GAIA introduces a geometry-adaptive integral autoencoder that unifies forward, boundary-value, and inverse PDE operator learning on arbitrary domains via geometry tokens and cross-attention.
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Transformer for partial differential equations’ operator learning
25 Pith papers cite this work, alongside 46 external citations. Polarity classification is still indexing.
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representative citing papers
HS-FNO lifts the state to include history and decomposes updates into a learned future-slice predictor plus an exact shift-append transport, yielding lower rollout errors than standard or lag-stack FNO baselines on five non-Markovian PDE families.
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.
HAMNO introduces adaptive gating between local and global operators in a hierarchical setup, with PI-HAMNO adding PDE residual constraints, demonstrating better performance on Allen-Cahn, Cahn-Hilliard, and Swift-Hohenberg equations.
MeshTok uses AMR-inspired adaptive multiscale tokenization to improve the efficiency-accuracy trade-off of Transformer models for PDEs over uniform-grid baselines.
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.
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.
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.
Scale-autoregressive modeling (SAR) samples fluid flow distributions hierarchically from coarse to fine resolutions on meshes, achieving lower distributional error and 2-7x faster runtime than diffusion or flow-matching baselines.
LGS pretrained on 2.5M trajectories across 16 systems matches deterministic baselines at one step and halves 20-step error while using far less compute and adapting to held-out higher-resolution flows.
DGPFM stacks GP-based linear and nonlinear transformations in function space via kernel integrals and inducing-point variational learning for function-on-function regression.
Self-attention mechanisms are used to build mesh-preserving neural surrogates that approximate PFEM dynamics for free-surface flows, delivering accurate transient predictions and improved scalability on 2D and 3D benchmarks.
EqGINO adds a spectral isotropy prior to FNOs to guarantee discrete equivariance and enable generalization to continuous SE(3) transformations on 3D PDEs with limited training data.
IV-Net is a multigrid-inspired convolutional neural operator that approximates solutions to linear elliptic PDEs with high-contrast coefficients and shows better accuracy than POD and other neural operators on heterogeneous coercive problems.
AOT-POT adaptively reshapes complex PDE solution operators via input-dependent transformations and parallel stream mixing to enable effective large-scale pre-training, yielding SOTA results on 12 benchmarks with minimal added parameters.
A fine-tuned video diffusion model becomes a fast, differentiable CFD surrogate for urban wind, enabling gradient-based building-layout optimization confirmed by ground-truth simulations.
ArGEnT adds self-, cross-, and hybrid-attention transformers to DeepONet to learn geometry-dependent operators from point-cloud inputs, yielding higher accuracy than standard DeepONet on fluid, solid, and electrochemical benchmarks.
DIANO builds coarse-grid latent spaces for fluid dynamics data via neural operator encoding and decoding while integrating a differentiable PDE solver directly in the latent space for end-to-end physics-constrained training.
Flow Marching jointly samples noise and physical time to learn a velocity field for generative PDE modeling, paired with a latent autoencoder and efficient transformer for large-scale pretraining on 2.5M trajectories.
TAE combines Tikhonov regularization with autoencoders and a data randomization strategy to learn forward and inverse surrogates from one sample, with linear error bounds and tests on heat inversion and Navier-Stokes reconstruction.
PI-DOSnet is a physics-informed operator learning framework using operator splitting that enables data-free long-time inference of evolution PDE solutions, with energy stability shown for the Allen-Cahn equation at large time steps.
FEDONet augments DeepONet with Fourier-embedded trunk networks using random Fourier features, yielding lower L2 reconstruction errors than standard DeepONet on Burgers', 2D Poisson, Eikonal, Allen-Cahn, and Kuramoto-Sivashinsky equations across dataset sizes and noise levels.
Empirical benchmarks indicate CViT-based PINOs with adapted PINN mitigations can match or exceed data-driven neural operators on parametric PDE tasks.
GeoTransolver applies geometry-aware operator learning and low-rank attention to predict high-fidelity crash dynamics on bumper and full-vehicle datasets, with one-shot temporal prediction achieving state-of-the-art accuracy and reduced overhead.
citing papers explorer
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GAIA: Geometry-Adaptive Operator Learning for Forward and Inverse Problems
GAIA introduces a geometry-adaptive integral autoencoder that unifies forward, boundary-value, and inverse PDE operator learning on arbitrary domains via geometry tokens and cross-attention.
-
HS-FNO: History-Space Fourier Neural Operator for Non-Markovian Partial Differential Equations
HS-FNO lifts the state to include history and decomposes updates into a learned future-slice predictor plus an exact shift-append transport, yielding lower rollout errors than standard or lag-stack FNO baselines on five non-Markovian PDE families.
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Temperature Field Reconstruction of Tungsten Monoblock Divertor on EAST using Physics-aware Neural Operator Transformer
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.
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HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems
HAMNO introduces adaptive gating between local and global operators in a hierarchical setup, with PI-HAMNO adding PDE residual constraints, demonstrating better performance on Allen-Cahn, Cahn-Hilliard, and Swift-Hohenberg equations.
