Chapter Four · failure evidence
What Fourier & Spectral Transformation got wrong, from 76 dissertations
The records document multiple settings where Fourier and spectral methods fail due to non-stationarity, boundary artifacts, computational bottlenecks, and spectral bias in neural architectures. Across various signal processing, graph learning, and operator modeling tasks, these transformations were frequently outperformed by spatial domain alternatives or simpler baselines. These records come from PhD theses at 24 institutions, 2021 to 2026. Each links to its thesis. They were extracted by language models reading the full text, so treat each as a lead to read, not a verdict.
Global Fourier transforms fail to capture localized features in non-stationary and transient signals
Standard Fourier and short-time Fourier transforms suffer from fixed windowing and global basis integration that eliminate essential time-domain localization for transient disturbances and non-stationary signals. Consequently, these transforms lose critical temporal phase details, induce spectral leakage across frequency bins, and degrade under dynamic operating conditions.
Considered and rejected
Considered and rejected: Rejected Fourier transform for rs-fMRI frequency decomposition due to global basis functions failing to handle local non-linearities without significant frequency leakage.
NEW APPROACHES FOR ASSESSING TIME-VARYING FUNCTIONAL BRAIN CONNECTIVITY USING FMRI DATA · JScholarship
Tried and failed
FFT magnitude features with PCA classification applied to aligned ultrasonic A-scan defect signals. Outcome: worse than baseline. Reason: Fourier magnitude spectrum lost phase and localized time-frequency defect details compared to wavelets or DCT
Ultrasonic signal processing and classification using principal component analysis · Iowa State
Tried and failed
Hilbert transform and classic spectrogram applied to instantaneous frequency estimation from sensor signals. Outcome: unstable. Reason: Edge effect artifacts and spectral leakage occurred without strict bandpass filtering
Considered and rejected
Considered and rejected: Standard Fourier Transform was rejected because it averages over the entire signal length and loses time localization of non-stationary flaw reflections.
Automatic ultrasound signal classification scheme · Iowa State
Considered and rejected
Considered and rejected: Rejected Fourier and Short-Time Fourier Transforms for signal processing due to loss of time localization and fixed windowing on non-stationary MZI signals.
Considered and rejected
Considered and rejected: Fast Fourier Transform (FFT) for frequency extraction was rejected in favor of time-varying Extended Kalman Filter due to FFT's limitations with irregular sampling, phase/amplitude shifts, and embedded real-time processing.
Psychological Monitoring: Detecting Real-Time Emotional Changes Real-Time Emotional Monitoring: A Constant Dimension Approach · Virginia Tech
Considered and rejected
Considered and rejected: Rejected pure Fourier Transform (FT) due to lack of time-domain localization for transient/non-stationary disturbances.
Power Quality Disturbance Classification in IEEE 9-Bus System using STFT and Deep Learning · Texas Tech
Considered and rejected
Considered and rejected: Rejected discrete Fourier transform / standard Fourier transform for time-varying seismic signal analysis due to complete lack of temporal localisation
Seismic characterisation based on time-frequency spectral analysis · Imperial
Considered and rejected
Considered and rejected: Rejected fixed-window short-time Fourier transform (STFT) due to the rigid trade-off between temporal and spectral resolution and spectral leakage from window side lobes
Seismic characterisation based on time-frequency spectral analysis · Imperial
Considered and rejected
Considered and rejected: Rejected Discrete Fourier Transform (DFT) for ramp-rate power reference filtering because it loses time localization of frequency components.
Energy storage for complementary services in grid-tied PV systems · University of Nottingham Repository
Considered and rejected
Considered and rejected: Rejected standard FFT spectrogram analysis due to poor resolution for abrupt frequency/time shifts, adopting 3D Time Fast Fourier Transform (TFFT) instead.
A comparative study of the creative processes in selected works of Roxy Music’s Phil Manzanera and Andy Mackay · De Montfort Open Research Archive (DORA)
Considered and rejected
Considered and rejected: Rejected Fourier transform in favor of continuous wavelet transform due to its inability to assess time-varying, non-stationary periodicities in noisy hydrological datasets.
