We introduce a new class of hybrid preconditioners for solving parametric linear systems of equations. The proposed preconditioners are constructed by hybridizing the deep operator network, namely DeepONet, with standard iterative methods. Exploiting the spectral bias, DeepONet-based components are harnessed to address low-frequency error components, while conventional iterative methods are employed to mitigate high-frequency error components. Our preconditioning framework comprises two distinct hybridization approaches: direct preconditioning (DP) and trunk basis (TB) approaches. In the DP approach, DeepONet is used to approximate an action of an inverse operator to a vector during each preconditioning step. In contrast, the TB approach extracts basis functions from the trained DeepONet to construct a map to a smaller subspace, in which the low-frequency component of the error can be effectively eliminated. Our numerical results demonstrate that utilizing the TB approach enhances the convergence of Krylov methods by a large margin compared to standard non-hybrid preconditioning strategies. Moreover, the proposed hybrid preconditioners exhibit robustness across a wide range of model parameters and problem resolutions.
The Deep Operator Networks~(DeepONet) is a fundamentally different class of neural networks that we train to approximate nonlinear operators, including the solution operator of parametric partial differential equations (PDE). DeepONets have shown remarkable approximation and generalization capabilities even when trained with relatively small datasets. However, the performance of DeepONets deteriorates when the training data is polluted with noise, a scenario that occurs very often in practice. To enable DeepONets training with noisy data, we propose using the Bayesian framework of replica-exch…
We extend the operator preconditioning framework [R. Hiptmair, Comput. Math. with Appl. 52 (2006), pp.~699--706] to Petrov-Galerkin methods while accounting for parameter-dependent perturbations of both variational forms and their preconditioners, as occurs when performing numerical approximations. By considering different perturbation parameters for the original form and its preconditioner, our bi-parametric abstract setting leads to robust and controlled schemes. For Hilbert spaces, we derive exhaustive linear and super-linear convergence estimates for iterative solvers, such as $h$-independ…
Deep operator networks (DeepONets) are receiving increased attention thanks to their demonstrated capability to approximate nonlinear operators between infinite-dimensional Banach spaces. However, despite their remarkable early promise, they typically require large training data-sets consisting of paired input-output observations which may be expensive to obtain, while their predictions may not be consistent with the underlying physical principles that generated the observed data. In this work, we propose a novel model class coined as physics-informed DeepONets, which introduces an effective r…
Parametric Analytical Preconditioning and its Applications to the Reduced Collocation Methods
University of Massachusetts Dartmouth（马萨诸塞大学达特茅斯分校）；Sorbonne Universités, UPMC Univ Paris 06（索邦大学，巴黎第六大学）；Institut Universitaire de France（法国高等研究院）；Brown University（布朗大学）
This paper deals with the fast solution of linear systems associated with the mass matrix, in the context of isogeometric analysis. We propose a preconditioner that is both efficient and easy to implement, based on a diagonal-scaled Kronecker product of univariate parametric mass matrices. Its application is faster than a matrix-vector product involving the mass matrix itself. We prove that the condition number of the preconditioned matrix converges to 1 as the mesh size is reduced, that is, the preconditioner is asymptotically equivalent to the exact inverse. Moreover, we give numerical evide…
Freezing Calderón: Robust Preconditioning for Maxwell Scattering under Shape Uncertainty
Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez（阿道夫·伊巴涅斯大学工程与科学学院）；Department of Mathematics and Digital Futures, KTH Royal Institute of Technology（瑞典皇家理工学院数学与数字未来系）；Inria Chile（法国国家信息与自动化研究所智利分部）
We investigate a range of techniques for the acceleration of Calderón (operator) preconditioning in the context of boundary integral equation methods for electromagnetic transmission problems. Our objective is to mitigate as far as possible the high computational cost of the barycentrically-refined meshes necessary for the stable discretisation of operator products. Our focus is on the well-known PMCHWT formulation, but the techniques we introduce can be applied generically. By using barycentric meshes only for the preconditioner and not for the original boundary integral operator, we achieve …
Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method
Toulouse-INP (ENSEEIHT)/IRIT（图卢兹INP（ENSEEIHT）/IRIT）；Artificial and Natural Intelligence Toulouse Institute (ANITI)（人工智能与自然智能图卢兹研究所（ANITI））；Division of Applied Mathematics, Brown University（应用数学系，布朗大学）
In this work we focus on the preconditioning of a Galerkin space-time isogeometric discretization of the heat equation. Exploiting the tensor product structure of the basis functions in the parametric domain, we propose a preconditioner that is the sum of Kronecker products of matrices and that can be efficiently applied thanks to an extension of the classical Fast Diagonalization method. The preconditioner is robust w.r.t. the polynomial degree of the spline space and the time required for the application is almost proportional to the number of degrees-of-freedom, for a serial execution. By i…
Discontinuous Galerkin finite element operator network for solving non-smooth PDEs
Department of Mathematics, Florida State University（数学系，佛罗里达州立大学）；Department of Mathematical Sciences, Seoul National University（数学科学系，首尔国立大学）；Department of AI, Chung-Ang University（人工智能系， Chung-Ang 大学）
