) due to parameter sharing fundamentally shape the learnability and generalization of the circuit; the FCC serves as a predictive metric for ansatz choice, outperforming conventional expressibility measures (Strobl et al., 28 Aug 2025).
• Scalability via Tensor Decompositions: In high-dimensional classical settings, tensor network parameterizations (e.g., low-rank CPD) of deterministic Fourier feature models overcome the curse of dimensionality with linear complexity in both sample size and dimension (Wesel et al., 2021, Saiapin et al., 2 Dec 2025).
• Limitations: FLMs requiring uniform spectral sampling (e.g., NFM) may underperform on irregular grids unless adapted, and the cost of classic tensor-product expansions can still lead to exponential growth in some settings without efficient CPD (Kim et al., 2024, Saiapin et al., 2 Dec 2025). In high-dimensional phase-shifted FLMs, the exponential width in $d$ needs to be managed.

6. Algorithmic Workflows and Implementation

FLMs are characterized by algorithmic strategies exploiting fast transforms and analytic solutions:

• Iterative Filtering for Regression, Classification, Clustering: Fourier regression and classification are performed via alternating domain–frequency filtering and clamping; model complexity is controlled by bandwidth growth, and stopping rules are based on validation metric convergence (Mehrabkhani, 2019, Mehrabkhani, 2020, Mehrabkhani, 2019).
• Block Coordinate Descent for Tensor FLMs: Alternating least squares updates for both coefficients and feature hyperparameters yield provably monotonic convergence and streamline high-dimensional model fitting (Saiapin et al., 2 Dec 2025, Wesel et al., 2021).
• Closed-Form Experience Solutions: In single-layer FLMs (e.g., ELM with Fourier features), the output weights admit closed-form solutions; spectral hyperparameters are adaptively refined by direct monitoring of output weight usage (Ren et al., 14 Oct 2025).
• Quantum Circuit Training: Fourier weights in quantum FLMs are extracted by sampling model outputs at a grid of inputs and performing DFT; training is performed via hybrid classical-quantum optimization with loss gradients computed by the parameter-shift rule (Atchade-Adelomou et al., 2023).

7. Broader Implications and Outlook

FLMs constitute a unifying abstraction linking kernel methods, neural operators, modern deep learning, and quantum machine learning:

• Interpretability and Inductive Structure: Many FLMs natively provide interpretable models, where each basis function or filter has explicit frequency, amplitude, and phase correspondence; nonharmonic FLMs enable spectral dictionaries optimized for the target task (Rubel et al., 10 Sep 2025, Verma, 2023).
• Resolution Invariance and Local-Global Information: FLMs such as NFM exhibit remarkable invariance to changes in sampling rate, while LOGLO-FNO demonstrates the necessity of combining local and global spectral information for full reconstructive accuracy (Kim et al., 2024, Kalimuthu et al., 5 Apr 2025).
• Expanding the FLM Paradigm: Ongoing and future research targets richer adaptive routing (mixtures of learned bases), continuous-time and irregular-sampling extensions, spectral attention mechanisms, and integration with transformer architectures for sequence modeling (Verma, 2023).
• Universal Function Learning Across Platforms: Both classical and quantum FLMs reveal that the capacity to efficiently learn and generalize is intricately tied to the interplay between spectral expressivity, architectural constraints, and the modular design of frequency-space computation (Strobl et al., 28 Aug 2025, Casas et al., 2023).

In total, Fourier Learning Machines provide a rigorous, algorithmically efficient, and spectrally transparent framework for high-fidelity modeling of functions and operators in scientific computing, signal processing, machine learning, and quantum information processing. The breadth of instantiations and theoretical clarity position FLMs as a foundational paradigm in modern computational mathematics and data-driven science.