Tried and failed
Short-time Fourier transform spectral feature extraction applied to non-stationary operational vibration signals. Outcome: did not generalise. Reason: Spectral features degraded under non-stationary ambient operating conditions compared to impulse excitation
Data-Driven Structural Health Monitoring of Wind Turbine Blades under Operational and Environmental Variability · Research Repository UCD
Considered and rejected
Considered and rejected: Fourier transforms rejected due to global support (small frequency edits alter entire time domain).
Modelling Bipedal Locomotion Using Wavelets for Figure Animation · De Montfort Open Research Archive (DORA)
Considered and rejected
Considered and rejected: Rejected standard Fourier and wavelet transforms for decomposing ionospheric EM field signals due to their inability to properly handle non-stationary geophysical signals.
Electromagnetic Characterization of the Ionosphere in the Framework of the Magnetosphere-Ionosphere-Lithosphere Coupling · IRIS - UNITN - prod
Considered and rejected
Considered and rejected: Fourier transform / Windowed Fourier transform was rejected due to lack of variable time-frequency scaling window resolution.
Application of Wavelet-based Denoising to Improve the Accuracy of Nanopore Sequencing Data · HARVEST
Fourier basis representations struggle with non-periodic structures and complex analytical constraints
Static Fourier series expansions and Fourier feature encodings lack the expressiveness or smoothness needed to represent non-periodic spatial profiles and dynamic systems compared to learned representations or domain-specific bases. In several settings, using pure Fourier expansions broke time-translation invariance, required strict spatial periodicity, or failed to provide closed-form analytical inverse transformations.
Tried and failed
Fourier basis functions instead of polynomial basis applied to Koopman operator linear dynamic models. Outcome: worse than baseline. Reason: Fourier basis did not yield significantly better linearization results than polynomial basis
Tried and failed
non-learned Fourier series basis functions applied to dynamic motion representation. Outcome: worse than baseline. Reason: insufficiently expressive and lacked smoothness compared to learned neural network representations
4D VISION: REPRESENT, RECONSTRUCT AND GENERATE THE DYNAMIC 3D WORLD · Penn
Tried and failed
Fourier transform input encoding for bound propagation applied to neural network robustness verification. Outcome: worse than baseline. Reason: offered limited benefit over pixel-space verification while adding unnecessary complexity for high-frequency perturbations
Expressive specifications for neural network verification and certified training · Imperial
Tried and failed
Fourier series expansion for spatial profile representation applied to radial non-uniform flow profile reconstruction. Outcome: worse than baseline. Reason: yielded higher truncation error than domain-specific orthonormal basis functions without significant improvement at higher orders
Considered and rejected
Considered and rejected: Rejected analyzing the coupled system using weighted Fourier norms due to severe incompatibility with the linear translation operator in the Grad system.
Invariant manifold theory and the relationship between kinetic theory and continuum models · OpenBU
Considered and rejected
Considered and rejected: Rejected using a single shared basis (e.g., pure Fourier features) for both prior and posterior updates due to severe variance starvation
Decision-making with gaussian processes: sampling strategies and monte carlo methods · Imperial
Considered and rejected
Considered and rejected: Rejected fast-transformation Fourier MZI network architecture because it lacks arbitrary unitary universality
An Electronic Control Architecture for a Photonic Integrated Circuit · IRIS - UNITN - prod
Considered and rejected
Considered and rejected: Rejected relying on Fourier-based frequency transfer function analysis for NBI power deposition due to invalidity under long slowing-down times.
Refined interpretation of T e̳ response to neutral beam injection at DIII-D · UT Austin
Considered and rejected
Considered and rejected: Rejected standard discrete Fourier transforms and piece-wise segmentation for mutating materials because breaking time-translation invariance causes systematic over- and under-prediction errors.
