Machine Learning Discovery of Optimal Quadrature Rules for Isogeometric Analysis

We propose the use of machine learning techniques to find optimal quadrature rules for the construction of stiffness and mass matrices in isogeometric analysis (IGA). We initially consider 1D spline spaces of arbitrary degree spanned over uniform and non-uniform knot sequences, and then the generated optimal rules are used for integration over higher-dimensional spaces using tensor product sense. The quadrature rule search is posed as an optimization problem and solved by a machine learning strategy based on gradient-descent. However, since the optimization space is highly non-convex, the success of the search strongly depends on the number of quadrature points and the parameter initialization. Thus, we use a dynamic programming strategy that initializes the parameters from the optimal solution over the spline space with a lower number of knots. With this method, we found optimal quadrature rules for spline spaces when using IGA discretizations with up to 50 uniform elements and polynomial degrees up to 8, showing the generality of the approach in this scenario. For non-uniform partitions, the method also finds an optimal rule in a reasonable number of test cases. We also assess the generated optimal rules in two practical case studies, namely, the eigenvalue problem of the Laplace operator and the eigenfrequency analysis of freeform curved beams, where the latter problem shows the applicability of the method to curved geometries. In particular, the proposed method results in savings with respect to traditional Gaussian integration of up to 44% in 1D, 68% in 2D, and 82% in 3D spaces.

A Deep Double Ritz Method (D$^2$RM) for solving Partial Differential Equations using Neural Networks

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. In the context of neural networks, we can address this min-max approach by employing one network to seek the trial minimum, while another network seeks the test maximizers. However, the resulting method is numerically unstable as we approach the trial solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. We call the resulting scheme the Deep Double Ritz Method (D$^2$RM), which combines two neural networks for approximating trial functions and optimal test functions along a nested double Ritz minimization strategy. Numerical results on several 1D diffusion and convection problems support the robustness of our method, up to the approximation properties of the networks and the training capacity of the optimizers.

A Deep Double Ritz Method for solving Partial Differential Equations

Residual minimization is a widely used technique for solving Partial Differential Equations in variational form. It minimizes the dual norm of the residual, which naturally yields a saddle-point (min-max) problem over the so-called trial and test spaces. Such min-max problem is highly non-linear, and traditional methods often employ different mixed formulations to approximate it. Alternatively, it is possible to address the above saddle-point problem by employing Adversarial Neural Networks: one network approximates the global trial minimum, while another network seeks the test maximizer. However, this approach is numerically unstable due to a lack of continuity of the text maximizers with respect to the trial functions as we approach the exact solution. To overcome this, we reformulate the residual minimization as an equivalent minimization of a Ritz functional fed by optimal test functions computed from another Ritz functional minimization. The resulting Deep Double Ritz Method combines two Neural Networks for approximating the trial and optimal test functions. Numerical results on several 1D diffusion and convection problems support the robustness of our method up to the approximability and trainability capacity of the networks and the optimizer.

Deep-Learning Inversion Method for the Interpretation of Noisy Logging-While-Drilling Resistivity Measurements

Deep Learning (DL) inversion is a promising method for real time interpretation of logging while drilling (LWD) resistivity measurements for well navigation applications. In this context, measurement noise may significantly affect inversion results. Existing publications examining the effects of measurement noise on DL inversion results are scarce. We develop a method to generate training data sets and construct DL architectures that enhance the robustness of DL inversion methods in the presence of noisy LWD resistivity measurements. We use two synthetic resistivity models to test three approaches that explicitly consider the presence of noise: (1) adding noise to the measurements in the training set, (2) augmenting the training set by replicating it and adding varying noise realizations, and (3) adding a noise layer in the DL architecture. Numerical results confirm that the three approaches produce a denoising effect, yielding better inversion results in both predicted earth model and measurements compared not only to the basic DL inversion but also to traditional gradient based inversion results. A combination of the second and third approaches delivers the best results. The proposed methods can be readily generalized to multi dimensional DL inversion.

Modeling extra-deep EM logs using a deep neural network

Modern geosteering is heavily dependent on real-time interpretation of deep electromagnetic (EM) measurements. This work presents a deep neural network (DNN) model trained to reproduce the full set of extra-deep real-time EM logs consisting of 22 measurements per logging position. The model is trained in a 1D layered environment and has sensitivity for up to seven layers with different resistivity values. A commercial simulator provided by a tool vendor is utilized to generate a training dataset. The impossibility of parallel execution of the simulator effectively limits the permissible dataset size. Therefore, the geological rules and geosteering specifics supported by the forward model are embraced when designing the dataset. It is then used to produce a fully parallel EM simulator based on a DNN without access to the proprietary information about the EM tool configuration or the original simulator source code. Despite a relatively small training set size, the resulting DNN forward model is quite accurate for synthetic geosteering cases, yet independent of the logging instrument vendor. The observed average evaluation time of 0.15 milliseconds per logging position makes it also suitable for future use as part of evaluation-hungry statistical and/or Monte-Carlo inversion algorithms.