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.
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
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:2607. 22215v1 Announce Type: new Abstract: In this study, we introduce latent PDE mapping, a broadly applicable physics-informed learning technique designed to enable efficient geometric generalization with sparse training data.
By Ingvild Askim Adde, Mary M. Maleckar, Gabriel Balaban
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
arXiv:2608. 04778v1 Announce Type: new Abstract: Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets.
By Xujia Chen, Xinyue Hu, Letian Chen, Yi Liu, Wenhui Fan
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:2608. 07053v1 Announce Type: new Abstract: Pretrained partial differential equation (PDE) foundation models can generalize across different equations, but adapting them to unseen PDE systems typically requires dense solution data, which is often expensive or unavailable.
By Ziye Song, Zhao Wei, Xin Yu, Ivor Tsang, Yueming Lyu
The paper investigates how different attention mechanisms affect the performance of DeepONet neural operators. Five variants—varying in cross‑attention, self‑attention, tokenization, and attention depth—are trained in both data‑driven and physics‑informed settings on one‑ and two‑dimensional PDE benchmarks. Results show that per‑sensor tokenization with cross‑attention consistently reduces error, while branch self‑attention helps only in complex spatial problems, and deeper cross‑attention yields diminishing returns with higher cost.
By Amar Alem Koric, Qibang Liu, Seid Koric
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
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:2605. 08318v2 Announce Type: replace Abstract: We study the problem of \emph{architecture selection} for deep learning models trained to solve partial differential equations (PDEs), asking when transformer-based architectures with learned attention outperform Fourier-domain neural operators.
By Brandon Yee, Pairie Koh, Jack Rodriguez, Mihir Tekal