Deflation-based Identification of Nonlinear Excitations of the 3D Gross--Pitaevskii equation

We present previously unknown solutions to the 3D Gross--Pitaevskii equation describing atomic Bose-Einstein condensates. This model supports elaborate patterns, including excited states bearing vorticity. The discovered coherent structures exhibit striking topological features, involving combinations of vortex rings and multiple, possibly bent vortex lines. Although unstable, many of them persist for long times in dynamical simulations. These solutions were identified by a state-of-the-art numerical technique called deflation, which is expected to be applicable to many problems from other areas of physics.Comment: 9 pages, 11 figure

Two-Component 3D Atomic Bose-Einstein Condensates Support Complex Stable Patterns

We report the computational discovery of complex, topologically charged, and spectrally stable states in three-dimensional multi-component nonlinear wave systems of nonlinear Schrödinger type. While our computations relate to two-component atomic Bose-Einstein condensates in parabolic traps, our methods can be broadly applied to high-dimensional, nonlinear systems of partial differential equations. The combination of the so-called deflation technique with a careful selection of initial guesses enables the computation of an unprecedented breadth of patterns, including ones combining vortex lines, rings, stars, and “vortex labyrinths”. Despite their complexity, they may be dynamically robust and amenable to experimental observation, as confirmed by Bogolyubov-de Gennes spectral analysis and numerical evolution simulations

Improved comprehensibility and reliability of explanations via restricted halfspace discretization

Abstract. A number of two-class classification methods first discretize each attribute of two given training sets and then construct a propositional DNF formula that evaluates to True for one of the two discretized training sets and to False for the other one. The formula is not just a classification tool but constitutes a useful explanation for the differences between the two underlying populations if it can be comprehended by humans and is reliable. This paper shows that comprehensibility as well as reliability of the formulas can sometimes be improved using a discretization scheme where linear combinations of a small number of attributes are discretized

Internet of Things for Sustainable Mining

The sustainable mining Internet of Things deals with the applications of IoT technology to the coupled needs of sustainable recovery of metals and a healthy environment for a thriving planet. In this chapter, the IoT architecture and technology is presented to support development of a digital mining platform emphasizing the exploration of rock–fluid–environment interactions to develop extraction methods with maximum economic benefit, while maintaining and preserving both water quantity and quality, soil, and, ultimately, human health. New perspectives are provided for IoT applications in developing new mineral resources, improved management of tailings, monitoring and mitigating contamination from mining. Moreover, tools to assess the environmental and social impacts of mining including the demands on dwindling freshwater resources. The cutting-edge technologies that could be leveraged to develop the state-of-the-art sustainable mining IoT paradigm are also discussed

Predictors of poor retention on antiretroviral therapy as a major HIV drug resistance early warning indicator in Cameroon: results from a nationwide systematic random sampling

Retention on lifelong antiretroviral therapy (ART) is essential in sustaining treatment success while preventing HIV drug resistance (HIVDR), especially in resource-limited settings (RLS). In an era of rising numbers of patients on ART, mastering patients in care is becoming more strategic for programmatic interventions. Due to lapses and uncertainty with the current WHO sampling approach in Cameroon, we thus aimed to ascertain the national performance of, and determinants in, retention on ART at 12 months

Data-driven discovery of Green's functions

Discovering hidden partial differential equations (PDEs) and operators from data is an important topic at the frontier between machine learning and numerical analysis. Theoretical results and deep learning algorithms are introduced to learn Green's functions associated with linear partial differential equations and rigorously justify PDE learning techniques. A theoretically rigorous algorithm is derived to obtain a learning rate, which characterizes the amount of training data needed to approximately learn Green's functions associated with elliptic PDEs. The construction connects the fields of PDE learning and numerical linear algebra by extending the randomized singular value decomposition to non-standard Gaussian vectors and Hilbert--Schmidt operators, and exploiting the low-rank hierarchical structure of Green's functions using hierarchical matrices. Rational neural networks (NNs) are introduced and consist of neural networks with trainable rational activation functions. The highly compositional structure of these networks, combined with rational approximation theory, implies that rational functions have higher approximation power than standard activation functions. In addition, rational NNs may have poles and take arbitrarily large values, which is ideal for approximating functions with singularities such as Green's functions. Finally, theoretical results on Green's functions and rational NNs are combined to design a human-understandable deep learning method for discovering Green's functions from data. This approach complements state-of-the-art PDE learning techniques, as a wide range of physics can be captured from the learned Green's functions such as dominant modes, symmetries, and singularity locations.</p

Data-driven discovery of Green’s functions with human-understandable deep learning

There is an opportunity for deep learning to revolutionize science and technology by revealing its findings in a human interpretable manner. We develop a novel data-driven approach for creating a human-machine partnership to accelerate scientific discovery. By collecting physical system responses, under carefully selected excitations, we train rational neural networks to learn Green's functions of hidden partial differential equation. These solutions reveal human-understandable properties and features, such as linear conservation laws, and symmetries, along with shock and singularity locations, boundary effects, and dominant modes. We illustrate this technique on several examples and capture a range of physics, including advection-diffusion, viscous shocks, and Stokes flow in a lid-driven cavity

Computing with functions in the ball

A collection of algorithms in object-oriented MATLAB is described for numerically computing with smooth functions defined on the unit ball in the Chebfun software. Functions are numerically and adaptively resolved to essentially machine precision by using a three-dimensional analogue of the double Fourier sphere method to form “Ballfun" objects. Operations such as function evaluation, differentiation, integration, fast rotation by an Euler angle, and a Helmholtz solver are designed. Our algorithms are particularly efficient for vector calculus operations, and we describe how to compute the poloidal-toroidal and Helmholtz--Hodge decompositions of a vector field defined on the ball

Learning Elliptic Partial Differential Equations with Randomized Linear Algebra

AbstractGiven input–output pairs of an elliptic partial differential equation (PDE) in three dimensions, we derive the first theoretically rigorous scheme for learning the associated Green’s function G. By exploiting the hierarchical low-rank structure of G, we show that one can construct an approximant to G that converges almost surely and achieves a relative error of $$\mathcal {O}(\varGamma _\epsilon ^{-1/2}\log ^3(1/\epsilon )\epsilon )$$ O ( Γ ϵ - 1 / 2 log 3 ( 1 / ϵ ) ϵ ) using at most $$\mathcal {O}(\epsilon ^{-6}\log ^4(1/\epsilon ))$$ O ( ϵ - 6 log 4 ( 1 / ϵ ) ) input–output training pairs with high probability, for any 0<\epsilon <1 0 &lt; ϵ &lt; 1 . The quantity 0<\varGamma _\epsilon \le 1 0 &lt; Γ ϵ ≤ 1 characterizes the quality of the training dataset. Along the way, we extend the randomized singular value decomposition algorithm for learning matrices to Hilbert–Schmidt operators and characterize the quality of covariance kernels for PDE learning.</jats:p