Tried and failed
perturbing low-frequency Fourier components during data augmentation applied to image semantic segmentation models. Outcome: worse than baseline. Reason: corrupting low-frequency information destroys essential visual structure and semantics needed for generalization
Harnessing Synthetic Data for Robust and Reliable Vision · Georgia Tech
Considered and rejected
Considered and rejected: Rejected Fourier series approaches for correcting asteroid rotational light curve noise because they do not account for the variation in the rotation curve form as a function of phase angle.
Small Bodies and Large Surveys: What Modern Dynamics can teach us about the Solar System · Harvard
Tried and failed
analytical inverse Fourier transform of filter function applied to spatial contrast sensitivity function modeling. Reason: the fractional power term in the frequency-domain formulation lacked a closed-form analytical inverse transform
Tried and failed
Discrete Fourier transform unitary for multiport projection applied to multiport linear optical entanglement fusion. Outcome: worse than baseline. Reason: Did not improve success probability and introduced significant non-stabilizer failure rates
Linear optical fusion-based quantum computation with multi-way fusions · Imperial
Considered and rejected
Considered and rejected: Rejected standard Fourier Transform (FFT) for spatial mode identification due to its strict requirement for spatial periodicity
Data-driven reduced-order modelling of the plasma systems dynamics · Imperial
Fourier neural operators and spectral networks suffer from spectral bias and optimization difficulties
Physics-informed and Fourier neural operator architectures failed to strictly enforce physical conservation laws and suffered from spectral bias that prevented learning multi-scale high-frequency dynamics. In addition, higher-frequency Fourier coefficients exhibited low gradients and contour oscillations, while spectral normalization over-constrained value networks and harmed model capacity.
Tried and failed
Fourier neural operators applied to PDE gradient and Hessian operator learning. Outcome: worse than baseline. Reason: Standard UNet outperformed Fourier Neural Operator in both accuracy and GPU inference speed
Generative models for seismic imaging and inversion · UT Austin
Tried and failed
non-autoregressive Fourier neural operator for transient dynamics applied to time-dependent PDE inverse problems. Outcome: worse than baseline. Reason: lack of autoregressive modeling across sequential time channels caused elevated errors at early time steps
Physics Informed Deep Learning Application In High-Dimensional Groundwater Inverse Modeling Of Hydraulic Tomography · Georgia Tech
Tried and failed
physics-informed Fourier neural operator with soft regularization applied to current density continuity in heterogeneous media. Outcome: did not generalise. Reason: Soft penalty loss failed to strictly enforce flux conservation and zero-divergence constraints on unseen test data
Integration of machine learning for enhanced digital rock physics workflows · UT Austin
Tried and failed
fully-connected physics-informed neural networks with single Fourier features applied to multi-scale high-frequency differential equations. Outcome: did not converge. Reason: spectral bias prevents capturing multi-scale and high-frequency dynamics
PHYSICS-INFORMED MACHINE LEARNING: THEORY, ALGORITHMS AND APPLICATIONS · Penn
Tried and failed
Min-max normalization to [-1, 1] applied to Fourier neural operator input features. Outcome: overfit. Reason: Min-max scaling led to overfitting compared to standard z-score normalization during training
Direct pore-scale modeling of foam transport in porous media and related machine learning · UT Austin
Lost to a baseline
Spectral neural network had higher energy period RMSE (0.43 s) than the median transfer function benchmark (0.37 s)
Measuring waves in difficult places: New approaches to observing waves in hurricanes and sea ice · ResearchWorks
Lost to a baseline
Basic (phase-variant) convolution model outperformed spectral convolution models across total unbanded neural variance (EVR ~0.40 vs ~0.25 at rank 10).
