Artificial Neural Networks (ANNs) have aroused great interest of researchers as an option to solve or approximate complex engineering problems. In Anitescu et al. , it has been used to solve boundary value problem (BVP) for second order partial differential equations (PDEs) based on minimizing the combined error in PDE at multiple collocation points inside the domain and at the boundary conditions. In such an approach, ANNs have shown the potential of being an alternative of finite element method (FEM) which is commonly used for solving BVPs.
The boundary element method (BEM) has been a common alternative of FEM in solving BVPs. In many practical applications, BVPs could be transformed to a boundary integral equation (BIE) and then solved by BEM. BEM has several commendable advantages over domain-type methods, such as lowering the dimension of the problem to reduce the computational cost and avoiding domain truncation error in exterior domains. In Simpson et al. , an isogeometric boundary element method (IGABEM) was proposed, which combines BEM with Non-Uniform Rational B-Splines (NURBS) and further improves the accuracy of BEM.
In this work, we proposed a new approach for solving BVPs formulated as BIEs. The approach consists in using deep neural network to approximate the solution of BIEs. Loss function is set to estimate the error at the collocation points and is subsequently minimized by the appropriate weights and biases. The approach preserves the main advantage of IGABEM, i.e. using NURBS to describe the boundary, hence keeping the exact geometry and providing the tight link with CAD. The approach inherits the main advantages of the boundary-type methods and allows to reduce the computational cost by using collocation points on the boundary only, which is very beneficial in practical applications where only the boundary data measurements are available.
Application of the method to some benchmark problems for the Laplace equation is demonstrated. The results are presented in terms of the loss function convergence plots and error norms. A detailed parametric study is presented to evaluate the performance of the method. It is shown that the method is highly accurate for Dirichlet, Neumann and mixed problems in continuous domain, however the accuracy deteriorates in presence of corners.
In future studies, accuracy of the method can be improved by varying the choice of the activation function and the network architecture. The method can be extended to other applications by changing the BIE. In the current study, the method was implemented for 2D applications, but extension to 3D is also possible and it is an objective of future works.
 Anitescu et al. Artificial neural network methods for the solution of second order boundary value problems. 2019
 Simpson et al. A two-dimensional isogeometric boundary element method for elastostatic analysis. 2012
Physics-informed neural networks (PINNs) have been studied and used widely in the field of computational mechanics. In this study, we proposed a new approach to merge PINNs with (geometry-based) constrained neural networks. Using this method, we reconstruct the velocity fields of particles moving in the fixed bed. Following that, the trained neural network is used to predict the motion of particles for a period of time more than the training time, and the results are compared with the simulation data in terms of accuracy and CPU time.
Service now is an important tool in an industry that helps internal employees to report and solve their issues. In an organization that receives an average of 7000+ IT related incidents in six months, human assisted resolving of issues is cumbersome. Therefore, my thesis focuses on designing an artificial intelligence assisted chat-bot that uses machine learning algorithms to bring up a precise solution that could bring an automated response to the user. In this talk I will be discussing about the challenges that we face in an industrial setting, the roughness of data, prepossessing them and finally retrieving the relevant information from the data. I will be discussing about the libraries that was used to simplify the text data and the type conversions involved. I propose to discuss in this presentation about the input given to the Bert model and the working of the transformer model itself.
GPT models (Generative Pre-trained Transformer) have increasingly become a popular choice by researchers and practitioners. Their success is mostly due to the technology’s ability to move beyond single word predictions. Indeed, unlike in traditional neural network language models, GPT generates text by looking at the entirety of the input text. Thus, rather than determining relevance sequentially by looking at the most recent segment of input, GPT models determine each word’s relevance selectively. However, if on the one hand this ability allows the machine to ‘learn’ faster, the datasets used for training have to be fed as one single document, meaning that all metadata information is inevitably lost (e.g., date, authors, original source). Moreover, as GPT models are trained on crawled, English web material from 2016, these models are not only ignorant of the world prior to this date, but they also express the language as used exclusively by English-speaking users (mostly white, young males). They also expect data pristine in quality, in the sense that these models have been trained on digitally-born material which do not present the typical problems of digitized, historical content (e.g., OCR mistakes, unusual fonts). Although a powerful technology, these issues seriously hinder its application for humanistic enquiry, particularly historical. In this presentation, I discuss these and other problematic aspects of GPT and I present the specific challenges I encountered while working on a historical archive of Italian American immigrant newspapers.
