199 citations · 302 across the 61 of their papers we have counts for
12 papers · 1 filter
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…
A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers
Youngkyu Lee, Shanqing Liu, Jerome Darbon +1
We propose a novel neural preconditioned Newton (NP-Newton) method for solving parametric nonlinear systems of equations. To overcome the stagnation or instability of Newton iterat…
Safe cross-entropy-based importance sampling for rare event simulations
Zhiwei Gao, George Karniadakis
The Improved Cross-Entropy (ICE) method is a powerful tool for estimating failure probabilities in reliability analysis. Its core idea is to approximate the optimal importance-samp…
Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method
Alena Kopaničáková, Youngkyu Lee, George Em Karniadakis
We propose a new deflation strategy to accelerate the convergence of the preconditioned conjugate gradient(PCG) method for solving parametric large-scale linear systems of equation…
Automatic discovery of optimal meta-solvers for time-dependent nonlinear PDEs
Youngkyu Lee, Shanqing Liu, Jerome Darbon +1
We present a general and scalable framework for the automated discovery of optimal meta-solvers for the solution of time-dependent nonlinear partial differential equations after ap…
Automatic discovery of optimal meta-solvers via multi-objective optimization
Youngkyu Lee, Shanqing Liu, Jerome Darbon +1
We design two classes of ultra-fast meta-solvers for linear systems arising after discretizing PDEs by combining neural operators with either simple iterative solvers, e.g., Jacobi…
Enhancing training of physics-informed neural networks using domain-decomposition based preconditioning strategies
Alena Kopaničáková, Hardik Kothari, George Em Karniadakis +1
We propose to enhance the training of physics-informed neural networks (PINNs). To this aim, we introduce nonlinear additive and multiplicative preconditioning strategies for the w…