7 research outputs found
Efficient Bayesian inference using physics-informed invertible neural networks for inverse problems
In the paper, we propose a novel approach for solving Bayesian inverse
problems with physics-informed invertible neural networks (PI-INN). The
architecture of PI-INN consists of two sub-networks: an invertible neural
network (INN) and a neural basis network (NB-Net). The invertible map between
the parametric input and the INN output with the aid of NB-Net is constructed
to provide a tractable estimation of the posterior distribution, which enables
efficient sampling and accurate density evaluation. Furthermore, the loss
function of PI-INN includes two components: a residual-based physics-informed
loss term and a new independence loss term. The presented independence loss
term can Gaussianize the random latent variables and ensure statistical
independence between two parts of INN output by effectively utilizing the
estimated density function. Several numerical experiments are presented to
demonstrate the efficiency and accuracy of the proposed PI-INN, including
inverse kinematics, inverse problems of the 1-d and 2-d diffusion equations,
and seismic traveltime tomography
Domain-decomposed Bayesian inversion based on local Karhunen-Lo\`{e}ve expansions
In many Bayesian inverse problems the goal is to recover a spatially varying
random field. Such problems are often computationally challenging especially
when the forward model is governed by complex partial differential equations
(PDEs). The challenge is particularly severe when the spatial domain is large
and the unknown random field needs to be represented by a high-dimensional
parameter. In this paper, we present a domain-decomposed method to attack the
dimensionality issue and the method decomposes the spatial domain and the
parameter domain simultaneously. On each subdomain, a local Karhunen-Lo`eve
(KL) expansion is constructed, and a local inversion problem is solved
independently in a parallel manner, and more importantly, in a
lower-dimensional space. After local posterior samples are generated through
conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel
projection procedure is developed to effectively reconstruct the global field.
In addition, the domain decomposition interface conditions are dealt with an
adaptive Gaussian process-based fitting strategy. Numerical examples are
provided to demonstrate the performance of the proposed method
Domain-decomposed Bayesian inversion based on local Karhunen-Loève expansions
In many Bayesian inverse problems the goal is to recover a spatially varying random field. Such problems are often computationally challenging especially when the forward model is governed by complex partial differential equations (PDEs). The challenge is particularly severe when the spatial domain is large and the unknown random field needs to be represented by a high-dimensional parameter. In this paper, we present a domain-decomposed method to attack the dimensionality issue and the method decomposes the spatial domain and the parameter domain simultaneously. On each subdomain, a local Karhunen-Loève (KL) expansion is constructed, and a local inversion problem is solved independently in a parallel manner, and more importantly, in a lower-dimensional space. After local posterior samples are generated through conducting Markov chain Monte Carlo (MCMC) simulations on subdomains, a novel projection procedure is developed to effectively reconstruct the global field. In addition, the domain decomposition interface conditions are dealt with an adaptive Gaussian process-based fitting strategy. Numerical examples are provided to demonstrate the performance of the proposed method