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MeshTok: Efficient Multi-Scale Tokenization for Scalable PDE Transformers
MeshTok uses AMR-inspired adaptive multiscale tokenization to improve the efficiency-accuracy trade-off of Transformer models for PDEs over uniform-grid baselines.
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CATO: Charted Attention for Neural PDE Operators
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.
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QuadNorm: Resolution-Robust Normalization for Neural Operators
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.
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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.
-
One Scale at a Time: Scale-Autoregressive Modeling for Fluid Flow Distributions
Scale-autoregressive modeling (SAR) samples fluid flow distributions hierarchically from coarse to fine resolutions on meshes, achieving lower distributional error and 2-7x faster runtime than diffusion or flow-matching baselines.
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Latent Generative Solvers for Generalizable Long-Term Physics Simulation
LGS pretrained on 2.5M trajectories across 16 systems matches deterministic baselines at one step and halves 20-step error while using far less compute and adapting to held-out higher-resolution flows.
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Deep Gaussian Processes for Functional Maps
DGPFM stacks GP-based linear and nonlinear transformations in function space via kernel integrals and inducing-point variational learning for function-on-function regression.
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Attention mechanism for scalable mesh-based neural surrogates of free-surface fluids
Self-attention mechanisms are used to build mesh-preserving neural surrogates that approximate PFEM dynamics for free-surface flows, delivering accurate transient predictions and improved scalability on 2D and 3D benchmarks.
-
EqGINO: Equivariant Geometry-Informed Fourier Neural Operators for 3D PDEs
EqGINO adds a spectral isotropy prior to FNOs to guarantee discrete equivariance and enable generalization to continuous SE(3) transformations on 3D PDEs with limited training data.
-
IV-Net: A neural network for elliptic PDEs with random and highly varying coefficients
IV-Net is a multigrid-inspired convolutional neural operator that approximates solutions to linear elliptic PDEs with high-contrast coefficients and shows better accuracy than POD and other neural operators on heterogeneous coercive problems.
-
AOT-POT: Adaptive Operator Transformation for Large-Scale PDE Pre-training
AOT-POT adaptively reshapes complex PDE solution operators via input-dependent transformations and parallel stream mixing to enable effective large-scale pre-training, yielding SOTA results on 12 benchmarks with minimal added parameters.
-
Pretrained Video Models as Differentiable Physics Simulators for Urban Wind Flows
A fine-tuned video diffusion model becomes a fast, differentiable CFD surrogate for urban wind, enabling gradient-based building-layout optimization confirmed by ground-truth simulations.
-
ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning
ArGEnT adds self-, cross-, and hybrid-attention transformers to DeepONet to learn geometry-dependent operators from point-cloud inputs, yielding higher accuracy than standard DeepONet on fluid, solid, and electrochemical benchmarks.
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Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling
DIANO builds coarse-grid latent spaces for fluid dynamics data via neural operator encoding and decoding while integrating a differentiable PDE solver directly in the latent space for end-to-end physics-constrained training.
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Flow marching for a generative PDE foundation model
Flow Marching jointly samples noise and physical time to learn a velocity field for generative PDE modeling, paired with a latent autoencoder and efficient transformer for large-scale pretraining on 2.5M trajectories.
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TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems
TAE combines Tikhonov regularization with autoencoders and a data randomization strategy to learn forward and inverse surrogates from one sample, with linear error bounds and tests on heat inversion and Navier-Stokes reconstruction.
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PI-DOSnet: A Physics-Informed Deep Operator-Splitting Network for Evolution Partial Differential Equations
PI-DOSnet is a physics-informed operator learning framework using operator splitting that enables data-free long-time inference of evolution PDE solutions, with energy stability shown for the Allen-Cahn equation at large time steps.
-
FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning
FEDONet augments DeepONet with Fourier-embedded trunk networks using random Fourier features, yielding lower L2 reconstruction errors than standard DeepONet on Burgers', 2D Poisson, Eikonal, Allen-Cahn, and Kuramoto-Sivashinsky equations across dataset sizes and noise levels.
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On the training of physics-informed neural operators for solving parametric partial differential equations
Empirical benchmarks indicate CViT-based PINOs with adapted PINN mitigations can match or exceed data-driven neural operators on parametric PDE tasks.
-
High-Fidelity Industrial Crash Dynamics Prediction via Geometry-Aware Operator Learning with Memory-Efficient Low-Rank Attention
GeoTransolver applies geometry-aware operator learning and low-rank attention to predict high-fidelity crash dynamics on bumper and full-vehicle datasets, with one-shot temporal prediction achieving state-of-the-art accuracy and reduced overhead.
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RETO: A Rotary-Enhanced Transformer Operator for High-Fidelity Prediction of Automotive Aerodynamics
RETO achieves relative L2 errors of 0.063 on ShapeNet and 0.089/0.097 on DrivAerML surface pressure/velocity, outperforming Transolver and other baselines.