MEEC equips point clouds with a discrete exterior calculus that satisfies exact conservation and is differentiable in point positions, allowing a single trained kernel to produce compatible physics on unseen geometries and parameters.
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Neural Operator: Graph Kernel Network for Partial Differential Equations
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abstract
The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces. We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. The kernel integration is computed by message passing on graph networks. This approach has substantial practical consequences which we will illustrate in the context of mappings between input data to partial differential equations (PDEs) and their solutions. In this context, such learned networks can generalize among different approximation methods for the PDE (such as finite difference or finite element methods) and among approximations corresponding to different underlying levels of resolution and discretization. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.
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representative citing papers
Fourier Neural Operator parameterizes integral kernels in Fourier space to learn parametric PDE solution operators, delivering up to 1000x speedups and zero-shot super-resolution on turbulent Navier-Stokes flows.
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.
Zero-shot super-resolution is information-theoretically impossible for some simple operators but possible under Hölder smoothness of outputs, accompanied by generalization bounds.
Functional Attention replaces pairwise softmax attention with structured linear operators inspired by geometric functional maps to produce compact, resolution-invariant representations for operator learning.
Neural networks parameterize finite-rank generators for ODEs on the orthogonal Lie group, allowing optimization of orthonormal bases in function space with a universality result that rank-2 generators suffice for density.
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.
Proposes a vector-valued RKHS framework for Bayesian optimization with structured measurements, deriving concentration bounds and UCB-based regret guarantees that recover sublinear rates.
QuadNorm uses quadrature-based moments instead of uniform averaging in normalization layers, achieving O(h²) consistency across resolutions and better cross-resolution transfer in neural operators.
HIN-LRI augments a low-regularity integrator with a latent-manifold neural correction trained end-to-end on trajectory error to improve accuracy on nonlinear dispersive equations with rough data.
A multilinear operator learned on PCA coefficients maps time-since-ignition inputs to smoke outputs, matching Monte Carlo accuracy with half the model calls and outperforming prior classifiers on holdout data.
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.
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.
Parameterizing the temporal derivative in PINNs and reconstructing via Volterra integral yields 100-200x lower errors on advection, Burgers, and Klein-Gordon equations while proving equivalence to the original PDE.
PGOT uses physics-geometry attention and spatially adaptive computation paths to achieve state-of-the-art results on complex PDE benchmarks and industrial geometry tasks.
WHNO using Walsh-Hadamard transforms outperforms Fourier Neural Operators on PDEs with discontinuous coefficients, and optimal WHNO-FNO ensembles cut mean-squared error by 35-40 percent.
SAE-NOs extend sparse autoencoders to function spaces via Fourier neural operators with concept and domain sparsity, learning localized patterns more efficiently and generalizing across discretizations on vision data.
PVD-ONet combines multi-network DeepONet modules with Prandtl and Van Dyke matching conditions to map initial data to solution operators for families of singularly perturbed boundary-layer problems and to infer scaling exponents from sparse observations.
PCA-RaNN recasts latent neural operator learning as PCA-reduced random-feature linear regression, achieving 1-3 orders faster training than standard methods on PDE benchmarks while adding conformal uncertainty quantification.
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LFNO is a dual-branch neural operator combining Laplace and Fourier methods to explicitly decompose and model transient and steady-state dynamics, outperforming baselines on ODE benchmarks and remaining competitive on PDEs.
PINN failure modes are overfitting to collocation points; regularization and double backpropagation over full residuals fix them, achieving SOTA with up to 23x fewer points on standard benchmarks.
Normalized semigroup error is introduced as a diagnostic for learned simulators on 1D heat and Burgers equations; it correlates with rollout degradation (Spearman ρ=0.635) while regularization shows mixed results.
citing papers explorer
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Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale
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Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations
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Enabling Real-Time Training of a Wildfire-to-Smoke Map with Multilinear Operators
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Hybrid Fourier Neural Operator-Lattice Boltzmann Method
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Learning Neural Operator Surrogates for the Black Hole Accretion Code
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Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks
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LFNO: Bridging Laplace and Fourier via Transient-Steady Decomposition
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PINNs Failure Modes are Overfitting
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Semigroup Consistency as a Diagnostic for Learned Physics Simulators
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NOWS: Neural Operator Warm Starts for Accelerating Iterative Solvers
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Multiscale Physics-Informed Neural Network for Complex Fluid Flows with Long-Range Dependencies
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Scalable Mechanistic Neural Networks for Differential Equations and Machine Learning
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Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equations
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Multi-scale Dynamic Wake Modeling and Prediction of Floating Offshore Wind Turbines via Physics-Informed Neural Networks and Fourier Neural Operators
Fourier Neural Operators outperform Physics-Informed Neural Networks in computational efficiency, convergence speed, and fidelity to multi-scale wake structures including meandering frequencies when trained on LES-AL data for floating offshore wind turbines.
- Kernel Neural Operators (KNOs) for Scalable, Memory-efficient, Geometrically-flexible Operator Learning