arXiv:2610. 05337v2 Announce Type: replace-cross Abstract: We identify a mismatch between the physical role of transport fields in many PDEs and their usual role in neural operators: PDEs use them to select read coordinates, whereas neural operators typically treat them only as input values.
arXiv:2606. 11963v1 Announce Type: new Abstract: Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space.
arXiv:2606. 27354v1 Announce Type: cross Abstract: Neural surrogate models offer fast approximate mappings from PDE parameters to solutions, but they typically treat solving as a purely statistical task: once trained, they struggle to correct their own constraint violations and extrapolate beyond the training distribution.
arXiv:2604. 07366v2 Announce Type: replace Abstract: Partial differential equations (PDEs) govern nearly every physical process in science and engineering, but solving them at scale remains prohibitively expensive.
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
The paper introduces Align Latent Rollout (ALR), a regularization technique for neural PDE solvers that encourages consistency across different rollout origins. ALR uses a second‑order Taylor analysis to define two objectives: zeroth‑order consistency aligning latent states at a shared target time, and first‑order consistency aligning local evolution responses to perturbations. The method can be applied to various neural PDE solvers without altering their architectures and demonstrates improved prediction accuracy and reduced discrepancies across rollout origins in multiple benchmarks.
arXiv:2512. 19643v2 Announce Type: replace Abstract: Numerical simulation of time-dependent partial differential equations (PDEs) is central to scientific and engineering applications, but high-fidelity solvers are often prohibitively expensive for long-horizon or time-critical settings.
The paper introduces a framework for evaluating how well learned states from neural PDE solvers can be reused in new contexts. By separating solution accuracy, learning contribution, and numerical utility through paired state comparisons and cost accounting, the authors audit 12 papers and conduct experiments on linear PDE families, Fourier neural operators, and convolutional networks. Their results show that matched state comparisons and downstream computational evidence can lead to significant savings in conjugate‑gradient iterations and demonstrate finite‑budget pretraining value.
arXiv:2610.06952v1 Announce Type: new Abstract: Neural PDE solvers can achieve low prediction errors, but do they reproduce the dynamics of the systems they model? Prediction scores alone offer an in...
arXiv:2608.21677v1 Announce Type: cross Abstract: Recent mesh-based simulation advances have, in no small part, relied on neural surrogates of two distinct families: global models that route informat...
arXiv:2609.24947v1 Announce Type: new Abstract: Neural operators evaluate parametric partial differential equations cheaply but degrade sharply outside their training distribution. Physics-informed n...
Transolver‑σ is a neural PDE solver that jointly models spectral and physical subspaces to improve accuracy in both one‑step and autoregressive rollouts. The method uses adaptive physical-state interactions, Slice‑Residual Physics‑Attention, and an axis‑factorized Fourier operator to enable information exchange between representations. Across five standard PDE benchmarks, Transolver‑σ reduces benchmark‑averaged relative error by 33.4% compared to the strongest baseline and shows strong performance on coupled multiphysics systems and real‑world fluid and combustion data.
arXiv:2605. 25413v3 Announce Type: replace-cross Abstract: Neural operators learn mappings from function-dependent inputs to solutions, providing an effective framework for solving partial differential equations (PDEs).