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