Fourier Neural Operator

Also known as FNO

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4 papers in the last four weeks, against 2 the four weeks before. 0.0% of all new papers.

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Latest papers 42

Oct 7, 2026cs.LG

CAFE+FNO: Fourier Kernel Generation via Multiplicative Feature Composition

The Fourier Neural Operator (FNO) learns solution operators of partial differential equations (PDEs) through Fourier-space kernel parameterization, but frequency truncation can limit the learning of high-frequency variations. AM-FNO and SirenFNO generate kernels for all grid modes from spectral coordinates using shared networks, making coordinate encoding and generator design important. Recent work on implicit neural representations (INRs) has proposed constructing frequency interactions through explicit feature composition rather than relying on subsequent MLPs to form them implicitly. Building on this approach, we propose CAFE+FNO, which incorporates Content-Aware Frequency Encoding+ (CAFE+) into Fourier kernel generation. CAFE+ combines Fourier--Chebyshev features through parallel affine branches and a Hadamard product, forming interactions within and across the two feature families. A kernel MLP maps the resulting representation of each normalized spectral coordinate to a complex channel-mixing matrix. Each layer shares its generator across all stored modes, making the number of trainable parameters independent of the number of modes for a fixed architecture. We compare CAFE+FNO with existing FNO variants on five PDE benchmarks and conduct ablation studies on basis configuration, multiplicative composition, and bandwidth learnability. Code and experimental configurations are available at https://github.com/fabsk101/CAFEPlusFNO.git.

Hyungjoon Juen, Minwoo Shin

Department of Software, Yonsei University (Mirae Campus), Wonju 26493, Republic of Korea

Fourier Neural Operator PDE Operator Learning Fourier Feature Embeddings Compositional Representation Learning

Oct 5, 2026cs.LG

Targeted search shows that random-device testing underestimates worst-case error in a simulated wave-based neural operator

Wave-based processors promise fast, energy-efficient Fourier layers for neural operators. They are usually validated on randomly sampled devices, but using them requires knowing how large their error can become under fabrication and alignment variation. In a stylised numerical case study, a hybrid Fourier neural operator runs its four spectral layers on simulated coherent 4f processors with 32 toleranced knobs, whose half-widths are representative rather than calibrated. For 120 models (four tasks, six training methods, five seeds), we compared the worst of N random in-spec devices with a searched one. On a deterministic simulator with one frozen draw of the random static errors, the searched device's held-out error was 1.08-3.10 times the maximum over 200 Monte Carlo devices and 1.06-2.71 times that over 1000. With 20 fresh static draws, it still exceeded the maximum over 200 random devices in 116 of 120 models. Under uniform sampling, the probability of drawing such a device is at most 0.37% per model (two-sided 95% Clopper-Pearson), which says nothing about how large its error is. The gap persisted with uniform or Sobol' sampling at the search's budget, shared knobs, a second crosstalk model, box scales of 0.25-2 and a pixel-level device model. Models trained only with random static errors reached 3.7-39.9 times their nominal error on searched devices, and fine-tuning on random and gradient-searched devices gave the lowest searched error of the six in all 20 task-seed pairs. For two heat-exchanger quantities, a search targeted at each exceeded the worst of 1000 random devices in all 39 models, and hence the Wilks 95/95 limit (worst of 59). For the mean pressure of 11 models, no random device exceeded a 1% error threshold, but the searched device did. Random testing estimates how often errors exceed a threshold; worst-device search gives a lower bound on how large they can be.

Samrendra Roy, Jason Yoo, Souvik Chakraborty +1

Department of Nuclear, Plasma, and Radiological Engineering, University of Illinois Urbana-Champaign, Urbana, IL 61801, USA · Department of Applied Mechanics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India · Yardi School of Artificial Intelligence (ScAI), Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India +1

Fourier Neural Operator Uncertainty Quantification Neural Network Robustness Photonic Computing Neural Operators

Sep 29, 2026cs.AI

Transolver- $σ$ : Joint Spectral-Physical Subspace Modeling for Neural PDE Solving

Neural solvers offer efficient surrogates for numerical simulation of partial differential equations (PDEs). For time-dependent problems, strong one-step accuracy does not necessarily translate into reliable autoregressive rollout. We observe that a solver based only on physical-state modeling can achieve lower one-step error, whereas its spectral-only counterpart can become more accurate at later rollout steps. Motivated by this observation, we present Transolver-

σ

, a neural PDE solver based on joint spectral--physical subspace modeling. Within each block, adaptive physical-state interactions and spectral transformations are modeled in dedicated latent subspaces, whose responses are recomposed to enable information exchange between the two representations. Within the physical subspace, we introduce Slice-Residual Physics-Attention (SRPA), which preserves an explicit slice-space identity path while retaining learnable cross-slice interaction. In parallel, an axis-factorized Fourier operator captures global spectral structure. Across five well-established PDE benchmarks spanning steady-state prediction and time-dependent dynamics, Transolver-

σ

achieves state-of-the-art with a benchmark-averaged relative error reduction of 33.4% over the strongest baseline for each metric, while consistently improving autoregressive rollout over single-operator counterparts. Transolver-

σ

further delivers strong gains on coupled multiphysics systems and real-world fluid and combustion measurements from RealPDEBench, demonstrating its effectiveness beyond standard simulation benchmarks.

