Fourier Learning Machine (FLM) Overview

• FLM is a family of machine learning models that uses Fourier representations—via FFT methods, tensor decompositions, or cosine-activated neural networks—as primary tools for learning.
• The framework spans iterative spectral filtering, deterministic Fourier feature maps with CPD, and explicit multidimensional Fourier series in neural architectures.
• FLM methods offer practical gains in training efficiency and interpretability but face challenges with dimensional scaling and hyperparameter initialization.

Fourier Learning Machine (FLM) denotes a family of machine-learning constructions in which Fourier representations are primary objects of approximation, optimization, or regularization rather than merely auxiliary preprocessing steps. In the literature, the term encompasses at least three substantive strands: FFT-based nonparametric learning procedures for regression and classification, tensor-structured models built from deterministic or tensor-product Fourier features, and a recent neural architecture that explicitly represents a multidimensional nonharmonic Fourier series with trainable frequencies, amplitudes, and phase shifts. The literature also shows that the acronym is context-dependent: in one unrelated 2026 indoor occupancy paper, “FLM” denotes Feed-forward Likelihood Maximization rather than Fourier Learning Machine (Mehrabkhani, 2019, Mehrabkhani, 2020, Wesel et al., 2021, Li et al., 2021, Rubel et al., 10 Sep 2025, Chen et al., 19 Jun 2026).

1. Terminological scope and lineage

The designation “Fourier Learning Machine” has not been restricted to a single canonical architecture. Early work presented Fourier-transform-based supervised learning algorithms for regression and classification in which learning proceeds by iterative spectral filtering on sampled data, with no least-squares solve and no explicit kernel construction (Mehrabkhani, 2019, Mehrabkhani, 2020). Later work shifted the emphasis toward Fourier feature maps, especially deterministic Fourier features and tensor-product constructions whose dimensional explosion is controlled by low-rank tensor decompositions (Wesel et al., 2021, Saiapin et al., 2 Dec 2025). A more specialized recent usage defines the FLM as a feedforward neural network with cosine activations that is designed to represent a multidimensional nonharmonic Fourier series in a direct and interpretable way (Rubel et al., 10 Sep 2025).

Usage of “FLM”	Core mechanism	Representative paper
FFT-based Fourier learning	Iterative band filtering and spectral smoothing	(Mehrabkhani, 2019, Mehrabkhani, 2020)
Fourier-feature FLM	Tensor-product Fourier features with CPD/ALS	(Wesel et al., 2021, Saiapin et al., 2 Dec 2025)
Neural FLM	Cosine-activation NN with trainable spectral parameters	(Rubel et al., 10 Sep 2025)
Unrelated acronym usage	Feed-forward Likelihood Maximization	(Chen et al., 19 Jun 2026)

The historical progression is not strictly linear, because these strands address different bottlenecks. The FFT-based line focuses on unified training/evaluation and automatic complexity control; the tensor-decomposition line addresses the curse of dimensionality in deterministic Fourier features; the neural FLM line emphasizes an interpretable spectral parameterization within an MLP-like architecture. This suggests that “FLM” is best understood as a family resemblance term rather than a single standardized model class.

2. Mathematical representations

A defining feature across FLM variants is that the target function is represented in a spectral basis or in a structure that is explicitly equivalent to one. In the FFT-based regression formulation, learning is expressed through repeated discrete Fourier transformation, band filtering, and inverse transformation:

$$\tilde{y} = \mathcal{F}\{ \hat{y} \}, \qquad \tilde{y}_f = \tilde{y}\,\operatorname{rect}\left(\frac{f}{B}\right), \qquad \hat{y}_f = \mathcal{F}^{-1}\{ \tilde{y}_f \}.$$

The iteration overwrites training locations with observed responses and updates non-training locations with the filtered reconstruction, while bandwidth is gradually increased (Mehrabkhani, 2019).

In tensor-structured Fourier-feature FLMs, the feature map itself is a tensor product. For input $\mathbf{x} \in \mathbb{C}^D$,

$$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$

The model response is

$$f(\mathbf{x}) = \phi_\theta(\mathbf{x})^\top \mathbf{w},$$

with the weight tensor represented in Canonical Polyadic Decomposition (CPD) form,

$$\operatorname{vec}(\mathcal{W}) = \sum_{r=1}^R w^{(D)}_r \otimes \cdots \otimes w^{(1)}_r.$$

This replaces exponential parameter growth with a representation whose storage and computation are linear in dimension, basis size, and rank (Saiapin et al., 2 Dec 2025, Wesel et al., 2021).