Studying flows in random porous media leads to consider a permeability tensor which directly depends on the pore geometry. The latter can be characterised through the computation of various morphological parameters: Delaunay triangulation characteristics, nearest neighbour distance,… The natural question is: which morphological parameters provide the best statistical description of permeability? This question can be difficult to answer for several reasons: non-linear correlation between input parameters, non-linear correlation between inputs and outputs, small dataset, variability,…
A method of feature selection based on Gaussian Process Regression has been proposed. It can be applied to a wide range of applications where the parameters that best explain a given output are sought among a set of correlated features. The method uses anisotropic kernel that associates a hyperparameter to each feature. These hyperparameters can be interpreted as correlation lengths providing an estimation of the weight of each feature w.r.t the output.
This seminar talk addresses certain challenges associated with data-driven modelling of advanced materials — with special interest in the non-linear deformation response of rubber-like materials, soft polymers or biological tissue. The underlying (isotropic) hyperelastic deformation problem is formulated in the principal space, using principal stretches and principal stresses. The sought data-driven constitutive relation is expressed in terms of these principal quantities and to be captured by a non-parametric representation using a trained artificial neural network (ANN).
The presentation investigates certain physics-motivated consistency requirements (e.g. limit behaviour, monotonicity) for the ANN-based prediction of principal stresses for given principal stretches, and discusses the implications on the architecture of such constitutive ANNs. The neural network is exemplarily constructed, trained and tested using PyTorch.
The computational embedding of the data-driven material descriptor is demonstrated for the open-source finite element framework FEniCS which builds on the symbolic representation of the constitutive ANN operator in the Unified Form Language (UFL). We discuss the performance of the overall formulation within the non-linear solution process and will explain some future directions of research.
Recently, with the increase in GPU computational power, deep learning started to revolutionise several fields, in particular computer vision, language processing, and image processing. Deep learning engineering can be split into three main categories: Dataset, Network Architecture and Learning policy. Setting the network architecture to its simplest form, we will modify the dataset and learning policy formulation using the Finite Element Method to improve the training.
The Finite Element Method is often used as the numerical method of reference for solving the PDE associated with non-linear object deformations. In order to solve the resulting energy minimisation equations, root-finding algorithms such as the Newton-Raphson method are used. During its iterative process, the Newton-Raphson reveals important information about the state of the system which can be used in both the dataset formulation and the training policy.
The numerical simulation of physical systems typically involves the solution of large-scale systems of equations resulting from the discretization of PDEs. Model order reduction techniques are very advantageous to overcome this computational hurdle. Based on the proper orthogonal decomposition approach, the talk will present a model order reduction approach for parametric high dimensional convection-diffusion-reaction partial differential equations. The proper orthogonal decomposition requires solving the high dimensional model for some training parameters to obtain the reduced basis. In this work, the training parameters are chosen based on greedy sampling approaches. We propose an adaptive greedy sampling approach that utilizes an optimized search based on surrogate modeling for the selection of the training parameter set. The work also presents an elegant approach for monitoring the convergence of the developed greedy sampling approach along with its error and sensitivity analysis.
The developed techniques are analyzed, implemented, and tested on industrial data of a floater with caps and floors under the one-factor Hull-White model. The results illustrate that the model order reduction approach provides a significant speedup with excellent accuracy for short-rate models.
This talk will introduce two new tools for summarizing a probability distribution more effectively than independent sampling or standard Markov chain Monte Carlo thinning:
1. Given an initial n point summary (for example, from independent sampling or a Markov chain), kernel thinning finds a subset of only square-root n points with comparable worst-case integration error across a reproducing kernel Hilbert space.
2. If the initial summary suffers from biases due to off-target sampling, tempering, or burn-in, Stein thinning simultaneously compresses the summary and improves the accuracy by correcting for these biases.
These tools are especially well-suited for tasks that incur substantial downstream computation costs per summary point like organ and tissue modeling in which each simulation consumes 1000s of CPU hours.
Bayesian inference provides a unified framework for quantifying uncertainty in probabilistic models with latent variables. However, exact inference algorithms generally scale poorly with the dimensionality of the model and the size of the data. To overcome the issue of scaling, the ML community has turned to approximate inference. For the Big Data case, the most prominent method is Variational inference (VI), which uses a simpler parametric model to approximate the target distribution of the latent variables. In recent years, Stein’s method has caught the attention of the ML community as a way to formulate new schemes for performing variational inference. Stein’s method provides a fundamental technique for approximating and bounding distances between probability distributions. The kernel Stein discrepancy underlies Stein Variational Gradient Descent (SVGD) which works by iteratively transporting particles sampled from a simple distribution to the target distribution. We introduce the ELBO-within-Stein algorithm that combines SVGD and VI to alleviate issues due to high-dimensional models and large data sets. The ELBO-within-Stein algorithm is available in our computational framework EinStein distributed with the deep probabilistic programming language NumPyro. We will draw upon our framework to illustrate key concepts with examples. EinStein is currently freely available on GitHub and will be available in NumPyro from the next release. The framework is an efficient inference tool for practitioners and a flexible and unified codebase for researchers.
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