Haonan Shangguan, Hang Zhou, Haixu Wu +3

School of Software, BNRist, Tsinghua University · MIT CSAIL

Fourier Neural Operator Neural PDE Solvers PDE Surrogate Modeling PDE Operator Learning

Sep 23, 2026cs.CE

KATOsuper: Surrogate-accelerated neural topology optimization with sensitivity-consistent Fourier neural operators

Topology optimization (TO) remains computationally intensive due to repeated finite element analysis (FEA) evaluations required at each iteration. While neural network-based surrogates offer potential acceleration, existing approaches often suffer from gradient inconsistency between predicted objectives and sensitivities, leading to optimization instability. This work presents KATOsuper, an objective-agnostic framework that couples neural-reparameterized topology optimization with a Sensitivity-Consistent Fourier Neural Operator (SC-FNO). The framework employs the forward_split architecture, which derives deployed sensitivities via automatic differentiation through the predicted objective field and thereby preserves consistency between the predicted objective and the gradient used for optimization. The case studies include three 2D benchmark problems and three 3D structures considering compliance or stress minimization. A physics-informed multi-channel input encoding with Fourier position embedding enables resolution-invariant learning, supporting zero-shot extrapolation beyond the training resolution, with useful performance at moderate scaling factors and topology-preserving exploration at up to 64x without retraining. The framework extends to 3D through KATO3D, featuring novel KANConv3D blocks with learnable B-spline activations. KATOsuper demonstrates 15--110x deployment-time speedup over MATLAB baselines while maintaining competitive optimality, with the clearest gains observed in complex 3D and stress-optimization cases. The insight that sensitivity direction matters more than magnitude enables robust optimization even with approximate physics evaluation, extensible to other differentiable physics-driven design objectives.

Shengyu Yan, Jasmin Jelovica

Department of Mechanical Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada · Department of Civil Engineering, The University of British Columbia, 6250 Applied Science Ln, Vancouver, V6T 1Z4, Canada

Neural Surrogate Modeling Topology Optimization Surrogate Modeling Fourier Neural Operator Surrogate-Assisted Optimization

Sep 14, 2026cs.LG

Physics-Informed Conformal Prediction: Embedding PDE Consistency into Distribution-Free Uncertainty Quantification for Neural Operators

Neural operators such as the Fourier Neural Operator (FNO) achieve remarkable accuracy in approximating solutions to partial differential equations (PDEs). However, providing rigorous uncertainty estimates remains an open challenge. We propose Physics-Informed Conformal Prediction (PI-CP), a framework that embeds PDE residuals into the nonconformity score of split conformal prediction, producing prediction intervals that are (i) distribution-free with provable coverage guarantees, and (ii) spatially adaptive when the PDE residual correlates with prediction error -- tighter where physics is well-satisfied, wider where it is violated. Additionally, we prove that FNO's translation equivariance creates a fundamental approximation barrier for PDEs with Dirichlet boundary conditions, and show that coordinate channels resolve this with up to 63x error reduction. We validate PI-CP across six physics scenarios -- heat conduction (2D/3D), structural mechanics (2D/3D), Darcy flow, and Navier-Stokes -- demonstrating consistent 89-91% coverage for all four Conformal methods, while MC Dropout and Deep Ensembles are unstable (82-100%). FNO outperforms CNN and DeepONet by 10-12x.

Michael Chin

Independent Researcher

Fourier Neural Operator Uncertainty Quantification Conformal Prediction PDE Operator Learning Neural Operators

Sep 8, 2026cs.LG

Learning Metamaterial Eigenmodes with Wavelet-Encoded Fourier Neural Operators

Machine learning surrogates based on neural operators have shown broad applicability in solving forward PDE problems. However, eigenvalue problems, in which an eigenparameter and one of several valid eigenmodes must be simultaneously solved, remain difficult because standard operator learning formulations assume a unique input-output map. This work demonstrates that Fourier Neural Operators (FNOs), combined with wavelet-based encodings of PDE inputs, can learn and predict multiple eigenmodes of the elastic wave equation, corresponding to deformation modes of acoustic waves propagating through arbitrary metamaterial geometries. We provide a mechanistic explanation and experimental evidence for why wavelet encodings are well matched to the dual spatial-spectral structure of the FNO, enabling deterministic mode selection on both continuous-valued and binary-valued geometries within a single model, and for why prediction accuracy varies with geometric discontinuities. For metamaterial design, the resulting surrogate accelerates the simulation stage of the design cycle by three orders of magnitude relative to finite element analysis on a consumer-grade CPU, while preserving high fidelity. These results also carry broader implications for designing input encodings in other multi-mode PDE solvers based on spectral neural operators.

Han Zhang, Alexander Ogren, Cynthia Rudin +2

Department of Mechanical Engineering and Materials Science, Duke University, Durham, NC, USA · Department of Mechanical Engineering, California Institute of Technology, Pasadena, CA, USA · Department of Electrical and Computer Engineering, Duke University, Durham, NC, USA

Fourier Neural Operator PDE Surrogate Modeling

Aug 30, 2026cs.LG

Selection, Representation, and Execution in Sparse Fourier Neural Operators

Sparse representations are often expected to make models smaller and also reduce inference cost. For Fourier Neural Operators (FNOs), these objectives are not equivalent or do not always align: removing parts of the learned operator can leave the underlying transforms and dense computations unchanged, while changing the grid on which the model is evaluated can introduce overhead of its own. We therefore distinguish sparsity in the representation, in the stored parameters, in the theoretical operation count, and in measured runtime, and present an empirical study of several routes toward sparse FNOs that tests each transition between them separately. Coarsening the execution grid reduces the theoretical cost without reducing measured latency, and adding a correction term recovers accuracy at the cost of making the model slower. Even an 83% parameter reduction remains slower than the dense baseline under ordinary execution. These results motivate a stricter definition of useful sparsity: the deployed operator must preserve solution accuracy and map its reduced support to a genuinely cheaper execution path.

