The paper introduces equation recast, a method that transforms parametric operator learning into learning a single canonical operator. By analytically deriving parameter-induced variations from the governing equations and incorporating them as effective sources, the approach enables zero‑shot predictions across new parameter regimes and supports extrapolation in multi‑parameter, nonlinear, and singular PDE settings. It also integrates sparse heterogeneous datasets, uses loss of convergence as an internal warning, and demonstrates unification of electron‑temperature data from multiple tokamak geometries in high‑fidelity nuclear fusion simulations.
By Qiyun Cheng, Valentin Duruisseaux, Cesar F. Clauser, Md Hossain Sahadath, Huihua Yang, Shaowu Pan, Nathaniel Ferraro, Anima Anandkumar, Wei Ji, Cristina Rea
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
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
The paper presents a reinforcement‑learning framework that dynamically selects between full‑order models (FOMs) and reduced‑order models (ROMs) in hybrid domain decomposition simulations using the overlapping Schwarz alternating method (O‑SAM). Offline‑trained Deep Q‑networks choose subdomain‑local FOMs or pre‑trained Operator Inference ROMs based on a reward that balances accuracy, computational cost, and model‑switching frequency, and the learned policies are deployed on unseen problem instances without needing a reference FOM solution. Experiments on a 1D advection‑diffusion problem and a 3D elastic wave propagation benchmark show that the RL‑guided policies adaptively allocate high‑fidelity resolution as features propagate, outperforming static FOM/ROM assignments and demonstrating the feasibility of predictive online fidelity adaptation in Schwarz‑based hybrid simulations.
By Trishit Mondal, Irina Tezaur, Anthony Gruber
The paper introduces Sensitivity‑Constrained Neural Operators (SC‑NOs), which augment standard neural operator training with sampled Jacobian supervision from differentiable solvers or discrete adjoints. By matching selected sensitivities during training, SC‑NOs improve forward prediction accuracy and significantly enhance gradient‑based inverse reconstruction for distributed fields. Experiments on advection–diffusion, RANS–Spalart–Allmaras, high‑dimensional gridded inputs, and a shallow‑water tsunami source‑inversion case demonstrate that SC‑NOs achieve a better accuracy–cost trade‑off and enable near‑real‑time wave‑propagation forecasting from sparse observations.
By Abdolmehdi Behroozi, Chaopeng Shen, Daniel Kifer, Kathryn Lawson
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.
By Haina Jiang, Liam Wang, Peng-Chen Chen, Min Seop Kwak, Seungryong Kim, Brian Bell, Jeong Joon Park
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 introduces a linearized Physics-Informed Neural Network (lPINN), a reduced‑order neural basis approach for solving forward and inverse differential equations. In an offline phase, lPINN learns continuous, differentiable neural basis functions from numerical solutions, which are then frozen for new problem instances; the online solution is obtained by minimizing the governing‑equation residual with additional constraints. Experiments on advection‑diffusion, Burgers', and nonlinear pendulum equations show that lPINN achieves lower solution and parameter errors than vanilla PINNs while reducing online inference times by up to three orders of magnitude, and its continuous representation generalizes to finer meshes without retraining.
By Wenhao Chen, Alexandre M. Tartakovsky
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
arXiv:2606. 08672v1 Announce Type: cross Abstract: Diffusion and flow generative models sample by integrating a learned ODE, but high quality still requires many sequential model evaluations.
By Sihyeon Kim, Seunghun Lee, Vikas Singh, Hyunwoo J. Kim
The paper introduces Linearized Subspace Refinement (LSR), a post‑training framework that uses the local linearized model of a trained neural network to compute a low‑dimensional correction via a reduced least‑squares problem. LSR is architecture‑agnostic and improves accuracy across tasks such as function approximation, operator learning, physics‑informed fine‑tuning, and noisy inverse problems, often achieving order‑of‑magnitude error reductions. The method reveals that standard training can leave significant accuracy plateaus due to numerical ill‑conditioning, and it offers a subspace rank that balances correction strength, stability, and noise sensitivity.
By Wenbo Cao, Weiwei Zhang