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On quadrature rules for solving Partial Differential Equations using Neural Networks
Get PDFNeural Networks have been widely used to solve Partial Differential Equations. These methods require to approximate definite integrals using quadrature rules. Here, we illustrate via 1D numerical examples the quadrature problems that may arise in these applications and propose several alternatives to overcome them, namely: Monte Carlo methods, adaptive integration, polynomial approximations of the Neural Network output, and the inclusion of regularization terms in the loss. We also discuss the advantages and limitations of each proposed numerical integration scheme. We advocate the use of Monte Carlo methods for high dimensions (above 3 or 4), and adaptive integration or polynomial approximations for low dimensions (3 or below). The use of regularization terms is a mathematically elegant alternative that is valid for any spatial dimension; however, it requires certain regularity assumptions on the solution and complex mathematical analysis when dealing with sophisticated Neural Networks
A Multidirectional Deep Neural Network for Self-Supervised Reconstruction of Seismic Data
Get PDFSeismic studies exhibit gaps in the recorded data due to surface obstacles. To fill in the gaps with self-supervised deep learning, the network learns to predict different events from the recorded parts of data and then applies it to reconstruct the missing parts of the same dataset. We propose two improvements to the task: a rearrangement of the data, and a new deep-learning approach. We rearrange the traces of a 2D acquisition line as 3D data cubes, sorting the traces by the source and receiver coordinates. This 3D representation offers more information about the structure of the seismic events and allows a coherent reconstruction of them. However, learning the structure of events in 3D cubes is more complicated than in 2D images while the size of the training dataset is limited. Thus, we propose a specific architecture and training strategy to take advantage of 3D data samples, while benefiting from the simplicity of 2D reconstructions. Our proposed multidirectional convolutional neural network has two parallel branches trained to perform 2D reconstructions along the vertical and horizontal directions and a small 3D part that combines their results. We use our method to reconstruct data gaps resulting from several missing shots in a benchmark synthetic and a real land dataset. Compared to a conventional 3D U-net, our network learns to reconstruct the events more accurately. Compared to 2D U-nets, our method avoids the discontinuities that arise from the 2D reconstruction of each trace of the missing shot gathers
VSD-MOEA: A Dominance-Based Multiobjective Evolutionary Algorithm with Explicit Variable Space Diversity Management
Get PDFMost state-of-the-art Multiobjective Evolutionary Algorithms (moeas) promote the preservation of diversity of objective function space but neglect the diversity of decision variable space. The aim of this article is to show that explicitly managing the amount of diversity maintained in the decision variable space is useful to increase the quality of moeas when taking into account metrics of the objective space. Our novel Variable Space Diversity-based MOEA (vsd-moea) explicitly considers the diversity of both decision variable and objective function space. This information is used with the aim of properly adapting the balance between exploration and intensification during the optimization process. Particularly, at the initial stages, decisions made by the approach are more biased by the information on the diversity of the variable space, whereas it gradually grants more importance to the diversity of objective function space as the evolution progresses. The latter is achieved through a novel density estimator. The new method is compared with state-of-art moeas using several benchmarks with two and three objectives. This novel proposal yields much better results than state-of-the-art schemes when considering metrics applied on objective function space, exhibiting a more stable and robust behavior
Deep Learning for Inverting Borehole Resistivity Measurements
Get PDFThere exist multiple traditional methods to solve inverse problems, mainly, gradient-based or statistics-based methods. However, these methods have severe limitations. In particular, they often need to compute the forward problem hundreds of times, which is computationally expensive in three-dimensional (3D) problems.
In this dissertation, we propose the use of Deep Learning (DL) techniques to solve inverse problems. Although the training stage of a Deep Neural Network (DNN) may be time-consuming, after the network is properly trained it can forecast the solution in a fraction of a second, facilitating real-time operations. In the first part of this dissertation, we investigate appropriate loss functions to train a DNN when dealing with an inverse problem.
Additionally, to properly train a DNN that approximates the inverse solution, we require a large dataset containing the solution of the forward problem. To create such dataset, we need to solve aPartial Differential Equation (PDE) thousands of times. Building a dataset may be time-consuming, especially for two and three-dimensional problems since solving PDEs using traditional methods, such as the Finite Element Method (FEM), is computationally expensive. Thus, we want to reduce the computational cost of building the database needed to train the DNN. For this, we propose the use of rIGA methods.