Tried and failed
all-layer spectral normalisation applied to deep reinforcement learning value networks. Outcome: worse than baseline. Reason: over-constraining network to be 1-Lipschitz degrades model capacity and harms performance
Left open
Investigate why higher frequency coefficients in Cartesian Fourier mask representations suffer from low gradients and cause oscillating contours. Blocker: None
Instance Segmentation and 3D Multi-Object Tracking for Autonomous Driving · Publikationssystem UB Tuebingen
Left open
Investigate why higher-frequency Fourier coefficients exhibit low gradients and oscillations when trained with Cartesian Chamfer loss in instance segmentation models. Blocker: None
Representations in Object Detection and Instance Segmentation · Publikationssystem UB Tuebingen
Spectral graph convolutions and spectral filtering degrade localized transitions by over-smoothing
Spectral graph convolutions and polynomial spectral filters rely on low-pass filtering assumptions that smooth out sharp localized transitions and cause isotropic blurring over graph topologies. Furthermore, projecting directly into the spectral domain breaks spatial locality, loses semantic information, and causes long-window dynamical models to converge to trivial fixed points.
Tried and failed
graph convolutional network on patch-induced graphs applied to spectrally diverse graph patches. Outcome: worse than baseline. Reason: fixed low-pass filtering assumptions conflicted with spectrally diverse graph patches
Towards Open World Graph Learning and Applications · Virginia Tech
Tried and failed
moving average filtering applied to power spectral density estimation. Outcome: worse than baseline. Reason: it over-smoothed spectral peaks compared to ensemble dataset averaging
Wind Tunnel Testing to Evaluate Noise Emissions from a Small Wind Turbine · Carleton University Institutional Repository
Tried and failed
Chebyshev graph convolutional neural networks applied to fault localization in power grids. Outcome: did not generalise. Reason: polynomial spectral interpolation smooths out sharp localized transitions between attacks and topology changes
Achieving Security and Reliability of Industrial Control Systems Using Data-Driven Models Informed by Physical Domain Knowledge · Georgia Tech
Lost to a baseline
LSGC+LSTM (MAE 3.16 mph) performed worse than vanilla LSTM (MAE 2.70 mph) on the LOOP dataset due to insufficient parameters in 1-layer localized spectral convolution.
Deep Learning for Short-term Network-wide Road Traffic Forecasting · ResearchWorks
Considered and rejected
Considered and rejected: Rejected spectral domain convolutions (projecting feature maps and kernels into spectral domain) because they lose semantic information
Motion and Activity Understanding in 360° Videos: An Egocentric Perspective · TXST Digital Repository
Considered and rejected
Considered and rejected: Decided against spectral graph convolutions in favor of spatial spiral convolutions to avoid isotropic blur and better capture fine surface details.
Advancing 3D hand reconstruction and modelling with geometric deep learning · Imperial
Considered and rejected
Considered and rejected: Rejected spectral graph neural networks relying on graph Laplacian eigendecomposition because eigenvectors break locality and are unstable across varying graph topologies
Tried and failed
ODE parameter inference from spectral time-series data applied to periodic dynamical systems. Outcome: did not generalise. Reason: Short windows yield transient models that diverge; long windows over-smooth and converge to trivial mean fixed points.
Robust spectral representations and model inference for biological dynamics · MIT
Eigendecompositions and unconstrained Fourier transforms create severe computational scaling bottlenecks
Full graph Laplacian eigendecompositions and direct Fourier summations over non-uniform points scale quadratically or exponentially as problem sizes grow. In dense matrix operations and trajectory or graph evaluations, spectral solvers proved significantly slower than finite-difference methods or specialized non-uniform fast Fourier libraries.
Lost to a baseline
Galerkin-collocation spectral methods were significantly slower than finite-difference methods due to dense matrix operations and inverses, without improving stability
Simulating Semiclassical Black Holes · University of Nottingham Repository
Considered and rejected
Considered and rejected: Standard spectral graph convolutions via full eigendecomposition rejected due to O(N^2) complexity, replaced by Chebyshev polynomial approximations
Network time series forecasting in photovoltaics power production · EPFL
Lost to a baseline
Standard Sage cospectral.graphs() command was too computationally slow for graphs on 9 and 10 vertices, requiring a custom two-step approximate hashing sieve.
The distance matrix and its variants for graphs and digraphs · Iowa State University Digital Repository
Considered and rejected
Considered and rejected: Rejected spectral clustering for frame clustering because runtime scaled exponentially with each additional frame, choosing k-means as a baseline instead.
