An Adaptive Surrogate Modeling Based on Deep Neural Networks for Large-Scale Bayesian Inverse Problems (Communications in Computational Physics, 2020)
In Bayesian inverse problems, surrogate models are often constructed tospeed up the computational procedure, as the parameter-to-data map can be veryexpensive to evaluate. However, due to the curse of dimensionality and the nonlinear concentration of the posterior, traditional surrogate approaches (such us thepolynomial-based surrogates) are still not feasible for large scale problems. To thisend, we present in this work an adaptive multi-fidelity surrogate modeling framework based on deep neural networks (DNNs), motivated by the facts that the DNNscan potentially handle functions with limited regularity and are powerful tools forhigh dimensional approximations. More precisely, we first construct offline a DNN-based surrogate according to the prior distribution, and then, this prior-based DNN-surrogate will be adaptively & locally refined online using only a few high-fidelitysimulations. In particular, in the refine procedure, we construct a new shallow neuralnetwork that views the previous constructed surrogate as an input variable – yieldinga composite multi-fidelity neural network approach. This makes the online computational procedure rather efficient. Numerical examples are presented to confirm that theproposed approach can obtain accurate posterior information with a limited numberof forward simulations.
Keywords: Bayesian inverse problems; deep neural networks; multi-fidelity surrogate modeling; Markov chain Monte Carlo.