Variational Inverting Network for Statistical Inverse Problems of Partial Differential Equations
Authors: Junxiong Jia, Yanni Wu, Peijun Li, Deyu Meng; 24(201):1−60, 2023.
Abstract
In order to address the uncertainties in inverse problems of partial differential equations (PDEs), a statistical inference approach using Bayes’ formula is formulated. Although infinite-dimensional Bayesian analysis methods have been developed to construct dimension-independent algorithms, there are challenges that need to be overcome. These challenges include the inefficient incorporation of prior information by prior measures acting as regularizers, the limited consideration of complex noises such as non-i.i.d. distributed noises, and the requirement of time-consuming forward PDE solvers to estimate posterior statistical quantities. To tackle these challenges, an infinite-dimensional inference framework called the Variational Inverting Network (VINet) is proposed based on the infinite-dimensional variational inference method and deep generative models. By introducing measure equivalence assumptions, the evidence lower bound in the infinite-dimensional setting is derived, and parametric strategies are provided to yield the VINet inference framework. This framework is capable of encoding prior and noise information from learning examples. Utilizing the power of deep neural networks, the posterior mean and variance can be efficiently and explicitly generated during the inference stage. Numerical experiments are conducted to demonstrate the effectiveness of the proposed inference framework, including linear inverse problems of an elliptic equation and the Helmholtz equation.
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