The neural FLM of 2025 makes the Fourier parameterization explicit at the architecture level. Its global output is

$$\widehat{f}_{NH}(\boldsymbol{x}) = \sum_{\boldsymbol{n} \in \mathcal{N}} \sum_{i=1}^{2^{m-1}} \widehat{A}_i^{(\boldsymbol{n})} \cos\left((\boldsymbol{e}^{(i)} \odot \boldsymbol{n}) \cdot \boldsymbol{x} - b_i^{(\boldsymbol{n})}\right),$$

where $\mathcal{N}$ is the set of trainable frequency vectors, $\widehat{A}_i^{(\boldsymbol{n})}$ are trainable amplitudes, $b_i^{(\boldsymbol{n})}$ are trainable phase shifts, and $\boldsymbol{e}^{(i)}$ are rows of the $\mathbf{x} \in \mathbb{C}^D$0-Lexi Sign Matrix (Rubel et al., 10 Sep 2025). The paper states that this architecture is the first able to represent a complete, separable Fourier basis in multiple dimensions using a standard Multilayer Perceptron-like architecture (Rubel et al., 10 Sep 2025).

A probabilistic density-based formulation also appears in Fourier classification. Each class is written as a signed density of Dirac masses, Fourier transformed, low-pass filtered by a Gaussian, and then normalized into class probabilities:

$\mathbf{x} \in \mathbb{C}^D$1

This preserves a Fourier-domain interpretation while producing probabilistic multiclass outputs (Mehrabkhani, 2020).

3. Learning mechanisms and optimization

The optimization procedures used by FLM variants are as distinctive as their representations. In the FFT-based regression method, there is no least-squares objective and therefore no inversion of ill-conditioned matrices. Model complexity is controlled by progressively widening the spectral filter, and stopping is determined by convergence of a performance metric such as $\mathbf{x} \in \mathbb{C}^D$2 (Mehrabkhani, 2019). In the companion classification method, smoothness is controlled by increasing the Gaussian low-pass bandwidth, and the selected iteration is the one at which the second derivative of the correlation between successive smoothed densities drops below a fixed threshold (Mehrabkhani, 2020).

In tensor-network FLMs, optimization is usually blockwise. The 2021 tensor-decomposition approach derives a monotonically converging block coordinate descent algorithm for regularized squared loss with CP-rank constraint (Wesel et al., 2021). The 2025 Feature Learning model extends this by representing tensor-product features as a learnable CPD and alternates between updating the CPD cores and updating feature coefficients $\mathbf{x} \in \mathbb{C}^D$3 via least squares, with regularization choices including L1, L2, and a fixed norm constraint (Saiapin et al., 2 Dec 2025). The resulting model learns feature hyperparameters jointly with model parameters rather than selecting them by cross-validation:

$\mathbf{x} \in \mathbb{C}^D$4

Learned Fourier embeddings also appear in deep reinforcement learning. There, the input is mapped as

$\mathbf{x} \in \mathbb{C}^D$5

with $\mathbf{x} \in \mathbb{C}^D$6 trainable and initialized from $\mathbf{x} \in \mathbb{C}^D$7. Infinite-width analysis shows that tuning $\mathbf{x} \in \mathbb{C}^D$8 changes the Neural Tangent Kernel,

$\mathbf{x} \in \mathbb{C}^D$9

which the paper interprets as functional regularization controlling the relative ease of fitting low- and high-frequency components (Li et al., 2021).

The neural FLM architecture is simpler in form but more explicit in spectral semantics. It uses cosine activation functions in hidden layers, with frequencies, amplitudes, and phase shifts all trainable. The paper emphasizes a one-to-one correspondence between Fourier coefficients and amplitudes and phase shifts, so that the network parameters have a direct interpretation in terms of a nonharmonic Fourier series (Rubel et al., 10 Sep 2025).

4. Architectural variants and neighboring models

The recent neural FLM is structurally close to a feedforward network, but it differs from generic sinusoidal architectures in what it guarantees about basis structure. The paper contrasts FLM with SIREN and states that, unlike previous Fourier-inspired neural models, FLM is able to represent a complete, separable Fourier basis in multiple dimensions using a standard MLP-like architecture (Rubel et al., 10 Sep 2025). In this architecture, each frequency mode is handled by a parallel sub-network, and each sub-network uses $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$0 symmetric phase-shifted cosine bases for an $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$1-dimensional input (Rubel et al., 10 Sep 2025).

The tensor-decomposition strand is architecturally different. It is not a neural network in the usual deep-learning sense, but a low-rank parameterization of a model linear in a very large Fourier feature space. Its main architectural object is the CPD factorization of the weight tensor, which turns full tensor-product deterministic Fourier features into a tractable large-scale learner (Wesel et al., 2021, Saiapin et al., 2 Dec 2025). The Feature Learning model adds a second learned layer of structure by assigning weights to several Fourier feature maps with different hyperparameters and learning those weights jointly (Saiapin et al., 2 Dec 2025).

