FourierPathFinder: Fourier-Based Path Methods

• FourierPathFinder is a cross-domain methodology that transforms complex search problems into tractable Fourier-domain representations.
• It leverages fixed-point dynamics and Fourier-based deformations to efficiently solve phase retrieval, tunnelling, and propagation challenges.
• Its versatility spans applications in quantum mechanics, PDE solvers, image analysis, and sparse signal recovery, demonstrating practical efficiency improvements.

Searching arXiv for the cited works and related uses of “FourierPathFinder”. First, I’ll verify the core paper on Fourier-domain fixed point algorithms and then gather the other papers that use “FourierPathFinder” as a descriptive label for Fourier-based path construction or propagation. “FourierPathFinder” appears in the supplied literature as a descriptive label for methods that use Fourier representations to construct, deform, propagate, or recover paths, trajectories, and solution manifolds. In different settings, the term refers to an algorithm that walks in Fourier space during phase retrieval, an endpoint-safe Fourier parametrization of tunnelling paths, a Fourier-flow generator of Euclidean Feynman paths, FFT-stitched real-time path-integral propagation, and related Fourier-domain search or detection procedures in PDEs, stochastic integration, sparse transforms, and image analysis (Fannjiang, 2014, Kannagi et al., 16 Jun 2026, Chen et al., 2022, Feldbrugge et al., 27 Jan 2025, Hatharasinghe et al., 29 Jul 2025, Gubinelli et al., 2014, Lawlor et al., 2012, Ruzicka et al., 2021). This suggests that FourierPathFinder is best understood as a cross-domain methodological motif rather than a single canonical algorithm.

1. Conceptual range of the term

Across the cited works, the common structure is the replacement of a direct search in physical, object, or configuration space by a search in a Fourier or Fourier-like representation. In phase retrieval, the iterate is a Fourier-domain variable constrained by magnitude and range projections (Fannjiang, 2014). In multi-field vacuum tunnelling, the unknown field-space curve is written as a straight-line interpolation plus Fourier sine deformations that vanish at the endpoints (Kannagi et al., 16 Jun 2026). In Euclidean quantum mechanics, paths are generated in Matsubara space and mapped back by inverse discrete Fourier transform (Chen et al., 2022). In real-time path integrals, a high-dimensional oscillatory integral is rewritten as a sequence of low-dimensional oscillatory integrals stitched together with fast Fourier transformations (Feldbrugge et al., 27 Jan 2025).

The same pattern recurs in adjacent areas. A Fourier approach to rough path integration expands continuous functions in Schauder functions and turns pathwise stochastic integrals into recursive scale-by-scale constructions (Gubinelli et al., 2014). Non-periodic Fourier propagation algorithms solve parabolic PDEs and SPDEs on uniform grids with fast discrete sine and cosine transforms (Hatharasinghe et al., 29 Jul 2025). Automated contour deformation for oscillatory integrals uses steepest-descent path construction in the complex plane and is implemented in software explicitly called PathFinder (Gibbs et al., 2023). A plausible implication is that “path finding” here denotes controlled navigation through a transformed representation in which geometry, decay, or boundary structure becomes simpler.

2. Fourier-space fixed points in phase retrieval

The most explicit “FourierPathFinder” picture in the supplied material is the Fourier-domain Difference Map for phase retrieval with two oversampled coded diffraction patterns (Fannjiang, 2014). The forward model uses two masks,

$\Psi_j = \Phi\,\mathrm{diag}(\mu_j),\quad j=1,2,$

and the combined operator

$$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$

with data $y_0=\Psi x_0$ and measured magnitudes $b=|y_0|$. The feasibility problem is posed directly in Fourier space as the intersection of the range of $\Psi$ with the magnitude set $\mathcal{Y}=\{y:|y|=b\}$ (Fannjiang, 2014).

