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.
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Fourier Neural Operator for Parametric Partial Differential Equations
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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.
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- 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
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representative citing papers
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Proposes residual-based physics-informed coarsening in multigrid GNNs to allocate capacity to high-activity regions for more stable solid mechanics surrogates.
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.
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.
PODiff performs conditional diffusion in a fixed, variance-ordered POD latent space to enable efficient probabilistic super-resolution of high-dimensional scientific fields with lower memory and better-calibrated uncertainty than pixel-space or dropout baselines.
Neural operators approximate continuous operators from H^s to H^t with O(N^{-s/d}) error in H^t norm; FNOs on Burgers achieve H^1 errors to 10^{-7} and follow a power-law scaling with exponent ~1.4.
Isotropic Fourier Neural Operators enforce spatial symmetries in Fourier layers, improving PDE-solving performance while reducing parameters by up to 16x in 2D and 96x in 3D.
A horizon-agnostic neural operator paired with a boundary control barrier function creates a real-time safety filter that raises safe trajectory rates by up to 22% on fluid manipulation tasks in simulation.
A physics-encoded Fourier neural operator (PeFNO) uses stress potentials to enforce divergence-free stress fields by architecture design, yielding better equilibrium satisfaction than physics-informed or physics-guided FNOs for polycrystalline materials under uniaxial tension.
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.
A safeguarded hybrid of Levenberg-Marquardt and learned operators achieves equivalent reconstruction quality for PGET in roughly one-third the iterations, with architecture-dependent robustness.
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.
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AI models of viscous fingering exhibit hallucinations from spectral bias; DeepFingers combines FNO and DeepONet with time-contrast conditioning to predict accurate finger dynamics while preserving mixing metrics.
A graph-based neural operator trained on expert-validated race-car CFD data reaches accuracy levels usable for early-stage interactive aerodynamic design exploration.
citing papers explorer
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Local neural operators on 3x3x3 patches, composed via Schwarz iteration, solve large-scale nonlinear elasticity on arbitrary geometries without domain-specific retraining.
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Approximation Theory of Laplacian-Based Neural Operators for Reaction-Diffusion System
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CATO: Charted Attention for Neural PDE Operators
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Physics-Informed Neural PDE Solvers via Spatio-Temporal MeanFlow
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PODiff: Latent Diffusion in Proper Orthogonal Decomposition Space for Scientific Super-Resolution
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Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equation
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Hybrid Fourier Neural Operator-Lattice Boltzmann Method
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Droplet-LNO: Physics-Informed Laplace Neural Operators for Accurate Prediction of Droplet Spreading Dynamics on Complex Surfaces
PI-LNO is a physics-informed neural operator that uses Laplace transforms and fluid physics constraints to accurately and rapidly predict droplet spreading dynamics on complex surfaces.
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AI models of unstable flow exhibit hallucination
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Faster by Design: Interactive Aerodynamics via Neural Surrogates Trained on Expert-Validated CFD
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DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting
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A Unified Multiscale Auxiliary PINN Framework for Generalized Phonon Transport
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