Abdul Qadir Ibrahim, Martin Burger

Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, 22607 Hamburg, Germany

Efficient Neural Network Inference Fourier Neural Operator Sparse Neural Networks Neural Network Compression

Aug 11, 2026cs.CV

Lesion-Aware Adaptive Fourier Neural Operator for CT-to-PSMA PET Synthesis in Prostate Cancer

Deep learning models that synthesize PET from CT or MRI can reduce patient dose and scanner demand, but are typically optimized with global losses such as L1 or mean squared error (MSE) that treat all voxels similarly. In whole-body PSMA-PET, tumor voxels occupy only a small fraction of the volume, yet carry the clinically relevant activity signal; as a result, models can achieve high structural similarity index measure (SSIM) and peak signal-to-noise ratio (PSNR) while still underestimating lesion activity or failing to preserve tumor-specific structure. Radiomics provides biologically meaningful descriptors of tumor intensity and texture, but direct radiomics conditioning is time-consuming because it requires feature extraction from delineated lesion regions. We propose LAFNO, a Lesion-Aware Adaptive Fourier Neural Operator for CT-to-PSMA-PET synthesis that replaces high-dimensional radiomics conditioning with two efficient CT-derived proxy channels. Motivated by radiomics analysis of PSMA-avid tumor core and peritumoral regions, LAFNO uses a contrast proxy for local density variation and a disorder proxy for local texture heterogeneity, both injected into the model bottleneck. LAFNO combines whole-volume reconstruction with lesion-level total lesion activity (TLA), tumor-core contrast, and peritumoral supervision. We evaluated LAFNO against four baseline architectures on the TCIA PSMA-PET-CT-Lesions dataset. LAFNO remained competitive on whole-volume image quality, achieving SSIM of 0.960 and 0.938 for 18F- and 68Ga-PSMA, respectively, while reducing per-patient TLA error to 48.3% and 64.0% for 18F- and 68Ga-PSMA, respectively, and achieving the highest tumor-core radiomics reproducibility across all feature classes for both tracers. Peritumoral reproducibility remained tracer-dependent, indicating that biological fidelity in synthetic PSMA-PET remains challenging.

Rashmi Bhaskara, Waleed M. Almutairi, Matthew Gopaulchan +5

aSchool of Health Sciences, Purdue University, West Lafayette, 47907, IN, USA · bDepartment of Radiology, Indiana University School of Medicine, Indianapolis, 46202, IN, USA · cDepartment of Urology, Indiana University School of Medicine, Indianapolis, 46202, IN, USA +1

Fourier Neural Operator PET/CT Imaging Medical Image Synthesis

Aug 9, 2026math.NA

ADEx-FNO: A Unified Ambient-Domain Framework for Fourier Neural Operators on Varying Geometries

Fourier neural operators (FNOs) provide efficient nonlocal spectral learning, but varying geometries and independently chosen discretizations remain difficult to accommodate. We introduce the ambient-domain extension Fourier neural operator (ADEx-FNO), a deterministic framework that incorporates geometry without modifying the defining Fourier-operator layers. Each physical domain is embedded in a fixed ambient hypercube and represented by a signed distance function. Inputs and solution fields are deterministically extended to the ambient domain, transferred to a common, potentially nonuniform rectilinear latent grid, processed by the FNO, then interpolated to an independently chosen target discretization and restricted to the physical domain. All geometry-transfer operations lie outside the optimization procedure and require no trainable graph, point-cloud, deformation, or geometry-decoding modules. ADEx-FNO achieves relative l2 errors of 0.32%-0.77% on held-out smooth-domain nonlinear Poisson and advection-reaction-diffusion problems in 2D and 3D, and is also evaluated on unseen nonsmooth geometries. A single ADEx-FNO inference is then used to initialize conventional CFD solvers. For all 29 converged 2D and 3D RANS cases, pseudo-time iterations decrease, with mean reductions of 44.17% and 43.03%, respectively, with comparable gains across three mesh resolutions. URANS cases reduce post-window physical-time advances by 18.52%-27.51%. In transfer from 2D URANS training data to DNS at different Mach and Reynolds numbers, the bootstrap interval decreases by 23.47%-48.21%, depending on the target statistic. In all CFD tests, ADEx-FNO provides only the initial field; the governing-equation solver controls the subsequent solution, while physical or statistical consistency is assessed separately from computational savings.

Roberto Nuca, Giovanni Testa, Luca Galimberti +1

Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia · Dipartimento di Scienze e Tecnologie Aerospaziali, Politecnico di Milano, Milan, Italy · Physical Science and Engineering Division, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

Fourier Neural Operator PDE Surrogate Modeling PDE Operator Learning Neural Operators

Jul 28, 2026physics.optics

Physics-Informed Neural Operator for Warm-Starting Background-Decomposed and Preconditioned PSFD: Enabling Scalable 3-D EUV Mask Simulation

We present a physics-informed neural operator (PINO) trained with pseudo-spectral frequency-domain (PSFD) equations for electromagnetic (EM) scattering problems in EUV lithography. The Fourier neural operator is factorized into a two-dimensional lateral (

xy

) branch and a one-dimensional axial (

z

) branch and is trained self-consistently with background decomposition.Thus, the full-vector coupling between the mask and the multilayer response is retained without invoking a finite-order Born approximation. In this way, the computational domain size is significantly reduced, thereby lowering the computational cost. The PINO is trained on approximately 16,000 mask designs from the LithoBench library sampled randomly at each training iteration without using precomputed EM field solutions. The PINO surrogate model yields predictions with a mean absolute error of about

7 \times 10^{-3}

for the scattered intensity of held-out mask patterns relative to the reference PSFD solution. Combined with spectral damping, the PINO warm-start initialization accelerates the background-decomposed PSFD solver on finer discretizations.