The two central projectors are the range projector

$\mathcal{P}_{\mathrm{o}} y = \Psi\Psi^* y$

and the magnitude projector

$\mathcal{P}_{\mathrm{m}} y = b\odot \frac{y}{|y|}.$

The Fourier-Domain Difference Map is a 3-parameter family,

$\mathcal{D}= I+\beta\,\Delta,$

with

$$\Delta=\mathcal{P}_{\mathrm{o}}\big((1+\gamma_2)\mathcal{P}_{\mathrm{m}}-\gamma_2 I\big)-\mathcal{P}_{\mathrm{m}}\big((1+\gamma_1)\mathcal{P}_{\mathrm{o}}-\gamma_1 I\big),$$

and includes Fourier-domain Hybrid-Projection-Reflection and Fourier-domain Douglas–Rachford as special cases (Fannjiang, 2014). For generic complex objects without any object constraint, the method yields a unique fixed point, after proper projection back to the object domain, which is the true solution up to a global phase factor (Fannjiang, 2014). In this usage, FourierPathFinder denotes a fixed-point dynamics whose trajectory is entirely organized by Fourier-domain projections.

3. Fourier-parameterized paths in tunnelling and quantum mechanics

In multi-field vacuum tunnelling, the term refers to an endpoint-safe Fourier method in which the tunnelling path is written as a straight-line interpolation between the false and true vacua plus sine-mode deformations that vanish at the endpoints (Kannagi et al., 16 Jun 2026): $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$0 The method turns path finding into a finite-dimensional optimization over $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$1 coefficients, implemented using automatic differentiation in the JAX numerical framework (Kannagi et al., 16 Jun 2026). It is used both as a standalone variational ansatz and as a preconditioner for existing bounce solvers. On the OptiBounce benchmark potential for $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$2 and on a nested random-coefficient potential family up to $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$3, the Fourier result agrees with FindBounce, OptiBounce, and CosmoTransitions at the sub-percent level in the regular benchmark cases, while requiring only a modest number of modes; in FindBounce point-injection tests, it gives large runtime improvements in the high-dimensional cases up to $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$4 (Kannagi et al., 16 Jun 2026).

A related use appears in generalized one-dimensional Feynman path integration, where fluctuations about a classical path are expanded in Fourier sine series,

$$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$5

or, more generally, in eigenfunctions of a Sturm–Liouville problem derived from the second variation of a reference action (Russell, 2018). There the Fourier basis acts as a parametrization of path space itself. In the machine-learning variant, the Fourier-Flow model generating Feynman paths uses a Fourier transformation to approach a Matsubara representation and then applies affine coupling layers to learn the Euclidean path distribution of harmonic and anharmonic oscillators (Chen et al., 2022). The reported outputs include accurate estimates of the ground-state wave function and low-lying energy levels (Chen et al., 2022).

4. Fourier-guided propagation and deformation

In real-time quantum dynamics, FourierPathFinder denotes a propagation strategy rather than a variational parametrization. The lattice-regularized real-time world-line path integral is rewritten as a series of low-dimensional oscillatory integrals, which are efficiently evaluated with Picard-Lefschetz theory or approximated with the eikonal approximation, and then stitched together with a series of fast Fourier transformations to recover the lattice regularized Feynman path integral (Feldbrugge et al., 27 Jan 2025). The method applies to theories where the potential is dominated by a quadratic at infinity and is described as directly applicable to quantum mechanics, the world-line quantization of quantum field theory, and quantum gravity (Feldbrugge et al., 27 Jan 2025).

An adjacent contour-deformation interpretation appears in the steepest-descent software PathFinder for oscillatory integrals with polynomial phase (Gibbs et al., 2023). There the input is simply the phase and amplitude functions, the endpoints and orientation of the original integration contour, and a small number of numerical parameters. The algorithm automates stationary-point detection, contour deformation, and numerical quadrature even when stationary points coalesce with each other or with endpoints, and it is used to evaluate cuspoid canonical integrals from scattering theory (Gibbs et al., 2023). Although that paper does not define “FourierPathFinder” as a formal term, the structural resemblance is clear: the path is not given a priori but is constructed algorithmically from transformed phase information.