In addition, we explore the possibility of using DL techniques to solve PDEs, which is the main computational bottleneck when solving inverse problems. Our main goal is to develop a fast forward simulator for solving parametric PDEs. As a first step, in this dissertation we analyze the quadrature problems that appear while solving PDEs using DNNs and propose different integration methods to overcome these limitations
The Euler characteristic as a topological marker for outbreaks in vector-borne disease
Get PDFAbstract. Epidemic outbreaks represent a significant concern for the current state of global health, particularly in Brazil, the epicentre of several vector-borne disease outbreaks and where epidemic control is still a challenge for the scientific community. Data science techniques applied to epidemics are usually made via standard statistical and modelling approaches, which do not always lead to reli- able predictions, especially when the data lacks a piece of reliable surveillance information needed for precise parameter estimation. In particular, dengue out- breaks reported over the past years raise concerns for global health care, and thus novel data-driven methods are necessary to predict the emergence of out- breaks. In this work, we propose a parameter-free approach based on geometric and topological techniques, which extracts geometrical and topological invariants as opposed to statistical summaries used in established methods. Specifically, our procedure generates a time-varying network from a time-series of new epidemic cases based on synthetic time-series and real dengue data across several dis- tricts of Recife, the fourth-largest urban area in Brazil. Subsequently, we use the Euler characteristic (EC) to extract key topological invariant of the epidemic time-varying network and we finally compared the results with the effective reproduction number (Rt) for each data set. Our results unveil a strong cor- relation between epidemic outbreaks and the EC. In fact, sudden changes in the EC curve preceding and/or during an epidemic period emerge as a warn- ing sign for an outbreak in the synthetic data, the EC transitions occur close to the periods of epidemic transitions, which is also corroborated. In the real dengue data, where data is intrinsically noise, the EC seems to show a better sign-to-noise ratio once compared to Rt. In analogy with later studies on noisy data by using EC in positron emission tomography scans, the EC estimates the number of regions with high connectivity in the epidemic network and thus has potential to be a signature of the emergence of an epidemic state. Our results open the door to the development of alternative/complementary topological and geometrical data-driven methods to characterise vector-borne disease outbreaks, specially when the conventional epidemic surveillance methods are not effective in a scenario of extreme noise and lack of robustness in the data
Pattern alternations induced by nonlocal interactions
Get PDFPattern formation is a visual understanding of the dynamics of complex systems. Patterns arise in many
ways, such as the segmentation of animals, bacterial colonies during growth, vegetation, chemical reactions,
etc. In most cases, the long-range diffusion occurs, and the usual reaction-diffusion (RD) model can not
capture such phenomena. The nonlocal RD model, on the other hand, can fill the gap. Analytical derivation
of the amplitude equations (AE) for an RD system is a valuable tool to predict the pattern selections, in
particular, the stationary Turing patterns when they occur. In this paper, we analyze the conditions for the
Turing bifurcation for the nonlocal model and also derive the AE for the nonlocal RD model near the Turing
bifurcation threshold to describe the reason behind the pattern selections. This derivation of the AE is not
only limited to the nonlocal prey-predator model, as shown in our representative example but also can be
applied to other nonlocal models near the Turing bifurcation threshold. The analytical prediction agrees
with numerical simulation near the Turing bifurcation threshold. Moreover, the analytical and numerical
results fit each other well even more remote from the Turing bifurcation threshold for the small values of
the nonlocal parameter but not for the higher values
Synthesis of 3D jigsaw puzzles over freeform 2-manifolds
Get PDFWe present a simple algorithm for synthesizing 3D jigsaw puzzles from arbitrary 3D freeform 2-manifold geometric models represented with trimmed NURBS surfaces. The construction algorithm is based on a few conventional geometric operations on freeform curves and surfaces. In particular, we need to compute the offset of freeform NURBS surfaces (for thickening the 2-manifold surfaces) and the functional composition of a univariate curve representation to a bivariate rational surface (for breaking up a 3D model into curved jigsaw tiles). It is thus almost straightforward to convert the proposed algorithm to a practical system using standard tools available in B-rep based geometric modeling systems, that employ trimmed NURBS surfaces. We demonstrate the effectiveness of the proposed approach by fabricating several test sets of 3D jigsaw puzzles for freeform solids consisting of trimmed NURBS surface models
On the motion of a large number of small rigid bodies in a viscous incompressible fluid
No full textWe consider the motion of N rigid bodies – compact sets (S1
ε ,· · ·
,SN
ε )ε>0 – immersed
in a viscous incompressible fluid contained in a domain in the Euclidean space Rd
, d = 2,3.
We show the fluid flow is not influenced by the presence of the infinitely many bodies in the
asymptotic limit ε → 0 and N= N(ε) → ∞ as soon as
diam[Si
ε] → 0 as ε → 0, i = 1,· · · ,N(ε).
The result depends solely on the geometry of the bodies and is independent of their mass
densities. Collisions are allowed and the initial data are arbitrary with finite energy
Cross immunity protection and antibody-dependent enhancement in a distributed delay dynamic model
Get PDFDengue fever is endemic in tropical and subtropical countries, and certain important features of the spread of dengue fever continue to pose challenges for mathematical modelling. Here we propose a system of integro-differential equations (IDE) to study the disease transmission dynamics that involve multi-serotypes and cross immunity. Our main objective is to incorporate and analyze the effect of a general time delay term describing acquired cross immunity protection and the effect of antibody-dependent enhancement (ADE), both characteristics of Dengue fever. We perform qualitative analysis of the model and obtain results to show the stability of the epidemiologically important steady solutions that are completely determined by the basic reproduction number and the invasion reproduction number. We establish the global dynamics by constructing a suitable Lyapunov functional. We also conduct some numerical experiments to illustrate bifurcation structures, indicating the occurrence of periodic oscillations for a specific range of values of a key parameter representing ADE.Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brazil (CAPES) - Finance Code 001
LIAM - Laboratory for Industrial and Applied Mathematics,
Department of Mathematics and Statistics, York University-CA