An adaptive surrogate modeling based on deep neural networks for large-scale Bayesian inverse problems
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It is popular approaches to use surrogate models to speed up the computational procedure for Bayesian inverse problems (BIPs). For instance, the polynomial chaos expansions (PCE) are often combined with the Markov chain Monte Carlo sampling to accelerate the online computations. While this approach can be very efficient, there exist, however, two main limitations: (i) the PCE surrogate admits limitations to handle problems with low regularity; (ii) the PCE surrogate suffers from the so called curse of dimensionality. To this end, we present in this work an adaptive multi-fidelity deep neural networks (DNNs) based surrogate modeling for large-scale BIPs, motivated by the facts that the DNNs can potentially handle functions with limited regularity and are powerful tools for high dimensional problems. More precisely, we begin with a low fidelity DNN-surrogate and then correct it adaptively using online high fidelity data. The key idea is to view the low fidelity surrogate as an input variable into the DNN-surrogate of the next iteration – yielding a composite DNN that combine two surrogates between two iterations. By doing this, the online correction procedure can be made very efficient. Numerical experiments confirm that the proposed approach can obtain accurate posterior information with limited number of forward simulations.
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