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Spectrally Safe Neural Operator Warm-Starts for Large-Scale Newton Solvers
Jaemin Oh, Youngkyu Lee, Jerome Darbon +1
Neural operators are increasingly used to warm-start Newton solvers for nonlinear PDEs, on the premise that a low test error places the initial guess inside the basin of attraction…
Adjoint Method versus Physics-Informed Neural Networks in PDE-Constrained Inverse Problems
Zhen Zhang, Alessandro Alla, George Em Karniadakis
Inverse problems governed by partial differential equations (PDEs) are central to computational mechanics and are commonly solved by adjoint-based optimization, while physics-infor…
Spectral Audit of In-Context Operator Networks
Zhiwei Gao, Liu Yang, George Em Karniadakis
Existing evaluations of neural operators and in-context operator learning rely primarily on prediction error, but accurate output prediction does not guarantee the correct local dy…
Proposal-Guided Greedy Surrogate Refinement for PDE-Driven High-Dimensional Rare-Event Estimation
Zhiwei Gao, George Karniadakis
Accurate surrogate construction for PDE-driven high-dimensional rare-event simulation is challenging when performance evaluations are expensive. Since a globally accurate surrogate…
NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces
Francesc Levrero-Florencio, Youngkyu Lee, Jay Pathak +1
The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, th…
Nonlinear parametrization solver for fractional Burgers equations
Haojun Qin, Zhiwei Gao, Jinye Shen +1
Fractional Burgers equations pose substantial challenges for classical numerical methods due to the combined effects of nonlocality and shock-forming nonlinear dynamics. In particu…