5. Fourier path construction in PDEs and rough integration

For parabolic PDEs and SPDEs on non-periodic domains, the relevant construction is a Fourier method that employs fast discrete sine transforms or and discrete cosine transforms on a uniform grid with non-periodic boundaries (Hatharasinghe et al., 29 Jul 2025). The paper implements this in two ways: a Fourier spectral derivative method and a Fourier interaction picture approach. These methods treat vector fields with combinations of Dirichlet and or Neumann boundary conditions in one or more space dimensions, and on the 1D heat equation problem the interaction-picture method is accurate up to the machine precision (Hatharasinghe et al., 29 Jul 2025). Here the “path” is the evolution operator itself, propagated through boundary-adapted spectral coordinates.

In rough path integration, a Fourier approach to rough path integration is based on the series decomposition of continuous functions in terms of Schauder functions and leads to recursive algorithms for the calculation of pathwise stochastic integrals, both of Itô and of Stratonovich type (Gubinelli et al., 2014). The construction decomposes the integral into a paraproduct term, a symmetric smooth term, and an antisymmetric Lévy-area term, and applies the resulting machinery to solve stochastic differential equations in a pathwise manner (Gubinelli et al., 2014). In this setting, FourierPathFinder denotes a scale-recursive reconstruction of integrals and controlled paths from a Fourier-like basis rather than a geometric path in space.

6. Detection, sparse recovery, and scope

The same label extends to Fourier-guided detection and sparse search. In image analysis, a general blueprint is given for detecting directed, quasi-periodic linear structures by transforming an image to the 2D frequency domain, analyzing angular sector power, estimating dominant wavelengths, and reconstructing filtered structures by inverse FFT (Ruzicka et al., 2021). In time-series analysis, the fractional period spectrum is computed through the congruence derivative sequence and self-shift sums, giving a direct method for evaluating Fourier spectra at fractional periods without computing the full transform (Wang et al., 2016). In sparse harmonic recovery, an adaptive sub-linear Fourier algorithm identifies $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$6 significant Fourier coefficients using short DFTs, tiny time shifts, and adaptive sampling decisions; the paper reports that the algorithm scales linearly with $$\Psi = \frac{1}{\sqrt{2}} \begin{bmatrix} \Psi_1\ \Psi_2 \end{bmatrix},$$7 in the average case and is empirically orders of magnitude faster than competing algorithms (Lawlor et al., 2012).

A common misconception would be to treat FourierPathFinder as the name of one specific arXiv method. The supplied literature does not support that reading. Instead, it uses the phrase to describe a recurring design principle: represent the relevant object in a Fourier or Fourier-like basis, use that basis to expose constraints or geometry, and then recover the desired solution, path, or spectrum by projection, optimization, propagation, or deformation (Fannjiang, 2014, Kannagi et al., 16 Jun 2026, Feldbrugge et al., 27 Jan 2025, Hatharasinghe et al., 29 Jul 2025). Another plausible implication is that the term is most useful when the underlying problem has either periodic structure, boundary-compatible trigonometric structure, or an oscillatory integral structure for which Fourier coordinates simplify the dominant operator.

Within that shared pattern, the underlying tasks remain distinct. Phase retrieval seeks a fixed point in the intersection of a magnitude set and a forward-model range (Fannjiang, 2014). Vacuum tunnelling optimizes a finite Fourier ansatz for a field-space curve (Kannagi et al., 16 Jun 2026). Generative quantum models learn the Euclidean path distribution in Matsubara space (Chen et al., 2022). Real-time path-integral solvers use FFT-based stitching of low-dimensional oscillatory building blocks (Feldbrugge et al., 27 Jan 2025). PDE solvers diagonalize linear propagation by DST or DCT (Hatharasinghe et al., 29 Jul 2025). FourierPathFinder therefore names a family resemblance: a Fourier-organized route through otherwise difficult path or propagation problems.