The paper introduces an end‑to‑end equivariant surrogate model for three‑dimensional Rayleigh‑Bénard convection, combining an equivariant convolutional autoencoder with an equivariant convolutional LSTM that employs $G$‑steerable kernels. The architecture exploits $D_4$‑steerable kernels in vertically stacked layers and partial kernel sharing in the vertical direction to respect the system’s E(2) equivariance in the horizontal plane while handling broken translational symmetry vertically. Experiments show notable improvements in sample and parameter efficiency and better scaling to more complex dynamics.
By Fynn Fromme, Hans Harder, Christine Allen-Blanchette, Sebastian Peitz
The paper introduces the Local Gradient Neural Operator (LGNO), a lightweight and interpretable neural operator designed for field temporal evolution prediction and source identification in mechanical problems. LGNO leverages nonlinear gradient discretization priors and multilayer perceptron convolutional layers to learn translation‑invariant local kernels resembling discrete stencils, with a zero‑consistent stencil factorization that separates coefficient learning from field reconstruction. Experiments on a range of PDE benchmarks—including linear, nonlinear, static, dynamic, low‑ and high‑dimensional cases—demonstrate that LGNO achieves comparable accuracy to global neural operators while using fewer parameters and maintaining rollout stability across diffusion, flow, and quantum phenomena.
By Baiming Zhang, Jinsong Tang, Ying Xu, Lihua Chen, Shiying Xiong
arXiv:2607. 28762v1 Announce Type: new Abstract: This work embeds feature interaction modules derived from factorization machines (FMs) into physics-informed neural networks (PINNs) and neural operator learning, to enhance model expressiveness for solution manifolds of parameterized partial differential equations (PDEs).
By Quan Gu, Hongxia Liu
Physics-Informed Conformal Prediction (PI‑CP) embeds PDE residuals into the nonconformity score of split conformal prediction, yielding distribution‑free prediction intervals with provable coverage that adapt spatially to physics violations. The method demonstrates consistent 89‑91% coverage across six physics scenarios, outperforming MC Dropout and Deep Ensembles, while Fourier Neural Operators (FNO) achieve superior accuracy over CNN and DeepONet. Additionally, the authors prove that FNO’s translation equivariance limits its ability to solve PDEs with Dirichlet boundary conditions, and show that adding coordinate channels can reduce error by up to 63×.
By Michael Chin
arXiv:2603. 04430v2 Announce Type: replace Abstract: We introduce Flowers, a neural architecture for learning PDE solution operators built entirely from multihead warps.
By Till Muser, Alexandra Spitzer, Matti Lassas, Maarten V. de Hoop, Ivan Dokmani\'c
arXiv:2603. 12676v3 Announce Type: replace Abstract: Generalizing neural surrogate models across different PDE parameters remains difficult because changes in PDE coefficients often make learning harder and optimization less stable.
By Zhangyong Liang, Huanhuan Gao
Field temporal prediction and source identification constitute canonical problems in dynamical systems. Conventional approaches to these problems depend on a thorough understanding of the governing pa...
arXiv:2501. 04339v2 Announce Type: replace-cross Abstract: We introduce the Deep Convolutional Interpreter for Time Series (DCIts), a deep-learning architecture for nonlinear multivariate time series that provides sample-specific, locally interpretable descriptions of the underlying interaction structure.
By Domjan Baric, Davor Horvatic
arXiv:2606. 03260v1 Announce Type: cross Abstract: Deep learning surrogates for 3D Partial Differential Equations (PDEs) often fail to generalize across geometric transformations because they depend heavily on specific coordinate systems.
By Sungwon Kim, Juho Song, Seungmin Shin, Guimok Cho, Sangkook Kim, Chanyoung Park
arXiv:2512. 01370v2 Announce Type: replace-cross Abstract: Diffusion-based solvers for partial differential equations (PDEs) are often bottle-necked by slow gradient-based test-time optimization routines that use PDE residuals for loss guidance.
By Medha Sawhney, Abhilash Neog, Mridul Khurana, Anuj Karpatne
arXiv:2410. 10137v5 Announce Type: replace Abstract: We develop Riemannian approaches to variational autoencoders (VAEs) for PDE-type ambient data with regularizing geometric latent dynamics, which we refer to as VAE-DLM, or VAEs with dynamical latent manifolds.
By Andrew Gracyk
The paper presents a method to recover unknown functional terms in partial differential equations (PDEs) by embedding neural networks into standard parameter estimation workflows. By training on data, the approach learns interaction kernels and external potentials in nonlocal aggregation‑diffusion equations, achieving high accuracy. The study systematically investigates how reconstruction accuracy depends on solution diversity, sampling density, and measurement noise.
By Torkel E. Loman, Yurij Salmaniw, Antonio Leon Villares, Jose A. Carrillo, Ruth E. Baker