Surrogate strategies are used widely for uncertainty quantification of groundwater models in order to improve computational efficiency. However, their application to dynamic multiphase flow problems is hindered by the curse of dimensionality, the saturation discontinuity due to capillarity effects, and the time-dependence of the multi-output responses. In this paper, we propose a deep convolutional encoder-decoder neural network methodology to tackle these issues. The surrogate modeling task is transformed to an image-to-image regression strategy. This approach extracts high-level coarse features from the high-dimensional input permeability images using an encoder, and then refines the coarse features to provide the output pressure/saturation images through a decoder. A training strategy combining a regression loss and a segmentation loss is proposed in order to better approximate the discontinuous saturation field. To characterize the high-dimensional time-dependent outputs of the dynamic system, time is treated as an additional input to the network that is trained using pairs of input realizations and of the corresponding system outputs at a limited number of time instances. The proposed method is evaluated using a geological carbon storage process-based multiphase flow model with a 2500-dimensional stochastic permeability field. With a relatively small number of training data, the surrogate model is capable of accurately characterizing the spatio-temporal evolution of the pressure and discontinuous CO2 saturation fields and can be used efficiently to compute the statistics of the system responses.
This paper presents a semisupervised system responses evaluations, easy-to-evaluate surrogate models learning framework for dimension reduction and have been utilized as substitutes for computationally expensive reliability analysis. An autoencoder is first adopted for mapping simulations or experiments. Popular choices for surrogate the high-dimensional space into a low-dimensional latent space, models in the literature include, support vector machines (SVM) which contains a distinguishable failure surface. Then a deep [4-7], Kriging models [8-10], and artificial neural networks [11-feedforward neural network (DFN) is utilized to learn the 14]. Given a set of training data, surrogate models can be mapping relationship and reconstruct the latent space, while the constructed and then MCS can be directly carried out for Gaussian process (GP) modeling technique is used to build the reliability analysis. Research efforts have been devoted to surrogate model of the transformed limit state function. During developing adaptive sampling strategies [15-18], which aim at the training process of the DFN, the discrepancy between the balancing the fidelity of the surrogate model and the costs of actual and reconstructed latent space is minimized through semisupervised function evaluations.
Characterization of a non-Gaussian channelized conductivity field in subsurface flow and transport modeling through inverse modeling usually leads to a high-dimensional inverse problem and requires repeated evaluations of the forward model. In this study, we develop a convolutional adversarial autoencoder (CAAE) network to parameterize the high-dimensional non-Gaussian conductivity fields using a low-dimensional latent representation and a deep residual dense convolutional network (DRDCN) to efficiently construct a surrogate model for the forward model. The two networks are both based on a multilevel residual learning architecture called residual-in-residual dense block. The multilevel residual learning strategy and the dense connection structure in the dense block ease the training of deep networks, enabling us to efficiently build deeper networks that have an essentially increased capacity for approximating mappings of very high-complexity. The CCAE and DRDCN networks are incorporated into an iterative local updating ensemble smoother to formulate an inversion framework. The integrated method is demonstrated using a synthetic solute transport model. Results indicate that CAAE is a robust parameterization method for the channelized conductivity fields with Gaussian conductivities within each facies. The DRDCN network is able to obtain an accurate surrogate model of the forward model with high-dimensional and highly-complex concentration fields using relatively limited training data. The CAAE paramterization approach and the DRDCN surrogate method together significantly reduce the number of forward model runs required to achieve accurate inversion results.
Non-intrusive reduced-order models (ROMs) have recently generated considerable interest for constructing computationally efficient counterparts of nonlinear dynamical systems emerging from various domain sciences. They provide a low-dimensional emulation framework for systems that may be intrinsically high-dimensional. This is accomplished by utilizing a construction algorithm that is purely data-driven. It is no surprise, therefore, that the algorithmic advances of machine learning have led to non-intrusive ROMs with greater accuracy and computational gains. However, in bypassing the utilization of an equation-based evolution, it is often seen that the interpretability of the ROM framework suffers. This becomes more problematic when black-box deep learning methods are used which are notorious for lacking robustness outside the physical regime of the observed data. In this article, we propose the use of a novel latent-space interpolation algorithm based on Gaussian process regression. Notably, this reduced-order evolution of the system is parameterized by control parameters to allow for interpolation in space. The use of this procedure also allows for a continuous interpretation of time which allows for temporal interpolation. The latter aspect provides information, with quantified uncertainty, about full-state evolution at a finer resolution than that utilized for training the ROMs. We assess the viability of this algorithm for an advection-dominated system given by the inviscid shallow water equations.
Driven by increased complexity of dynamical systems, the solution of system of differential equations through numerical simulation in optimization problems has become computationally expensive. This paper provides a smart data driven mechanism to construct low dimensional surrogate models. These surrogate models reduce the computational time for solution of the complex optimization problems by using training instances derived from the evaluations of the true objective functions. The surrogate models are constructed using combination of proper orthogonal decomposition and radial basis functions and provides system responses by simple matrix multiplication. Using relative maximum absolute error as the measure of accuracy of approximation, it is shown surrogate models with latin hypercube sampling and spline radial basis functions dominate variable order methods in computational time of optimization, while preserving the accuracy. These surrogate models also show robustness in presence of model non-linearities. Therefore, these computational efficient predictive surrogate models are applicable in various fields, specifically to solve inverse problems and optimal control problems, some examples of which are demonstrated in this paper.