Related Fourier-domain neural systems share motivations with FLMs without necessarily using the term. FourierMamba performs image deraining with Mamba in the Fourier space, using zigzag coding to scan spatial Fourier frequencies and direct Mamba scanning in the channel Fourier domain (Li et al., 2024). GFF-PIELM integrates a variation of Fourier feature mappings into a physics-informed extreme learning machine by replacing the activation with a cosine-based Fourier function and assigning per-neuron frequency coefficients (Ren et al., 14 Oct 2025). These systems are not presented as FLMs, but they reinforce a broader pattern: Fourier-parameterized representations are repeatedly used to mitigate spectral bias, improve efficiency, or encode domain structure.

5. Domains of application and empirical behavior

The application range associated with FLM-style methods is broad. FFT-based Fourier learning has been applied to supervised regression on noisy data and to highly nonlinear multiclass classification with overlapped classes (Mehrabkhani, 2019, Mehrabkhani, 2020). The tensor-decomposition line targets large-scale kernel learning and regression on real datasets of various dimensionality and scale (Wesel et al., 2021, Saiapin et al., 2 Dec 2025). Learned Fourier features in reinforcement learning are used in both state-based and image-based RL, especially for value-function approximation under noisy Bellman targets (Li et al., 2021). The neural FLM is evaluated on scientific computing problems including the 1D Heat equation, the 2D Poisson equation, the Generalized Black-Scholes equation, and a family of Optimal Control Problems (Rubel et al., 10 Sep 2025).

Several quantitative findings recur across this literature. In the 2019 Fourier regression paper, a noisy example with $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$2 equispaced points reached $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$3 after 43 iterations (Mehrabkhani, 2019). In the tensor-network Feature Learning model, joint optimization of model and feature weights trains 3–5 times faster than a standard cross-validated model while having prediction quality on par with it (Saiapin et al., 2 Dec 2025). The 2021 large-scale deterministic Fourier-feature model reports that it consistently outperforms random Fourier features and obtains the same performance as the corresponding nonparametric model (Wesel et al., 2021). In reinforcement learning, learned Fourier features improve sample efficiency, are robust to noise in Bellman targets, and can stabilize training to the extent that target networks can be omitted in some cases (Li et al., 2021).

The neural FLM paper reports that computational experiments show performance comparable, and often superior, to established architectures like SIREN and vanilla feedforward NNs (Rubel et al., 10 Sep 2025). For the Heat equation, the reported values are $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$4 and $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$5 for FLM, compared with $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$6 and $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$7 for SIREN-1 (Rubel et al., 10 Sep 2025). For the Optimal Control Problems, the reported Mean Absolute Percentage Error on the integral objective is typically around $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$8 (Rubel et al., 10 Sep 2025).

6. Limitations, misconceptions, and nomenclature

A recurring limitation of Fourier-centered learning is dimensional scaling. The FFT-based classifier requires mapping data to a uniform equidistant mesh, and the paper states that this leads to exponential dependency on the feature numbers, making the approach infeasible beyond low or moderate dimensions (Mehrabkhani, 2020). Deterministic tensor-product Fourier features also suffer from the curse of dimensionality before tensor decomposition is introduced; that is precisely the bottleneck addressed by the CPD-based models (Wesel et al., 2021, Saiapin et al., 2 Dec 2025). In the neural FLM, the number of cosine terms per sub-network grows as $$\phi_{\theta}(\mathbf{x}) = \psi_\theta^{(D)}(x_D) \otimes \cdots \otimes \psi_\theta^{(1)}(x_1), \qquad \psi_\theta^{(d)}(x_d) = \left[ e^{-2\pi j x_d k/\theta} \right]_{k=0}^{I_d-1}.$$9 with input dimension $$f(\mathbf{x}) = \phi_\theta(\mathbf{x})^\top \mathbf{w},$$0, which the paper identifies as a potential computational cost for very high-dimensional inputs (Rubel et al., 10 Sep 2025).

Initialization and hyperparameter selection remain important even when FLM formulations reduce other sources of tuning. The neural FLM paper notes that initializing frequencies at integer grid points and amplitudes at zero performs well, while also suggesting that improved initialization strategies remain an open question (Rubel et al., 10 Sep 2025). The GFF-PIELM work reaches a similar conclusion from a neighboring direction: directly concatenating multiple Fourier feature mappings can make frequency-related hyperparameters difficult to determine, motivating an initialization strategy based on monitoring the distribution of output weights (Ren et al., 14 Oct 2025).

A common misconception is that every paper using the acronym “FLM” refers to a Fourier Learning Machine. The 2026 occupancy-prediction paper explicitly states that “FLM” there does not stand for Fourier Learning Machine and does not employ Fourier-based methods; instead, it denotes Feed-forward Likelihood Maximization for primitive-based 3D occupancy modeling (Chen et al., 19 Jun 2026). The literature therefore supports a narrow and a broad reading of the term: narrowly, FLM may denote the nonharmonic cosine-activation neural architecture of 2025; broadly, it can describe machine-learning frameworks in which Fourier series, Fourier features, or Fourier-domain filtering are constitutive rather than incidental (Rubel et al., 10 Sep 2025, Mehrabkhani, 2019).