Physics-Informed Fourier Basis NN

• FBNN is a physics-informed neural PDE solver that explicitly integrates fixed Fourier basis functions with neural networks to improve high-frequency solution accuracy.
• The method augments standard PINNs with an additive neural-plus-Fourier decomposition and employs adaptive coefficient pruning to refine the spectral approximation.
• FBNNs demonstrate enhanced performance on oscillatory and multiscale PDE problems by leveraging explicit Fourier structure within the optimization process.

Searching arXiv for the core paper and closely related Fourier/PINN formulations to ground the article in current literature. A Physics-Informed Fourier Basis Neural Network (FBNN) is a physics-informed neural architecture in which Fourier basis functions enter the solution representation, feature representation, or physics-aware optimization in a structurally explicit way. In the literature surveyed here, the clearest direct instantiation is the “Fourier PINNs” formulation of Moseley and collaborators, which augments a standard PINN with an additive explicit Fourier basis in the output ansatz and trains it under PDE residual and boundary-condition losses (Cooley et al., 2024). Closely related formulations include shallow cosine-based Fourier neural networks for differential equations (Ngom et al., 2020), domain-aware Laplacian-eigenfunction embeddings for PINNs (Calero et al., 3 Mar 2026), cosine embeddings that hard-constrain Neumann boundary conditions (Straub et al., 1 Apr 2025), and several randomized, separable, recurrent, or operator-learning variants that are Fourier-based but differ materially from a canonical PINN-style output-basis construction (Yang et al., 29 May 2026, Wu et al., 22 Oct 2025, Gopakumar et al., 2023). Taken together, these works support a precise usage of “FBNN”: a neural PDE solver that remains physics-informed through PDE, boundary, and initial-condition constraints while introducing a Fourier basis as a first-class component of the approximation space rather than merely as generic preprocessing.

1. Definition and conceptual scope

In the most direct formulation, an FBNN can be identified with a physics-informed model whose approximation ansatz contains both a neural component and an explicit Fourier basis component. The strongest example is the “Fourier PINNs” model, whose output is

$$u_F(x;\theta) = \sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H) + \sum_{n=1}^{K} a_n \cos(2\pi \omega_n x) + b_n \sin(2\pi \omega_n x),$$

or equivalently

$u_F(x;\theta)=u_N(x)+u_B(x),$

with

$$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$

This makes the Fourier basis part of the output ansatz itself, trained under a PINN objective (Cooley et al., 2024).

That distinction is important. This formulation is explicitly described as closer to “Physics-Informed NN + explicit Fourier basis expansion in the output space” than to either Fourier feature input embeddings or sinusoidal hidden-layer networks (Cooley et al., 2024). A standard PINN surrogate in the same paper is written as

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$

with physics-informed training based on

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$

where

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$

An FBNN in this sense preserves the PINN structure of the loss while enlarging the trial space with explicit Fourier modes (Cooley et al., 2024).

A broader literature uses “Fourier” in weaker or different senses. Some methods place Fourier structure in the input embedding, as in domain-aware Fourier features derived from Laplacian eigenfunctions (Calero et al., 3 Mar 2026) or cosine-only embeddings for exact Neumann enforcement (Straub et al., 1 Apr 2025). Others use cosine activations to obtain shallow Fourier-like solution representations (Ngom et al., 2020), or use Fourier-domain recurrence or operator layers without explicit PDE-residual constraints (Gopakumar et al., 2023). This suggests that “FBNN” is most precise when reserved for models where the Fourier basis is structurally tied to the solution ansatz or to a mathematically explicit trial space, rather than simply appearing as a generic spectral ingredient.

2. Canonical output-space formulation

The canonical FBNN formulation in the present literature is the additive neural-plus-Fourier decomposition introduced as Fourier PINNs (Cooley et al., 2024). Its defining characteristics are a fixed candidate frequency set, learned Fourier coefficients, physics-informed training, and adaptive pruning of insignificant modes. The frequencies $\{\omega_n\}_{n=1}^K$ are pre-specified dense candidate frequencies, uniformly sampled from a range; in the one-dimensional description, the construction effectively uses $\{1,2,\dots,K\}$ (Cooley et al., 2024). The method therefore uses a fixed overcomplete Fourier dictionary rather than learned frequencies.