Doyun Kim, Werner Gillijns

IMEC, Kapeldreef 75, 3001 Leuven, Belgium

Fourier Neural Operator Semiconductor Manufacturing Physics-Informed Neural Operators Physics-Based Simulation

Jul 26, 2026cs.LG

Extending Fourier Neural Operators for Modeling Parameterized and Coupled PDEs

Parameterized and coupled partial differential equations (PDEs) are central to modeling phenomena in science and engineering, yet neural operator methods that address both aspects remain limited. We extend Fourier neural operators (FNOs) with minimal architectural modifications along two directions. For parameterized dynamics, we propose a hypernetwork-based modulation that conditions the operator on physical parameters. For coupled systems, we conduct a systematic exploration of architectural choices, examining how operator components can be adapted to balance shared structure with cross-variable interactions while retaining the efficiency of standard FNOs. Evaluations on benchmark PDEs, including the one-dimensional capacitively coupled plasma equations and the Gray-Scott system, show that our methods achieve up to 55-72% lower errors than strong baselines, demonstrating the effectiveness of principled modulation and systematic design exploration.

Cheng Jing, Uvini Balasuriya Mudiyanselage, Abhishek Verma +3

Arizona State University · Applied Materials Inc.

Fourier Neural Operator PDE Solving PDE Operator Learning

Jul 2, 2026cs.LG

Fourier Neural Operators for Rayleigh-Bénard Convection

We propose an improved Fourier Neural Operator (FNO) for modeling two-dimensional Rayleigh-Bénard convection by predicting time increments instead of full solutions, achieving higher accuracy than a standard FNO baseline. The resulting model is compact (314k parameters, 1.26 MB) and fast (7 ms inference), while maintaining similar accuracy as demonstrated in previous benchmarks. We show that although FNOs generalize to finer meshes, accuracy remains limited by the resolution of the training data.

Chelsea Maria John, Thibaut Lunet, Sebastian Götschel +3

Jülich Supercomputing Centre, Jülich, Germany · Hamburg University of Technology, Hamburg, Germany

Fourier Neural Operator PDE Surrogate Modeling Computational Fluid Dynamics

Jul 1, 2026stat.ML

From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operators

We establish approximation and learning guarantees for Fourier neural operators (FNOs) applied to time-

T

solution operators of dissipative evolution equations. The analysis builds on the premise that FNOs can efficiently approximate and learn solution operators whenever these operators admit stable and accurate spectral discretizations. To formalize this idea, we introduce classes of evolution operators defined through spectral methods and derive FNO approximation bounds and polynomial sample complexity guarantees for these classes. For equations with polynomial nonlinearities, the learning rates depend primarily on the smoothness of the input space and the dimension of the physical domain. Our results hold uniformly over broad families of dissipative equations, rather than for a single fixed PDE, and apply in particular to the Navier--Stokes, Allen--Cahn, and Cahn--Hilliard equations. For equations with non-polynomial smooth nonlinearities, we prove that polynomial sample complexity still holds with rates that now additionally depend on the smoothness of the nonlinear terms and the dissipation strength. Overall, we connect classical spectral approximation theory with modern operator learning and explain when FNOs can learn nonlinear evolution operators efficiently.

Nisha Chandramoorthy, Daniel Sanz-Alonso, Nathan Waniorek

University of Chicago

Dynamical Systems Fourier Neural Operator PDE Operator Learning Neural Network Approximation Theory Neural Operators

Jun 28, 2026math.NA

Fourier Neural Operators with Least-Squares Readout Refit for Learning Random Obstacle-to-Solution Maps

We study operator learning for random obstacle-to-solution maps arising from elliptic variational inequalities with finite-band self-affine random obstacle fields. Instead of introducing an explicit truncated stochastic parametrization of the random input, we learn the map directly from sampled obstacle realizations on a fixed grid. This problem is challenging because the solution is governed not only by the obstacle field itself, but also by the induced contact set and free-boundary geometry. We introduce a post-training least-squares readout refit for the Fourier neural operator (FNO). After the FNO is trained end to end, its nonlinear backbone is frozen and the final affine readout is recomputed by solving the induced linear least-squares problem over all training samples and grid points. The refit yields the empirical squared-error optimal readout for the learned frozen features while leaving the nonlinear representation unchanged. We compare vanilla DeepONet, POD-DeepONet, a two-stage DeepONet baseline, FNO, and FNO with least-squares readout refit (FNO-LS) on two obstacle ensembles with different amplitude levels. Numerical results show that FNO-LS achieves the strongest overall performance among the tested models, particularly for higher-amplitude obstacles with more complex contact geometry. The method improves average field accuracy, contact-set recovery, and obstacle-violation metrics at low additional cost, especially when the FNO backbone is informative but not fully converged. These results suggest that least-squares readout refit is a simple and effective post-training enhancement for learning random obstacle-to-solution maps.

Chenhui Zhu, Fei Wang

School of Mathematics and Statistics, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, P. R. China. · School of Mathematics and Statistics & State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, China.

Fourier Neural Operator PDE Solving PDE Operator Learning

Jun 27, 2026cs.CV

BREIT: A Framework for Brain Stroke Reconstruction using Multi-Frequency 3D EIT

Multi-Frequency Electrical Impedance Tomography (MF-EIT) is a non-invasive, low-cost modality that reconstructs electrical property distributions from boundary voltages. For stroke imaging, progress in 3D deep-learning reconstruction is limited by the lack of large-scale datasets with paired ground-truth (GT) volumes and by non-standardized pipelines for data generation, simulation, and evaluation. We introduce BREIT, a modular framework for 3D MF-EIT stroke reconstruction providing: (i) a neuroimaging-to-EIT pipeline that converts CT/MRI into frequency-dependent GT admittivity volumes; (ii) a self-contained Python 3D Complete Electrode Model (CEM) forward solver for simulating MF-EIT voltages; and (iii) a 3D D-bar implementation supporting non-uniform electrode layouts. Building on BREIT, we propose dFNO-bar, which integrates Fourier Neural Operators into D-bar by learning a mapping from scattering data

t(ξ)

to conductivity

σ(x){=}\Re\{γ\}

. We evaluate dFNO-bar against D-bar, Deep D-bar, and Gauss--Newton reconstructions on UCLH-matched synthetic data, and observe higher brain SSIM with comparable CC across noise settings.

