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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Transformer for partial differential equations’ operator learning
17 Pith papers cite this work. Polarity classification is still indexing.
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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.
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.
WinDiNet repurposes a 2B-parameter video diffusion model as a differentiable surrogate that generates 112-frame urban wind flow rollouts in under one second and enables direct gradient optimization of building positions.
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.
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.
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.
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.
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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.