The same paper is explicit that this is not frequency learning in the sense of optimizing $\omega_n$. Instead, the learned quantities are the Fourier coefficients $(a_n,b_n)$, and significance is determined by regularization and truncation (Cooley et al., 2024). This leads to a practical definition of significant frequencies: they are the modes whose learned coefficients survive coefficient shrinkage and threshold pruning.

For higher dimensions, the basis is extended by tensor products. For $u_F(x;\theta)=u_N(x)+u_B(x),$0, the one-dimensional basis vectors are

$u_F(x;\theta)=u_N(x)+u_B(x),$1

$u_F(x;\theta)=u_N(x)+u_B(x),$2

and the resulting two-dimensional ansatz is

$u_F(x;\theta)=u_N(x)+u_B(x),$3

The paper explicitly notes that this suffers from exponential growth with dimension and points to sparse basis constructions or low-rank tensor decompositions as future remedies (Cooley et al., 2024). This makes the multidimensional formulation conceptually straightforward but computationally fragile.

A closely related but more classical Fourier-basis viewpoint appears in shallow cosine networks. The Fourier neural network of Huang et al. uses

$u_F(x;\theta)=u_N(x)+u_B(x),$4

and interprets $u_F(x;\theta)=u_N(x)+u_B(x),$5 as mode indices, $u_F(x;\theta)=u_N(x)+u_B(x),$6 as amplitudes, $u_F(x;\theta)=u_N(x)+u_B(x),$7 as phase shifts, and $u_F(x;\theta)=u_N(x)+u_B(x),$8 as the constant mode (Ngom et al., 2020). That model becomes physics-informed when trained by residual minimization for differential equations, yielding what the paper explicitly calls a physics-informed Fourier neural network (Ngom et al., 2020). Relative to the Fourier PINNs construction, this is a learned-frequency shallow Fourier ansatz rather than a hybrid neural-plus-fixed-dictionary one.

A plausible implication is that current FBNN usage naturally splits into two subfamilies: fixed-dictionary output decompositions, exemplified by Fourier PINNs (Cooley et al., 2024), and shallow trainable-frequency Fourier ansätze, exemplified by cosine-activation Fourier neural networks (Ngom et al., 2020).

3. Physics-informed objectives and constraint handling

The physics-informed part of an FBNN remains the PDE-constrained training objective. In the baseline formulation,

$u_F(x;\theta)=u_N(x)+u_B(x),$9

with boundary conditions

$$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$0

and training proceeds by minimizing residual and constraint losses (Cooley et al., 2024). In Fourier PINNs, the loss is augmented with coefficient regularization,

$$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$1

where $$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$2 denotes the set of basis coefficients and $$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$3 is a user-chosen regularization strength (Cooley et al., 2024). The paper states that this “promotes sparsity,” although the written penalty is $$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$4, so sparsity is realized operationally through later truncation rather than directly through an $$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$5 objective (Cooley et al., 2024).

A distinct line of work uses Fourier structure to hard-code constraints into the architecture rather than penalize them softly. For Dirichlet conditions, the Fourier PINNs paper studies strong boundary-condition constructions of the form

$$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$6

where $$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$7 vanishes on the boundary, so the transformed ansatz satisfies the boundary exactly (Cooley et al., 2024). Two distance functions are considered: $$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$8 and

$$u_B(x;a,b)=\sum_{n=1}^{K} a_n\cos(2\pi\omega_n x)+b_n\sin(2\pi\omega_n x).$$9

with $$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$0 preset or optimized during training (Cooley et al., 2024). These constructions are exact for the considered 1D Dirichlet settings but difficult to generalize to many other boundary conditions and geometries (Cooley et al., 2024).

For Neumann conditions, a different Fourier-based hard constraint is available. Wu and coauthors replace the spatial coordinate by a cosine-only embedding such as

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$1

or its multiscale extension

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$2

The transformed model

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$3

then satisfies homogeneous Neumann conditions exactly by the chain rule because $$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$4 (Straub et al., 1 Apr 2025). For nonzero constant Neumann data, the architecture adds a polynomial correction: $$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$5 This construction is presented for intervals and hyperrectangles and for constant boundary flux values (Straub et al., 1 Apr 2025).