Djahid Abdelmoumene, Ishak Ayad, Maï K. Nguyen +1

ETIS (UMR 8051), CY Cergy Paris University, ENSEA, CNRS, Cergy France · AGM (UMR 8088), CY Cergy Paris University, CNRS, Cergy, France · Paris-Saclay University, CentraleSupélec, Inria, CVN, Gif-sur-Yvette, France

Medical Imaging Fourier Neural Operator Image Reconstruction Neuroimaging

Jun 26, 2026cs.CE

Higher-Order Fourier Neural Operator: Explicit Mode Mixer for Nonlinear PDEs

Neural operators provide deep neural networks for learning mappings between function spaces. Among them, the Fourier Neural Operator (FNO) is particularly effective: its spectral convolution relies on low-dimensional Fourier-domain representations and can handle inputs at different resolutions. This design aligns well with settings where the Fourier basis diagonalizes the underlying operator, such as linear, constant-coefficient PDEs on periodic domains, in which Fourier modes evolve independently. However, nonlinear PDEs may benefit from an additional inductive bias, as they exhibit structured interactions between modes, governed by polynomial nonlinearities. To capture this inductive bias, we introduce the Higher-Order Spectral Convolution, a spectral mixer that extends FNO from diagonal modulation to explicit n-linear mode mixing, aligned with the dynamics of nonlinear PDEs. Our experiments on standard benchmarks show that the proposed Higher-Order FNO (HO-FNO) retains the efficiency of FNO-based architectures and consistently improves over other spectral neural operators. HO-FNO also performs on par with or better than state-of-the-art transformers and state-space models on several datasets, with stronger gains in highly nonlinear regimes, such as the Poisson equation with polynomial forcing, where a single HO-FNO layer outperforms FNO models with up to 16 layers. We open-source our code for reproducibility at: https://github.com/AlexColagrande/HO-FNO.

Alex Colagrande, Paul Caillon, Eva Feillet +1

Miles Team, LAMSADE, Université Paris Dauphine-PSL, Paris, France · Université Paris-Saclay, CNRS, LISN, 91400, Orsay, France · ESPCI PSL, Paris, France

Fourier Neural Operator Neural PDE Solvers PDE Operator Learning

Jun 25, 2026cs.LG

Operator Learning for Cubic Nonlinear Schrödinger Equation on Periodic Domains

We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter

ω^2

. The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger

H^2

-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including

ω^2

improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.

Emmanuel E. Oguadimma, Victory C. Obieke, Xueying Yu

Fourier Neural Operator PDE Solving PDE Operator Learning

Jun 23, 2026cs.LG

Real vs. Complex Spectral Bases for Neural Operators: The Role of Green's Function Alignment

Fourier Neural Operators (FNO) learn solution operators of partial differential equations by parameterizing global convolutions in the complex Fourier domain. For real-valued PDE solutions, the complex FFT carries representational redundancy through conjugate symmetry. We introduce the Hartley Neural Operator (HNO), the exact real-valued mirror of FNO: it replaces the FFT with the purely real Discrete Hartley Transform and learns a single real multiplier per retained spectral mode, with no complex arithmetic. Because the real Hartley spectrum is not halved by conjugate symmetry, HNO retains twice as many frequency corners as FNO but one real weight where FNO carries a complex pair, so the two operators are iso-parametric at equal width and differ only in spectral basis. Our central thesis is that the best basis is a property of the operator. Self-adjoint elliptic operators (Poisson, biharmonic) have real, symmetric Green's functions that the real Hartley multiplier diagonalizes exactly, and HNO is favored there. Time-dependent operators carry phase, from oscillation in the wave equation to transport in advection, Burgers, and Navier-Stokes, which a real diagonal multiplier cannot represent, so FNO is favored there, and increasingly so with the operator's phase content, leaving the phaseless heat equation as the borderline case. Training both operators identically and benchmarking across PDE classes, initial-condition families, and boundary conditions, we find an elliptic-versus-time-dependent split that is monotone in operator phase content and matches the Green's-function theory we develop. Rather than a universal winner, our findings give a predictive rule: match the spectral basis to the symmetry of the solution operator.

Jason Sulskis, Sathya Ravi

Electronic Systems Laboratory, Applied Embedded Systems Division Georgia Tech Research Institute · Department of Computer Science University of Illinois at Chicago

Fourier Neural Operator PDE Operator Learning Spectral Methods Neural Operators

Jun 11, 2026cs.LG

How Much Memory Do We Need? Adaptive Memory Gate for Neural Operators

Neural operators have emerged as a powerful data-driven approach for solving time-dependent PDEs. Among recent advances, memory-augmented neural operators explicitly incorporate past states and have achieved remarkable performance under low-resolution observation settings. However, existing approaches apply a fixed memory weight regardless of observation conditions, such as resolution or physical parameters, limiting their adaptability. Our preliminary experiments reveal that optimal memory weight varies with resolution and viscosity, implying that a fixed memory weight cannot simultaneously optimize performance across diverse settings. We propose AMGFNO, which dynamically modulates memory weight through a learnable gate. On the Kuramoto-Sivashinsky and Burgers' equations, AMGFNO achieves 55-79% nRMSE reduction over at low resolution, with the learned gate value automatically decreasing from

\bar{g} \approx 0.7

to near-zero as resolution increases.

Jihyeon Hur, Yongseok Kwon, Min-Gi Jo +2

1KAIST, Republic of Korea.