A Computational Approach for Detailed Quantification of Mouse Parenting Behavior · Harvard
Considered and rejected
Considered and rejected: Rejected treating dynamic graph snapshots independently via static spectral clustering at each step because it is computationally time-consuming and lacks convergence guarantees under noise.
Statistical learning and change detection for dynamic networks · Georgia Tech
Tried and failed
direct Fourier transform of discrete point distributions applied to particle spatial spectral estimation. Outcome: too slow. Reason: direct summation of non-uniform delta functions scales poorly and cannot exploit FFT algorithms
Computational study of particle clustering and dispersion in turbulence · Imperial
Tried and failed
Fast Fourier Transform for few coefficients applied to evaluating numerical complex contour integrals. Outcome: too slow. Reason: computes all N Fourier coefficients when only two low-order coefficients are needed
An exact solution to Lambert’s problem · Iowa State
Lost to a baseline
fourier_toolkit spreading and interpolation was '2-10 times slower' than FINUFFT v2.2.0 for identical kernel widths.
Truncation, missing samples, and discretization induce spectral leakage and numerical instability
Transforming signals with truncated boundaries, missing data points, or finite time horizons causes severe spectral data leakage, aliasing, and Gibbs phenomenon artifacts. In deconvolution and integration tasks, these boundary and discretization errors accumulated over long horizons and produced numerical instability or spurious high-frequency noise.
Tried and failed
Fast Fourier transform deconvolution applied to solving Volterra integral equations. Outcome: worse than baseline. Reason: accumulated numerical errors and instability at long times compared to direct numerical quadrature
Tried and failed
standard fast Fourier transform applied to noisy time-series vibration signals. Reason: severe spectral data leakage and high stochastic noise corrupted the power spectral estimates
Structure-Borne Vehicle Interior Noise Estimation Using Accelerometer Based Intelligent Tires in Passenger Vehicles · Virginia Tech
Tried and failed
spectral domain transformation for fast inference applied to spatiotemporal state-space models with missing data. Reason: Fourier transformation breaks down when data contains missing values, preventing complexity reduction
Statistical spatio-temporal models with applications to natural processes · Georgia Tech
Lost to a baseline
Fourier-acoustics angular spectrum method produces numerical aliasing artifacts compared to direct 1D numerical integration of the exact Bessel-beam modal decomposition integral for circular pistons.
Scattering and diffraction of acoustic waves in three problems with broken symmetry · UT Austin
Tried and failed
Omitting corrupted samples prior to discrete Fourier transform applied to Fourier phase estimation from image data. Outcome: worse than baseline. Reason: Excluding bad pixels reduced measurement sensitivity compared to spatial domain interpolation
Considered and rejected
Considered and rejected: Rejected sliding discrete Fourier transform (SDFT) with partial substitution for separating coherent low-frequency modes because it produced spurious high-frequency noise and failed to converge harmonics unless sub-harmonics were used.
A fluid-solid coupled heat transfer methodology as applied to rotating cavities · Oxford
Considered and rejected
Considered and rejected: Rejected simple Fourier frequency-domain integration of Psi_4 due to spectral leakage and Gibbs phenomenon from the finite length and junk-radiation truncation of waveforms.
Dynamics of Mixed Binary Mergers using Numerical Relativity Simulations · Georgia Tech
Tried and failed
truncating spatial boundaries before Fourier transform applied to spatial dispersion extraction in small lattices. Outcome: worse than baseline. Reason: truncating limited spatial data points reduced frequency resolution more than removing boundary artifacts helped
Dynamic Behavior Of Periodic Media And Elastic Metamaterials · Penn
Spectral clustering and embedding methods struggle with partitioning constraints and cluster counts
Spectral clustering algorithms require specifying cluster counts in advance and face combinatorial explosion when searching over layer-wise cluster spaces. In embedding spaces, asymmetric perturbations cause candidate hyperplanes to overlap heavily on single clusters, while rounding methods violate distinctness constraints in multiway matching.