Another constraint-oriented formulation replaces generic Fourier features with domain-aware Laplacian eigenfunctions. PINN-DaFFs use eigenfunctions satisfying

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$6

so that geometry and boundary operators are embedded directly into the feature map (Calero et al., 3 Mar 2026). For rectangular domains with homogeneous Dirichlet conditions,

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$7

By removing biases in the downstream MLP, zero-valued encoded inputs at the boundary remain zero through the network, allowing the PDE residual to be optimized without separate boundary-condition losses in the homogeneous case (Calero et al., 3 Mar 2026).

These variants show that an FBNN can be physics-informed in at least three distinct ways: by standard soft PDE/BC penalties (Cooley et al., 2024), by hard-constrained Fourier-compatible embeddings for boundary operators (Straub et al., 1 Apr 2025), or by domain-eigenfunction front-ends that encode admissible function spaces before residual minimization (Calero et al., 3 Mar 2026).

4. Spectral motivation and optimization mechanisms

The main theoretical motivation for the modern FBNN literature is the difficulty of standard PINNs with high-frequency and multiscale targets. Fourier PINNs begins from the observation that strong Dirichlet constructions improve high-frequency learning, and explains this in the Fourier domain (Cooley et al., 2024). The boundary multiplier is expanded as a periodic Fourier series,

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$8

with coefficients such as

$$u_N(x;\theta)=\sum_{j=1}^{W} c_j \,\psi_j(x;\theta^H),$$9

and

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$0

The periodic extension has transform

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$1

and the hard-constrained ansatz satisfies

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$2

The interpretation given is that multiplication by $$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$3 in physical space corresponds to convolution with a rapidly decaying filter in frequency space, suppressing irrelevant high-frequency tails and helping the surrogate estimate the amplitudes of the true high-frequency components (Cooley et al., 2024).

That argument motivates the explicit Fourier basis augmentation. In the resulting Fourier PINNs method, adaptive basis selection is performed by alternating least-squares coefficient estimation, coefficient pruning, and joint gradient optimization (Cooley et al., 2024). The least-squares subproblem is written as

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$4

where $$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$5 contains evaluations of both neural and Fourier basis functions under the differential operator (Cooley et al., 2024). For nonlinear operators, the method freezes the nonlinear contribution at the current iterate; for Allen–Cahn, for example,

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$6

and the least-squares target uses

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$7

This yields a fixed-point-style linearization step (Cooley et al., 2024).

The adaptive algorithm alternates between least-squares coefficient estimation, pruning all coefficients satisfying $$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$8, joint Adam updates, and a final L-BFGS stage (Cooley et al., 2024). The reported hyperparameters are

$$\theta^*=\arg\min_\theta \;\lambda L_b(\theta)+L_r(\theta),$$9

although the text also mentions $$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$0 as a pruning level elsewhere, a minor inconsistency noted in the paper (Cooley et al., 2024). This alternating regression-plus-gradient strategy is one of the sharpest technical separations between Fourier PINNs and simpler “append Fourier modes and train end-to-end” models.

A different optimization lesson appears in adhesive contact mechanics. The mass-weighted Fourier PINN of Li et al. is not an output-basis FBNN in the strict sense; it uses Fourier-feature input encoding plus Fourier-space energy evaluation and spectral gradient preconditioning (Zhou et al., 5 Jul 2026). Its main contribution is a mass-weighting function

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$1

combined with a low-pass filter

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$2

to modify the gradient

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$3

This does not change the scalar energy objective $$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$4; it changes the descent direction by reweighting spectral components before backpropagation (Zhou et al., 5 Jul 2026). A plausible implication is that, for some Fourier-informed PINNs, trainability may depend as much on spectral conditioning of optimization as on basis choice.

