Physics-informed PointNet: A deep learning solver for steady-state incompressible flows and thermal fields on multiple sets of irregular geometries

We present a novel physics-informed deep learning framework for solving steady-state incompressible flow on multiple sets of irregular geometries by incorporating two main elements: using a point-cloud based neural network to capture geometric features of computational domains, and using the mean squared residuals of the governing partial differential equations, boundary conditions, and sparse observations as the loss function of the network to capture the physics. While the solution of the continuity and Navier-Stokes equations is a function of the geometry of the computational domain, current versions of physics-informed neural networks have no mechanism to express this functionally in their outputs, and thus are restricted to obtain the solutions only for one computational domain with each training procedure. Using the proposed framework, three new facilities become available. First, the governing equations are solvable on a set of computational domains containing irregular geometries with high variations with respect to each other but requiring training only once. Second, after training the introduced framework on the set, it is now able to predict the solutions on domains with unseen geometries from seen and unseen categories as well. The former and the latter both lead to savings in computational costs. Finally, all the advantages of the point-cloud based neural network for irregular geometries, already used for supervised learning, are transferred to the proposed physics-informed framework. The effectiveness of our framework is shown through the method of manufactured solutions and thermally-driven flow for forward and inverse problems

ChatGPT for Programming Numerical Methods

ChatGPT is a large language model recently released by the OpenAI company. In this technical report, we explore for the first time the capability of ChatGPT for programming numerical algorithms. Specifically, we examine the capability of GhatGPT for generating codes for numerical algorithms in different programming languages, for debugging and improving written codes by users, for completing missed parts of numerical codes, rewriting available codes in other programming languages, and for parallelizing serial codes. Additionally, we assess if ChatGPT can recognize if given codes are written by humans or machines. To reach this goal, we consider a variety of mathematical problems such as the Poisson equation, the diffusion equation, the incompressible Navier-Stokes equations, compressible inviscid flow, eigenvalue problems, solving linear systems of equations, storing sparse matrices, etc. Furthermore, we exemplify scientific machine learning such as physics-informed neural networks and convolutional neural networks with applications to computational physics. Through these examples, we investigate the successes, failures, and challenges of ChatGPT. Examples of failures are producing singular matrices, operations on arrays with incompatible sizes, programming interruption for relatively long codes, etc. Our outcomes suggest that ChatGPT can successfully program numerical algorithms in different programming languages, but certain limitations and challenges exist that require further improvement of this machine learning model

Physics-informed PointNet: On how many irregular geometries can it solve an inverse problem simultaneously? Application to linear elasticity

Regular physics-informed neural networks (PINNs) predict the solution of partial differential equations using sparse labeled data but only over a single domain. On the other hand, fully supervised learning models are first trained usually over a few thousand domains with known solutions (i.e., labeled data) and then predict the solution over a few hundred unseen domains. Physics-informed PointNet (PIPN) is primarily designed to fill this gap between PINNs (as weakly supervised learning models) and fully supervised learning models. In this article, we demonstrate that PIPN predicts the solution of desired partial differential equations over a few hundred domains simultaneously, while it only uses sparse labeled data. This framework benefits fast geometric designs in the industry when only sparse labeled data are available. Particularly, we show that PIPN predicts the solution of a plane stress problem over more than 500 domains with different geometries, simultaneously. Moreover, we pioneer implementing the concept of remarkable batch size (i.e., the number of geometries fed into PIPN at each sub-epoch) into PIPN. Specifically, we try batch sizes of 7, 14, 19, 38, 76, and 133. Additionally, the effect of the PIPN size, symmetric function in the PIPN architecture, and static and dynamic weights for the component of the sparse labeled data in the loss function are investigated

Real-Time Well Log Prediction From Drilling Data Using Deep Learning

The objective is to study the feasibility of predicting subsurface rock properties in wells from real-time drilling data. Geophysical logs, namely, density, porosity and sonic logs are of paramount importance for subsurface resource estimation and exploitation. These wireline petro-physical measurements are selectively deployed as they are expensive to acquire; meanwhile, drilling information is recorded in every drilled well. Hence a predictive tool for wireline log prediction from drilling data can help management make decisions about data acquisition, especially for delineation and production wells. This problem is non-linear with strong ineractions between drilling parameters; hence the potential for deep learning to address this problem is explored. We present a workflow for data augmentation and feature engineering using Distance-based Global Sensitivity Analysis. We propose an Inception-based Convolutional Neural Network combined with a Temporal Convolutional Network as the deep learning model. The model is designed to learn both low and high frequency content of the data. 12 wells from the Equinor dataset for the Volve field in the North Sea are used for learning. The model predictions not only capture trends but are also physically consistent across density, porosity, and sonic logs. On the test data, the mean square error reaches a low value of 0.04 but the correlation coefficient plateaus around 0.6. The model is able however to differentiate between different types of rocks such as cemented sandstone, unconsolidated sands, and shale

Computation of effective elastic moduli of rocks using hierarchical homogenization

This work focuses on computing the homogenized elastic properties of rocks from 3D micro-computed-tomography (micro-CT) scanned images. The accurate computation of homogenized properties of rocks, archetypal random media, requires both resolution of intricate underlying microstructure and large field of view, resulting in huge micro-CT images. Homogenization entails solving the local elasticity problem computationally which can be prohibitively expensive for a huge image. To mitigate this problem, we use a renormalization method inspired scheme, the hierarchical homogenization method, where a large image is partitioned into smaller subimages. The individual subimages are separately homogenized using periodic boundary conditions, and then assembled into a much smaller intermediate image. The intermediate image is again homogenized, subject to the periodic boundary condition, to find the final homogenized elastic constant of the original image. An FFT-based elasticity solver is used to solve the associated periodic elasticity problem. The error in the homogenized elastic constant is empirically shown to follow a power law scaling with exponent -1 with respect to the subimage size across all five microstructures of rocks. We further show that the inclusion of surrounding materials during the homogenization of the small subimages reduces error in the final homogenized elastic moduli while still respecting the power law with the exponent of -1. This power law scaling is then exploited to determine a better approximation of the large heterogeneous microstructures based on Richardson extrapolatio

Do the Seismic Velocities Depend on

The objective of our study is to test whether seismic velocities of rock depend on time and temperature index (TTI). This study is motivated by the observations that overpressure and reservoir qualities depend on temperature and time in many sedimentary basins. TTI, an important thermal maturity indicator, is directly linked with oil and gas generation and combines the effects of temperature and time. However, there is no existing model (theoretical, empirical, and numerical) to predict TTI from observed seismic velocities. Our study identifies an empirical-numerical relation between TTI and seismic velocities. In order to obtain this relation, we perform petroleum system modeling at a well location. The well is located in deep-water petroleum system at Rio Muni Basin, West Africa. The essential petroleum system elements and TOC are identified based on petrophysical and rock-physics analysis. The TTIs obtained from finite-element modeling of petroleum system are then compared with velocities measured at the same well location. We find that both Vp and Vs increase exponentially with TTI. The results can be applied to predict TTI, and thereby thermal maturity, from observed seismic velocities.