PDE-PFN is a prior‑data fitted neural solver that uses in‑context learning to approximate the posterior predictive mean of partial differential equation (PDE) solutions. It builds on a Transformer‑based PFN architecture with self‑ and cross‑attention, and is pre‑trained on noisy approximate solutions generated by physics‑informed neural networks. Experiments on various two‑dimensional PDEs show that PDE‑PFN generalizes across heterogeneous equations, remains robust to noisy priors, and can perform zero‑gradient‑update inference while achieving competitive or better predictive accuracy compared to baselines.
By Jaehyeon Park, Mingu Kang, Dongseok Lee, Woojin Cho, Kookjin Lee, Anthony Gruber, Youngjoon Hong, Noseong Park
arXiv:2602. 12706v2 Announce Type: replace Abstract: Neural operators have emerged as fast surrogate solvers for parametric partial differential equations (PDEs).
By Heechang Kim, Qianying Cao, Hyomin Shin, Seungchul Lee, George Em Karniadakis, Minseok Choi
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.
By Yilong Dai, Shengyu Chen, Xiaowei Jia, Runlong Yu
arXiv:2610.08475v1 Announce Type: new
Abstract: In many-query scenarios, data-driven surrogate models provide an efficient alternative to high-fidelity solvers for simulating physical systems governe...
By Stefano Maria Pizzamiglio, Stefano Pagani, Francesco Regazzoni
The paper introduces Neptune, a method that uses independent coordinate neural networks to infer parameter fields in multi-physics PDEs from sparse measurements. Neptune can accurately estimate parameters with nonlinear, spatiotemporal variations, outperforming existing techniques by reducing estimation errors by up to two orders of magnitude and improving dynamic response predictions by a factor of ten. It also demonstrates strong physical extrapolation, enabling reliable predictions beyond the training data.
By Xuyang Li, Mahdi Masmoudi, Rami Gharbi, Nizar Lajnef, Vishnu Naresh Boddeti
PDE-PFN is a prior-data fitted neural solver that uses in-context learning to approximate the posterior predictive mean of partial differential equation (PDE) solutions. It builds on a Transformer-based PFN architecture with self- and cross-attention, and is pre‑trained on noisy approximate solutions generated by physics‑informed neural networks. Experiments on various two‑dimensional PDEs show that PDE‑PFN generalizes across heterogeneous equations, remains robust to noisy priors, and can perform zero‑gradient‑update inference while achieving competitive or improved predictive accuracy compared to baselines.
arXiv:2606. 24999v1 Announce Type: new Abstract: High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging.
By Yanwei Jia, Du Ouyang, Huy\^en Pham, Xun Yu Zhou
arXiv:2602. 09708v2 Announce Type: replace-cross Abstract: We propose physics-informed spectral diffusion (PISD), a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of partial differential equations (PDEs) conditioned on partial observations, which includes, in particular, forward and inverse PDE problems.
By Davide Gallon, Philippe von Wurstemberger, Patrick Cheridito, Arnulf Jentzen
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
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:2608.24049v1 Announce Type: new
Abstract: Neural operators provide efficient surrogates for spatiotemporal PDE systems, but purely data-driven formulations often accumulate substantial errors d...
By Jihao Zhang, Junyi Guo, Jian-Xun Wang
The paper introduces a two‑stage physics‑informed deep learning framework for solving inverse problems in partial differential equations with jump discontinuities in coefficients. The first stage uses a dual‑network architecture to approximate the PDE solution and a relaxed continuous surrogate of the coefficient field, followed by Bayesian inference with Gaussian mixture and birth‑death Markov chain models to estimate coefficient regimes and transition regions. The second stage reformulates the inverse problem as a constrained estimator with a hard piecewise‑constant coefficient representation, achieving accurate parameter estimation with acceptable computational costs across various PDE types.
By Zhikun Zhang, Guanyu Pan, Xiangjun Wang, Yong Xu, Guangtao Zhang