5. Empirical behavior and problem classes

The main empirical evidence for explicit output-space FBNNs comes from Fourier PINNs (Cooley et al., 2024). The method is tested on linear and nonlinear PDEs in one and two dimensions, including 1D Poisson, 2D Poisson, 1D steady-state Allen–Cahn, 2D steady-state Allen–Cahn, and the 1D one-way wave equation (Cooley et al., 2024). The experiments focus on high-frequency and multiscale manufactured solutions such as

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$5

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$6

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$7

and in two dimensions,

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$8

$$L_b(\theta)=\frac{1}{M}\sum_{i=1}^{M}\Big(\mathcal{B}[u_N](x_b^i)-g(x_b^i)\Big)^2, \qquad L_r(\theta)=\frac{1}{N}\sum_{i=1}^{N}\Big(\mathcal{L}[u_N](x_r^i)-f(x_r^i)\Big)^2.$$9

$\{\omega_n\}_{n=1}^K$0

The paper reports that, across all tested problems, standard PINN, weighted PINN, and adaptive-activation PINN typically fail on these high-frequency examples, with relative errors around $\{\omega_n\}_{n=1}^K$1, while Fourier PINNs consistently achieve relative $\{\omega_n\}_{n=1}^K$2 errors around $\{\omega_n\}_{n=1}^K$3 to $\{\omega_n\}_{n=1}^K$4 (Cooley et al., 2024).

The same paper compares against standard PINN, RFF-PINN with random Fourier features and NTK-based dynamic loss reweighting, weighted PINN, adaptive-activation PINN, and a classical spectral method using the same Fourier basis family (Cooley et al., 2024). It reports that RFF-PINN can occasionally work but is highly sensitive to the number and scale of Gaussian variances, with roughly $\{\omega_n\}_{n=1}^K$5 of the 20 tested RFF configurations failing (Cooley et al., 2024). The spectral method also fails to match Fourier PINNs, which the authors take to indicate that a pure Fourier linear expansion without the neural component is insufficient (Cooley et al., 2024).

Constraint-aware Fourier embeddings also show strong empirical gains. For Neumann boundary conditions on the 1D diffusion equation, the cosine-embedding hard-constraint method outperforms existing hard-constraining methods and classical PINNs, particularly in multiscale and high-frequency scenarios (Straub et al., 1 Apr 2025). Reported relative improvements over the best vanilla baseline are $\{\omega_n\}_{n=1}^K$6 for low frequency, $\{\omega_n\}_{n=1}^K$7 for a third-order polynomial case, $\{\omega_n\}_{n=1}^K$8 for a fourth-order polynomial case, $\{\omega_n\}_{n=1}^K$9 for high frequency, and $\{1,2,\dots,K\}$0 for the multiscale case (Straub et al., 1 Apr 2025). A clear pattern is reported: single low-frequency embeddings work best for low-frequency problems, whereas larger cosine dictionaries work better for high-frequency and multiscale problems (Straub et al., 1 Apr 2025).

Domain-aware Fourier features produce similarly strong gains. For the Kirchhoff–Love plate benchmark, PINN-DaFFs achieve training loss $\{1,2,\dots,K\}$1 and validation loss $\{1,2,\dots,K\}$2, compared with $\{1,2,\dots,K\}$3 and $\{1,2,\dots,K\}$4 for vanilla PINNs and $\{1,2,\dots,K\}$5 and $\{1,2,\dots,K\}$6 for PINN-RFFs (Calero et al., 3 Mar 2026). For the Helmholtz benchmark, reported validation losses are $\{1,2,\dots,K\}$7 for vanilla PINNs, $\{1,2,\dots,K\}$8 for PINN-RFFs, and $\{1,2,\dots,K\}$9 for PINN-DaFFs (Calero et al., 3 Mar 2026). These results are tied to single-objective residual training once homogeneous constraints are built into the basis (Calero et al., 3 Mar 2026).

