Approximation of Bayesian inverse problems for PDEs [PDF]
SIAM Journal on Numerical Analysis, 2009 Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability.
A. M. Stuart +3 more
core +14 more sources
Bi-level iterative regularization for inverse problems in nonlinear PDEs [PDF]
Inverse Problems, 2023 We investigate the ill-posed inverse problem of recovering unknown spatially dependent parameters in nonlinear evolution partial differential equations (PDEs).
Tram Thi Ngoc Nguyen
semanticscholar +5 more sources
Fully probabilistic deep models for forward and inverse problems in parametric PDEs [PDF]
Journal of Computational Physics, 2022 We introduce a physics-driven deep latent variable model (PDDLVM) to learn simultaneously parameter-to-solution (forward) and solution-to-parameter (inverse) maps of parametric partial differential equations (PDEs). Our formulation leverages conventional PDE discretization techniques, deep neural networks, probabilistic modelling, and variational ...
Arnaud Vadeboncoeur +4 more
semanticscholar +8 more sources
A deep neural network approach for parameterized PDEs and Bayesian inverse problems
Machine Learning: Science and Technology, 2023 We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems.
Harbir Antil +3 more
doaj +2 more sources
Asymptotic expansion regularization for inverse source problems in two-dimensional singularly perturbed nonlinear parabolic PDEs [PDF]
CSIAM Transactions on Applied Mathematics, 2022 In this paper, we develop an asymptotic expansion-regularization (AER) method for inverse source problems in two-dimensional nonlinear and nonstationary singularly perturbed partial differential equations (PDEs).
Dmitrii Chaikovskii +2 more
openalex +3 more sources
Uncertainty Quantification for Forward and Inverse Problems of PDEs via Latent Global Evolution [PDF]
AAAI Conference on Artificial IntelligenceDeep learning-based surrogate models have demonstrated remarkable advantages over classical solvers in terms of speed, often achieving speedups of 10 to 1000 times over traditional partial differential equation (PDE) solvers.
Tailin Wu +4 more
openalex +2 more sources
Optimal experimental design for infinite-dimensional Bayesian inverse problems governed by PDEs: a review [PDF]
, 2021 We present a review of methods for optimal experimental design (OED) for Bayesian inverse problems governed by partial differential equations with infinite-dimensional parameters.
Alen Alexanderian
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The Hadamard-PINN for PDE inverse problems: Convergence with distant initial guesses
Examples and CounterexamplesThis paper presents the Hadamard-Physics-Informed Neural Network (H-PINN) for solving inverse problems in partial differential equations (PDEs), specifically the heat equation and the Korteweg–de Vries (KdV) equation.
Yohan Chandrasukmana +2 more
doaj +2 more sources
Kernel-Adaptive PI-ELMs for Forward and Inverse Problems in PDEs with Sharp Gradients [PDF]
arXiv.orgThis paper introduces the Kernel Adaptive Physics-Informed Extreme Learning Machine (KAPI-ELM), an adaptive Radial Basis Function (RBF)-based extension of PI-ELM designed to solve both forward and inverse Partial Differential Equation (PDE) problems ...
Vikas Dwivedi +3 more
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An Adaptive Sampling Algorithm with Dynamic Iterative Probability Adjustment Incorporating Positional Information [PDF]
EntropyPhysics-informed neural networks (PINNs) have garnered widespread use for solving a variety of complex partial differential equations (PDEs). Nevertheless, when addressing certain specific problem types, traditional sampling algorithms still reveal ...
Yanbing Liu +3 more
doaj +2 more sources