Memory-Augmented Neural Networks Fourier Neural Operator PDE Operator Learning Neural Operators

Jun 9, 2026cs.LG

SirenFNO: Efficient and Full Frequency Learning of Fourier Neural Operators

Fourier neural operators (FNOs) are effective and efficient surrogates for approximating solutions of PDEs and generalize across discretizations. However, owing to the reliance on frequency truncation to maintain learning efficiency of FNOs, empirical studies suggest that FNOs exhibit spectral bias toward low-frequency information, which may hinder the learning capability especially for certain PDEs with strong high-frequency oscillations. To address this limitation, we propose SirenFNO, a novel framework that leverages sinusoidal representation networks (SIRENs) to learn implicit neural representations and performs mode-wise kernel parameterization. Our SIREN parameterization learns a full-grid spectrum with a constant and discretization-independent parameter count, thereby eliminating the need for frequency truncation. We further extend SirenFNO with functional tensor decompositions to enhance parameter and learning efficiency. Empirical results show that our SirenFNO consistently outperforms FNO with approximately

4

to

15

times parameter reductions with preserved discretization invariance, and our functional decomposition variants obtain performance improvements with a maximum of

73

times fewer parameters across multiple PDE benchmarks.

Pengqing Shi, Jie Yin, Stephen Tierney +1

The University of Sydney, Australia

Fourier Neural Operator Spectral Bias PDE Operator Learning

Jun 7, 2026math.NA

Fourier Neural Operators with rank-1 lattice points and hyperbolic cross

The \emph{Fourier neural operator} (FNO) is a neural network architecture that learns mappings between function spaces. Its efficient implementation is based on the multi-dimensional Fourier transform. By deriving general regularity bounds for the FNO with respect to both the spatial and parametric variables, we prove that the generalization error of the FNO can be improved by replacing spatial tensor product grids with purpose-built rank-1 lattice points, and by using a second lattice carefully constructed as training points in the parametric space. We achieve more accurate and efficient approximations from fewer network parameters, fewer spatial points, and fewer training samples. In addition, the architecture is simplified, because the high-dimensional Fourier transform on rank-1 lattices requires only a \emph{one-dimensional fast Fourier transform}, and we can use a \emph{hyperbolic cross} frequency index set with lattice points. We demonstrate the benefits of our \emph{lattice-based hyperbolic-cross FNOs} for an elliptic PDE on the torus.

Jakob Dilen, Alexander Keller, Frances Y. Kuo +1

Neural Network Generalization Fourier Neural Operator PDE Operator Learning

Jun 6, 2026cs.LG

GENERIC-FNO: Embedding Energy Conservation and Entropy Production into Fourier Neural Operators

We introduce GENERIC-FNO, the first neural operator to embed the full GENERIC (metriplectic) structure of nonequilibrium thermodynamics -- reversible, energy-conserving dynamics and irreversible, entropy-producing dynamics coupled through the degeneracy conditions -- directly in function space. Existing structure-preserving neural operators enforce at most a single conservation law or reversible (Hamiltonian) structure, while thermodynamically consistent learning has been confined to finite-dimensional, graph, or particle systems. GENERIC-FNO closes this gap: it learns the energy and entropy functionals as neural operators and parameterizes the Poisson and friction operators as diagonal Fourier multipliers sandwiched between rank-one projections that enforce the degeneracy conditions exactly, by construction, with no penalty term, update projection, or residual. The degeneracy identities hold to machine precision (residuals ~10^-13) for any initialization, dimension, or resolution, so the continuous-time dynamics conserve the learned energy and produce entropy exactly; the explicit time stepping adds only a small O(dt^2) drift (per-step residual ~10^-6). We further note that the (E,S,L,M) decomposition of a given flow is not unique, and introduce a gauge-invariant dissipation diagnostic separating reversible from dissipative dynamics independently of the learned functionals. Across three operator backbones (1D/2D FNOs and DeepONet) and four PDEs spanning reversible, dissipative, and mixed regimes, GENERIC-FNO preserves its exact structural guarantees zero-shot across a 4x super-resolution range (64 to 256), recovers the ground-truth ordering of physical dissipation, and is competitive with strong unconstrained and energy-penalized baselines, outperforming them on several dissipative and mixed problems at comparable or fewer parameters.

Jason Sulskis, Sathya Ravi

Department of Computer Science University of Illinois at Chicago · Electronic Systems Laboratory, Applied Embedded Systems Division Georgia Tech Research Institute

Fourier Neural Operator Physics-Informed Neural Operators Neural Operators

Jun 2, 2026cs.LG

EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs

Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems. While equivariant networks offer a solution, they typically rely on local operations in the spatial domain, making the global receptive field, which is essential for PDE dynamics, computationally expensive. Conversely, Fourier Neural Operators (FNOs) efficiently capture global interactions, yet establishing 3D equivariance within them remains impractical due to the prohibitive cost of spectral group convolutions. To bridge this gap, we introduce EqGINO, a geometrically robust framework that enforces isotropy in the spectral domain. By design, EqGINO guarantees exact equivariance to the discrete symmetries inherent to the discretized computational domain. Beyond this discrete guarantee, our structural prior enables effective generalization to arbitrary continuous orientations even with a limited number of SE(3)-transformed training samples. Consequently, our method robustly models coordinate-invariant physical laws on complex irregular 3D geometries. Our code is available at https://github.com/sung-won-kim/EqGINO

Sungwon Kim, Juho Song, Seungmin Shin +3

Graduate School of Data Science, KAIST, Daejeon, Republic of Korea · Mathematical Sciences, KAIST, Daejeon, Republic of Korea · LG Electronics, Pyeong-taek, Republic of Korea +1

Fourier Neural Operator PDE Surrogate Modeling PDE Operator Learning Neural Operators