Lost to a baseline
Spectral truncation loses to Tikhonov regularisation when filter functions align with higher-frequency/tail eigenfunctions (shape A).
Sparsely Observed Functional Time Series: Theory and Applications · EPFL
Tried and failed
single-stage hyperplane partitioning in spectral embedding applied to graph clustering. Reason: asymmetric perturbations cause multiple candidate hyperplanes to overlap heavily on a single cluster
Considered and rejected
Considered and rejected: Rejected classical clustering methods (such as Spectral Clustering and K-means) for recall initiator extraction because they require specifying cluster numbers in advance, which is impractical for hundreds or thousands of unstructured records.
Blockchain, Recommender System, and Artificial Intelligence: Leveraging advanced digital technology for disruption risk mitigation in the medical device closed-loop supply chain · Research Repository UCD
Considered and rejected
Considered and rejected: Rejected k-means rounding on spectral graph embeddings for multiway matching because k-means violates the distinctness constraint.
Data Association Algorithms and Representations for Robust Geometric Perception · MIT
Considered and rejected
Considered and rejected: Rejected K-means++ and spectral clustering for channel pruning due to combinatorial search spaces over layer-wise cluster counts.
Robust Efficient Edge AI: New Principles and Frameworks for Empowering Artificial Intelligence on Edge Devices · Georgia Tech
Left open by the authors
Problems the authors named and did not get to.
Left open
Implement Fourier Neural Operator architectures within equation-based optimization frameworks like OMLT for NLP solver compatibility. Blocker: None
Merging First-Principles with Machine Learning for the Optimization of Process and Energy Systems · Georgia Tech
Left open
Train Fourier Neural Operators as surrogate models for physics-based gradient evaluations to accelerate multi-fiducial ASPIRE. Blocker: None
Generative Models for Uncertainty of Medical and Seismic Imaging · Georgia Tech
Left open
Integrate Fourier Neural Operator surrogates with generative models like VAEs or GANs for inverse modeling in complex non-Gaussian random fields. Blocker: None
Physics Informed Deep Learning Application In High-Dimensional Groundwater Inverse Modeling Of Hydraulic Tomography · Georgia Tech
Left open
Replace finite-difference forward solvers with Fourier Neural Operators or PINNs in physics-guided seismic inversion, and evaluate GANs or Vision Transformers as encoders. Blocker: None
Regularizing seismic inverse problems : transdimensional and machine learning based strategies · UT Austin
Left open
Integrate neural operator surrogates, such as Fourier Neural Operators, to accelerate PDE evaluations during variational transport map training. Blocker: None
Gradient-based dimension reduction for Bayesian inverse problems and simulation-based inference · MIT
Left open
Develop a multi-fidelity Fourier Neural Operator combining coarse and refined temporal discretizations to forecast longer time horizons under GPU memory constraints. Blocker: None
Direct pore-scale modeling of foam transport in porous media and related machine learning · UT Austin
Left open
Extend randomized prior ensemble uncertainty quantification to Fourier Neural Operators and attention-based operator architectures using dimension reduction techniques. Blocker: None
Deep Learning And Uncertainty Quantification: Methodologies And Applications · Penn
Left open
Implement and evaluate algebraic multigrid techniques and Kron reduction for graph coarsening in spectral graph convolutional networks. Blocker: None
Leveraging topology, geometry, and symmetries for efficient Machine Learning · EPFL
Left open
Modify the ANN loss function to penalize underestimation errors more heavily to mitigate spectral saturation at high soil moisture levels. Blocker: Requires the private multi-modal UAV and GPR field datasets paired with in-situ soil moisture measurements collected in the thesis
Left open
Prove a polynomial lower bound on lambda_2(U_{I - z_j W}) for periodic strongly connected graphs with doubly stochastic W. Blocker: Requires theoretical mathematical proof / spectral graph theory research rather than routine software engineering.
Sparsity Bounds for Spectral Approximations of Graphs · Harvard
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