A systematic approach maps any-dimensional invariant functions to a unique function on an infinite-dimensional limit space admitting a topology with compact sets where universality holds, with examples of non-universal architectures and fixes.
super hub Mixed citations
Fourier Neural Operator for Parametric Partial Differential Equations
Mixed citation behavior. Most common role is background (63%).
221
Pith papers citing it
Background
63% of classified citations
abstract
The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fourier space, allowing for an expressive and efficient architecture. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. The Fourier neural operator is the first ML-based method to successfully model turbulent flows with zero-shot super-resolution. It is up to three orders of magnitude faster compared to traditional PDE solvers. Additionally, it achieves superior accuracy compared to previous learning-based solvers under fixed resolution.
hub tools
citation-role summary
background 26
method 12
baseline 2
other 1
citation-polarity summary
claims ledger
- abstract The classical development of neural networks has primarily focused on learning mappings between finite-dimensional Euclidean spaces. Recently, this has been generalized to neural operators that learn mappings between function spaces. For partial differential equations (PDEs), neural operators directly learn the mapping from any functional parametric dependence to the solution. Thus, they learn an entire family of PDEs, in contrast to classical methods which solve one instance of the equation. In this work, we formulate a new neural operator by parameterizing the integral kernel directly in Fou
authors
co-cited works
representative citing papers
KANs with learnable univariate spline activations on edges achieve better accuracy than MLPs with fewer parameters, faster scaling, and direct visualization for scientific discovery.
PNOT combines graph attention on boundary heat flux with a physics-aware neural operator and gradient-constrained loss to reconstruct divertor temperature fields for real-time fusion control.
McMg is a phase-space multi-channel multigrid preconditioner that maps residuals to corrections while retaining unresolved wave information in extra channels, showing fewer iterations and lower runtime than classical and neural baselines on high-wavenumber 3D Helmholtz problems.
HO-FNO extends standard FNO with n-linear spectral mixing and shows improved accuracy on nonlinear PDE benchmarks, sometimes with a single layer beating deeper FNO models.
HNO using real Hartley transform outperforms FNO on elliptic PDEs while FNO wins on time-dependent PDEs, yielding a rule to match spectral basis to operator symmetry via Green's functions.
Optimizing training data via a differentiable SCM yields climate emulators that outperform those trained on six standard ScenarioMIP pathways while using less data and isolating distinct forcing responses.
Proves a weighted Nachbin theorem establishing universal approximation of differentiable maps from weighted infinite-dimensional manifolds to Banach spaces, including derivatives, with applications to non-anticipative path functionals and signature methods.
FreqNO-DPS corrects neural operator spectral bias in 3D elastic wavefield prediction by frequency-dependent guidance in diffusion posterior sampling conditioned on sparse observations, achieving near-zero bias at 2-5% sensor coverage.
TransportBench is a new high-fidelity dataset and standardized benchmark that evaluates neural architectures on non-equilibrium flow transport and finds performance depends strongly on specific flow characteristics.
Proposes residual-based physics-informed coarsening in multigrid GNNs to allocate capacity to high-activity regions for more stable solid mechanics surrogates.
GON uses 2-jet features and an anchor-and-variance objective to fix gauge freedom in ordinal predictability scoring, enabling pretrained initialization to outperform scratch training on held-out dynamical systems.
Therm-FM adapts a pretrained PDE foundation model using thermal-equivalent multi-fidelity training to achieve up to 10.6x lower error in 3D-IC thermal simulation with under 20% of typical training data and strong cross-design transfer.
A framework pretrained on authentic binary occlusion masks uses guided sampling and intersection-based partitioning to train diffusion models on incomplete physical observations without zero-query regions.
Presents a likelihood-free transport map learned by minimizing an averaged energy-distance objective to amortize Bayesian inference for inverse problems, including PDE-constrained cases with neural operator representations.
Fine-tuning neural PDE operators to regime endpoints reveals a physical direction in weight space that CCM uses to compose accurate merged models for new or extrapolated regimes from metadata or short prefixes.
Local neural operators on 3x3x3 patches, composed via Schwarz iteration, solve large-scale nonlinear elasticity on arbitrary geometries without domain-specific retraining.
Laplacian eigenfunction-based neural operators approximate the solution operator of the generalized Gierer-Meinhardt reaction-diffusion system with error bounds that imply only polynomial growth in parameters as accuracy improves.