May 30, 2026cs.LG

Limits of Resolution Equivariance in Fourier Neural Operators

Fourier Neural Operators are often assumed to generalize across spatial resolutions, enabling training on a coarse grid and deployment on a finer grid. We test this assumption by contrasting two inference-time choices when moving from training resolution

s

to test resolution

S>s

: running FNO directly at

S

, or running at

s

and upsampling the prediction to

S

via Fourier zero-padding. On Darcy flow, we observe that direct fine-grid inference is not reliably beneficial and can be worse than the low-grid-plus-upsampling baseline. We further analyze layerwise spectra and find that, under Fourier truncation, intermediate representations increasingly concentrate energy in low frequencies, with high-frequency output produced mainly by late nonlinear/decoder stages. This offers a mechanistic explanation for why FNO can perform well while retaining few modes, yet remain sensitive under resolution shifts. Our findings highlight a simple but strong baseline for cross-resolution evaluation and point to nonlinear aliasing as a key obstacle to zero-shot resolution equivariance.

Alex Colagrande, Paul Caillon, Eva Feillet +1

Miles Team, LAMSADE, Universit´e Paris Dauphine-PSL, Paris, France · Universit´e Paris-Saclay, CNRS, LISN, France · ESPCI PSL, Paris, France

Fourier Neural Operator Equivariant Neural Networks Neural Operators

May 27, 2026cs.LG

Sequential Physics-Constrained Neural Operator Forward Modeling for the $\textit{Norne}$ Reservoir System

We develop a comprehensive mathematical and computational framework for sequential surrogate modeling of three-phase black-oil reservoir dynamics using neural operators, with particular emphasis on Fourier Neural Operators (FNO) and their physics-informed variant (PINO). The application focus is the Norne benchmark reservoir, defined on a heterogeneous

46\times112\times22

grid (

N=113,344

cells), with a production history spanning

T=30

timesteps covering 3298 days. Our theoretical contributions are organized around four interlocking problems: (1) functional-analytic formulation in a product-Sobolev-space setting, including well-posedness of the implicit timestep map and sharp local Lipschitz estimates; (2) covariate shift quantification, proving that the Wasserstein-2 distance grows as

W_2 \leq \varepsilon(L^n-1)/(L-1)

, with exponential population-risk discrepancy for

L>1

; (3) physics-constrained spectral stability, showing PINO training with

λ_R \geq λ^*_R

reduces the learned Jacobian spectral radius to

ρ_F + Cλ_R^{-1/2}

, yielding uniform-in-time rollout error

|δ_n| \leq \varepsilon/(1-ρ)

; and (4)

K

-step TBPTT gradient analysis, deriving geometric bias decay

O(ρ^K)

, optimal window

K^ = O(\log(T/σ^2))

, and Adam convergence

O(1/\sqrt{t}) + O(ρ^{K^*})

. Empirical validation confirms all theoretical predictions: autoregressive PINO surrogates sustain

R^2>0.99

(oil),

R^2>0.90

(gas),

R^2\approx 0.80

(pressure), and monotonically improving

R^2

(water) across the full 3298-day horizon, trained on eight NVIDIA B200 GPUs in under one hour. A 1000-member ensemble runs in under one minute on a single B200 GPU, giving a

{\sim}10^4\times

wall-clock speedup over the OPM finite-volume simulator.

Clement Etienam, Juntao Yang, Oleg Ovcharenko +4

NVIDIA Corporation

Fourier Neural Operator PDE Surrogate Modeling Physics-Informed Neural Operators Computational Fluid Dynamics

May 26, 2026cs.LG

MTL-FNO: A Lightweight Multi-Task Fourier Neural Operator for Sparse Field Reconstruction

Efficient onboard multi-field sparse reconstruction is essential for the autonomous operation of aerospace vehicles. While existing deep learning models exhibit promise for single-field reconstruction, deploying multiple independent models leads to prohibitive model size growth and fails to exploit cross-field correlations, particularly under few-shot conditions. To address these challenges, we first propose a lightweight multi-task Fourier neural operator (MTL-FNO), an end-to-end joint training framework based on hard parameter sharing. In each layer, the parameters are divided into shared and task-specific components to capture common features across fields while preserving task-specific characteristics. Moreover, the task-specific fine-tuning parameters are implemented as low-rank terms, achieving substantial model compression. Second, to address the difficulty of co-optimizing shared and task-specific parameters along with their real and imaginary parts, we revisit the FNO's spectral weight from a polar-form perspective and devise a physically meaningful decoupled optimization scheme. Specifically, we apply polar decomposition to slice-wise disentangle the spectral weight into a unitary tensor encoding phase information and a positive semi-definite tensor characterizing amplitude. By decoupling the optimization of phase and amplitude, our method can effectively mitigate tasks conflict. Meanwhile, to preserve unitary geometric fidelity during training, the Cayley transform is introduced to reparameterize the unitary tensor, converting the constrained optimization problem to an unconstrained one. Finally, the effectiveness of the proposed method under few-shot conditions is validated on two representative engineering cases. Results show that MTL-FNO achieves accuracy comparable to or even surpassing that of standard FNO, while reducing total model size by 76% and 60%, respectively.

