Error analysis of kernel/GP methods for nonlinear and parametric PDEs
DOI10.1016/j.jcp.2024.113488MaRDI QIDQ6648398
Yi-Fan Chen, Andrew M. Stuart, Houman Owhadi, Bamdad Hosseini, Pau Batlle
Publication date: 4 December 2024
Published in: Journal of Computational Physics (Search for Journal in Brave)
Gaussian processes (60G15) Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs (65N75) Spectral, collocation and related methods for boundary value problems involving PDEs (65N35) Nonlinear boundary value problems for ordinary differential equations (34B15) Inverse problems for PDEs (35R30) Spline approximation (41A15) Kernel operators (47B34) Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs (65M75)
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Cites Work
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- Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity
- High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs
- Bayesian solution uncertainty quantification for differential equations
- Multiquadrics - a scattered data approximation scheme with applications to computational fluid-dynamics. I: Surface approximations and partial derivative estimates
- Convergence rates of best \(N\)-term Galerkin approximations for a class of elliptic SPDEs
- Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
- Convergence order estimates of meshless collocation methods using radial basis functions
- Solving partial differential equations by collocation using radial basis functions
- On \(n\)-widths for elliptic problems
- The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems
- Multiquadrics -- a scattered data approximation scheme with applications to computational fluid-dynamics. II: Solutions to parabolic, hyperbolic and elliptic partial differential equations
- Results on meshless collocation techniques
- Solving differential equations with radial basis functions: Multilevel methods and smoothing
- A nonlinear discretization theory
- Bayesian numerical methods for nonlinear partial differential equations
- Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs
- One-shot learning of stochastic differential equations with data adapted kernels
- Do ideas have shape? Idea registration as the continuous limit of artificial neural networks
- Solving and learning nonlinear PDEs with Gaussian processes
- Numerical methods for mean field games based on Gaussian processes and Fourier features
- Kernel flows: from learning kernels from data into the abyss
- Kernel-based reconstructions for parametric PDEs
- Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
- A computational tool for comparing all linear PDE solvers
- An extension of a bound for functions in Sobolev spaces, with applications to \((m, s)\)-spline interpolation and smoothing
- Solvability of partial differential equations by meshless kernel methods
- Gaussian measure in Hilbert space and applications in numerical analysis
- Bayesian Calibration of Computer Models
- Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
- Scattered Data Interpolation on Embedded Submanifolds with Restricted Positive Definite Kernels: Sobolev Error Estimates
- Certified Reduced Basis Methods for Parametrized Partial Differential Equations
- Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations
- ON THE OPTIMAL POLYNOMIAL APPROXIMATION OF STOCHASTIC PDES BY GALERKIN AND COLLOCATION METHODS
- Kernel techniques: From machine learning to meshless methods
- Meshless Collocation: Error Estimates with Application to Dynamical Systems
- Pushing the Limits of Contemporary Statistics: Contributions in Honor of Jayanta K. Ghosh
- Stable and Convergent Unsymmetric Meshless Collocation Methods
- Spectral Methods for Uncertainty Quantification
- A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
- An Anisotropic Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
- Elliptic Partial Differential Equations of Second Order
- On the small balls problem for equivalent Gaussian measures
- The Covering Radius of Randomly Distributed Points on a Manifold
- Kernel Mean Embedding of Distributions: A Review and Beyond
- $H^2$-Convergence of Least-Squares Kernel Collocation Methods
- Stochastic finite element methods for partial differential equations with random input data
- Approximation of stochastic partial differential equations by a kernel-based collocation method
- Can a finite element method perform arbitrarily badly?
- Consistency of empirical Bayes and kernel flow for hierarchical parameter estimation
- Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization
- Sparse Cholesky Factorization by Kullback--Leibler Minimization
- On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs
- Bayesian Probabilistic Numerical Methods
- Approximation of high-dimensional parametric PDEs
- Solving PDEs with radial basis functions
- Bayesian Numerical Homogenization
- Convergence of Unsymmetric Kernel‐Based Meshless Collocation Methods
- All well-posed problems have uniformly stable and convergent discretizations
- Cost-optimal adaptive iterative linearized FEM for semilinear elliptic PDEs
- A nonlinear discretization theory for meshfree collocation methods applied to quasilinear elliptic equations
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