NVQLS introduces the first hybrid quantum-classical unsupervised operator learning method for parametric PDEs via Legendre-Galerkin weak form, sign ambiguity resolution, and neural embedding.
A single-network implicit neural optimal transport method that solves the c-transform via proximal fixed-point iteration for stable, non-adversarial training.
A latent Structured Spectral Propagator enables stable autoregressive PDE forecasting by decoupling spatial details from recurrent modal dynamics.
A new Intent Fidelity Score and refinement loop verify that LLM-generated simulation code matches the intended PDEs, improving performance on a 220-case benchmark where execution alone fails to ensure correctness.
CATO learns a continuous latent chart for efficient axial attention on PDE meshes and adds derivative-aware supervision to improve accuracy and reduce oversmoothing on general geometries.
Spatio-Temporal MeanFlow adapts MeanFlow to PDEs by replacing the generative velocity field with the physical operator and extending the integral constraint to the spatio-temporal domain, yielding a unified solver for time-dependent and stationary equations with improved accuracy and generalization.
citing papers explorer
-
Hybrid Fourier Neural Operator-Lattice Boltzmann Method
Hybrid FNO-LBM accelerates porous media flow convergence by up to 70% via neural initialization and stabilizes unsteady simulations through embedded FNO rollouts, allowing small models to match larger ones in accuracy.
-
Learning Neural Operator Surrogates for the Black Hole Accretion Code
Physics-informed Fourier neural operators recover plasmoid formation in sparse SRRMHD vortex data where data-only models fail, and transformer operators approximate AMR jet evolution, marking first reported uses in these relativistic MHD settings.
-
SLE-FNO: Single-Layer Extensions for Task-Agnostic Continual Learning in Fourier Neural Operators
SLE-FNO achieves zero forgetting and strong plasticity-stability balance in continual learning for FNO surrogate models of pulsatile blood flow by adding minimal single-layer extensions across four out-of-distribution tasks.
-
Universal Neural Propagator: Learning Time Evolution in Many-Body Quantum Systems
The Universal Neural Propagator is a single neural model trained self-supervised to predict time evolution in driven quantum many-body systems across arbitrary protocols and initial states.
-
Large-eddy simulation nets (LESnets) based on physics-informed neural operator for wall-bounded turbulence
LESnets integrates LES equations and the law of the wall into F-FNO to enable data-free, stable long-term predictions of wall-bounded turbulence at Re_tau up to 1000 on coarse grids, matching traditional LES accuracy at higher efficiency.
-
Optimal-Transport-Guided Functional Flow Matching for Turbulent Field Generation in Hilbert Space
FOT-CFM generates turbulent fields in function space with superior high-order statistics and energy spectra on Navier-Stokes, Kolmogorov flow, and Hasegawa-Wakatani equations compared to baselines.
-
Resolution-Independent Machine Learning Heat Flux Closure for ICF Plasmas
A Fourier Neural Operator trained on PIC simulations yields a resolution-independent machine-learning closure for electron heat flux that reproduces temperature evolution when inserted into the energy equation and generalizes from coarse to fine grids.
-
MPDiT: Multi-Patch Global-to-Local Transformer Architecture For Efficient Flow Matching and Diffusion Model
MPDiT uses a hierarchical multi-patch design in transformers to lower computation in diffusion models by handling coarse global features first then fine local details, plus faster-converging embeddings.
-
Singularity Formation: Synergy in Theoretical, Numerical and Machine Learning Approaches
The work introduces a modulation-based analytical method for singularity proofs in singular PDEs and refines ML techniques like PINNs and KANs to identify blowup solutions, with application to the open 3D Keller-Segel problem.
-
A Multi-Scale ResNet-augmented Fourier Neural Operator Framework for High-Frequency Sequence-to-Sequence Prediction of Magnetic Hysteresis
Res-FNO predicts high-frequency magnetic hysteresis loops including ringing effects and minor loops with claimed strong generalization across materials from 79 to 3C90.
-
Elastomeric Strain Limitation for Design of Soft Pneumatic Actuators
Elastomeric strain limiters paired with electroadhesive clutches enable variable inflation trajectories in soft pneumatic actuators, supported by neural-network inverse design and experimental validation for safe human interaction.
-
Toward Artificial Intelligence Enabled Earth System Coupling
AI methods can strengthen cross-domain interactions and support more coherent multi-component representations in Earth system models.