Not all Fourier-related variants are standard PINNs. MS-SFNN, for example, is a separable cosine-basis collocation method with frozen hidden parameters and least-squares coefficient fitting rather than end-to-end PINN training (Yang et al., 29 May 2026). It reports extremely strong accuracy on high-frequency PDEs, but it is best understood as a physics-constrained random Fourier basis method rather than a classical physics-informed neural network (Yang et al., 29 May 2026). Likewise, IFeF-PINN augments learned latent features with random Fourier features and alternates coefficient regression with basis refinement, improving high-frequency Helmholtz and convection benchmarks dramatically over vanilla PINNs (Wu et al., 22 Oct 2025). These results suggest a broader theme: explicit or semi-explicit spectral structure consistently helps when the target solutions are oscillatory, multiscale, or otherwise difficult for standard PINN training.

6. Related architectures, distinctions, and limitations

Several neighboring architectures are relevant but should not be conflated with a canonical FBNN. A concise comparison is useful.

Method family	Fourier role	Physics-informed status
Fourier PINNs	Explicit Fourier basis in output ansatz	Yes (Cooley et al., 2024)
PINN-DaFFs	Domain-aware Laplacian eigenfunction input basis	Yes (Calero et al., 3 Mar 2026)
Neumann cosine embedding	Cosine embedding enforces boundary derivatives	Yes (Straub et al., 1 Apr 2025)
Fourier neural network	Cosine-basis shallow solver	Yes when trained by residual minimization (Ngom et al., 2020)
Fourier-RNN	Fourier operator layers inside recurrence	No explicit PDE-residual training (Gopakumar et al., 2023)
FD-PINN	Fourier-domain physics evaluation in loss	Yes, but not basis-explicit in output (Musgrave et al., 2024)

The Fourier-RNN is a strong conceptual neighbor but is not physics-informed in the PINN sense. Its recurrent update is

$\omega_n$0

and training is supervised on physics-relevant data rather than constrained by PDE residuals (Gopakumar et al., 2023). It therefore belongs to spectral scientific machine learning, but not to the strict FBNN category.

FD-PINN is also adjacent rather than identical. It predicts the field in physical coordinates and uses Fourier transforms in the loss to evaluate nonlocal convolution terms in the generalized pulse propagation equation (Musgrave et al., 2024). The Fourier domain is used for operator evaluation, not for the neural basis itself. The paper is explicit that model generalization and accurate prediction of high-frequency optical-field components under strong nonlinearity are not addressed by the described algorithm (Musgrave et al., 2024). This distinguishes Fourier-domain physics enforcement from Fourier-basis representation.

Important limitations recur across the literature. The output-space Fourier dictionary of Fourier PINNs suffers from tensor-product basis explosion in higher dimensions (Cooley et al., 2024). Constraint-oriented Fourier constructions are easiest on intervals, boxes, and simple geometries; the Neumann hard-constraint embedding is formulated for intervals and hyperrectangles, with constant flux data in the main treatment (Straub et al., 1 Apr 2025). Domain-aware eigenfunction features are most straightforward when the Laplacian eigensystem is available analytically or can be computed numerically at moderate cost (Calero et al., 3 Mar 2026). Fourier-based embeddings may also be inappropriate for problems dominated by discontinuities or strongly localized structures; RBF-PINN argues that Fourier feature mappings can suffer Gibbs-like artifacts near jump solutions and underperform RBF-based embeddings on several forward and inverse PDE problems (Zeng et al., 2024). This suggests that the appropriateness of an FBNN depends materially on the regularity, locality, geometry, and spectral content of the target problem.

A balanced conclusion is therefore possible. The literature supports a precise and reusable definition of a Physics-Informed Fourier Basis Neural Network as a physics-informed model in which Fourier basis functions are explicit elements of the solution space, constraint space, or basis-aware optimization mechanism. The clearest reusable components are the additive neural-plus-Fourier ansatz with fixed candidate frequencies and adaptive coefficient pruning (Cooley et al., 2024), boundary-compatible cosine embeddings for exact Neumann enforcement (Straub et al., 1 Apr 2025), and domain-aware Laplacian-eigenfunction features for embedding geometry and boundary operators directly into the representation (Calero et al., 3 Mar 2026). At the same time, the literature also indicates that “Fourier-informed” does not always mean “Fourier-basis” in the strict sense, and that spectral structure alone does not remove issues of dimensional scaling, geometry dependence, optimization conditioning, or mismatch between global trigonometric bases and localized solution features.