Ángel J. Omella

Ángel J. Omella, David Pardo

We introduce an $r-$adaptive algorithm to solve Partial Differential Equations using a Deep Neural Network. The proposed method restricts to tensor product meshes and optimizes the boundary node locations in one dimension, from which we build two- or three-dimensional meshes. The method allows the definition of fixed interfaces to design conforming meshes, and enables changes in the topology, i.e., some nodes can jump across fixed interfaces. The method simultaneously optimizes the node locations and the PDE solution values over the resulting mesh. To numerically illustrate the performance of our proposed $r-$adaptive method, we apply it in combination with a collocation method, a Least Squares Method, and a Deep Ritz Method. We focus on the latter to solve one- and two-dimensional problems whose solutions are smooth, singular, and/or exhibit strong gradients.

Sergey Alyaev, Mostafa Shahriari, David Pardo et al.

Modern geosteering is heavily dependent on real-time interpretation of deep electromagnetic (EM) measurements. We present a methodology to construct a deep neural network (DNN) model trained to reproduce a full set of extra-deep EM logs consisting of 22 measurements per logging position. The model is trained in a 1D layered environment consisting of up to seven layers with different resistivity values. A commercial simulator provided by a tool vendor is used to generate a training dataset. The dataset size is limited because the simulator provided by the vendor is optimized for sequential execution. Therefore, we design a training dataset that embraces the geological rules and geosteering specifics supported by the forward model. We use this dataset to produce an EM simulator based on a DNN without access to the proprietary information about the EM tool configuration or the original simulator source code. Despite employing a relatively small training set size, the resulting DNN forward model is quite accurate for the considered examples: a multi-layer synthetic case and a section of a published historical operation from the Goliat Field. The observed average evaluation time of 0.15 ms per logging position makes it also suitable for future use as part of evaluation-hungry statistical and/or Monte-Carlo inversion algorithms within geosteering workflows.