Siyu Ye, Shihang Li, Zhiqiang Gong +4

Defense Innovation Institute, Academy of Military Science, Beijing, 100071, China · Intelligent Game and Decision Laboratory, Beijing, 100071, China

Fourier Neural Operator Multi-Task Learning Neural Network Compression

May 18, 2026cs.RO

Neural Operators for Design-Space Surrogate Modeling of Tendon-Actuated Continuum Robots

Continuum robots enable dexterous manipulation in constrained environments, but require accurate and efficient models for real-time manipulation and control. Traditional physics-based models can be computationally expensive and may suffer from inaccuracies due to unmodeled effects, while current learning-based methods often generalize poorly beyond the specific robot on which they are trained. We present a formulation of surrogate modeling for tendon-driven continuum robots as an operator learning problem that maps robot design parameters and tendon actuation inputs to resulting configurations. This formulation enables a single trained model to generalize across a large class of robot designs. We develop four novel neural operator architectures--two based on Deep Operator Networks (DeepONets) and two based on Fourier Neural Operators (FNOs)--and train them on simulation data to predict robot configurations. All architectures achieve good accuracy while allowing for fast and accurate generalization across designs. Our results demonstrate that operator learning provides an effective and generalizable surrogate for continuum robot mechanics in the design space, enabling fast modeling for control, planning, and design optimization in surgical and industrial applications.

Branden Frieden, James M. Ferguson, Alan Kuntz +1

The Robotics Center and the Kahlert School of Computing at the University of Utah, Salt Lake City, UT 84112, USA. · The Departments of Computer Science and Electrical and Computer Engineering at Vanderbilt University, Nashville, TN, 37203, USA.

Neural Surrogate Modeling Fourier Neural Operator Continuum Robot Control Tendon-Driven Actuation Deep Operator Networks Neural Operators

May 18, 2026cs.LG

PACE-FNO: Physics-Aligned Canonical Equivariance for Fourier Neural Operators

Neural operators are often tested on states that differ physically from training data. A distinct failure occurs when the physical dynamics are unchanged but the observed coordinate frame differs from training. PACE-FNO addresses this case by estimating the frame, predicting after pulling the field to a canonical representative, and restoring the requested terminal frame. The default inference path uses one forward prediction; optional test-time adaptation (TTA) updates only the low-dimensional coordinate. On translated and Galilean-shifted Burgers and shallow-water systems, PACE-FNO lowers out-of-distribution (OOD) relative error by up to

12\times

relative to a data-augmented Fourier Neural Operator (FNO+Aug). A matched-estimator control supports attributing the gain to prediction in the canonical frame rather than added estimator capacity. Our analysis separates the canonical approximation error from the two alignment residuals. We also test regional WeatherBench2 time-series forecasting with the fifth-generation ECMWF reanalysis (ERA5) under natural time OOD, where PACE-FNO and FNO achieve nearly identical performance. Experiments with approximate rotation, other backbones, irregular domains, rollouts, and image data delineate the conditions under which the mechanism is effective.

Jiaxiao Xu, Changhong Mou, Yeyu Zhang +1

Shanghai University of Finance and Economics · Utah State University · University of Edinburgh

Fourier Neural Operator OOD Generalization PDE Operator Learning Neural Operators

May 17, 2026cs.LG

Stability and Discretization Error of State Space Model Neural Operators

Neural operators have emerged as a powerful, discretization-invariant framework for solving partial differential equations (PDEs). Although established approaches like the Deep Operator Network (DeepONet) have successfully achieved universal approximation for operators, and architectures such as Fourier Neural Operators (FNOs) have shown algebraic convergence rates, a precise theoretical connection between the continuous theory and its discrete numerical implementation remains a challenge. Specifically, the relationship between the continuous formulation and the discrete numerical stability has yet to be fully explored. In this paper, we address this gap by establishing theoretical guarantees for the discretization error and stability of neural operator approximation schemes. We prove analytical bounds that link solution regularity to input discretization, providing a formal quantification of neural operator accuracy under real-world numerical constraints. We derive these bounds to the specific cases of State Space Model-based Neural Operators (SS-NOs) and FNOs, thus providing a new discretization error theorem for these models. Additionally, through an input-to-state stability (ISS) analysis, we formally assess the impact of discretization on the stability of SS-NOs results obtained in the continuous domain. Our empirical experiments on 1D and 2D benchmarks validate our theoretical bounds and show the robustness of SS-NOs under varying resolutions.

Abderrahim Bendahi, Adrien Fradin, Johan Peralez +2

École polytechnique Paris, France · Université Claude Bernard Lyon 1, CNRS, LAGEPP UMR 5007 Villeurbanne, France · CNRS, UniversitéJulieLyonDigne1, INSA Lyon, LIRIS Lyon, France +1

Fourier Neural Operator Convergence Analysis Neural Operators

May 13, 2026cs.LG

Frequency Bias and OOD Generalization in Neural Operators under a Variable-Coefficient Wave Equation

Neural operators learn to map initial conditions to the terminal solution of partial differential equations (PDEs), providing a surrogate for the full operator mapping. This enables rapid prediction across different input configurations. While recent neural operator architectures have demonstrated strong performance on diverse PDE tasks, their behavior under structured distribution shifts remains insufficiently understood. To investigate this, we study operator learning in a wave propagation setting governed by a one-dimensional variable-coefficient wave equation, using two representative architectures, the Fourier Neural Operator (FNO) and the Deep Operator Network (DeepONet). To examine their generalization under distribution shifts, we consider structured out-of-distribution (OOD) settings that independently vary input frequency and coefficient smoothness. The results show that under smoothness shifts, both models maintain stable performance, with FNO achieving lower error. In contrast, under frequency shifts, FNO exhibits a sharp increase in error under unseen high-frequency inputs, whereas DeepONet shows milder degradation despite higher overall error. Our analysis reveals that these differences arise from how each architecture represents and responds to variations in frequency structure. Together, these findings highlight a fundamental gap between strong in-distribution performance and generalization under distribution shifts in operator learning, underscoring the role of architectural representation bias in developing more reliable neural operators for physics-based PDE simulations beyond the training distribution.

Runlong Xie, An Luo

Independent Researcher · School of Statistics, University of Minnesota, MN, USA.

Fourier Neural Operator PDE Solving OOD Generalization Spectral Bias PDE Operator Learning Deep Operator Networks