We consider the interfacial growth morphologies of crystals growing in contact in crystal mushes, with a specific application to those formed by the solidification of basaltic magma. We focus on the particular case of an augite (pyroxene) crystal growing between two plagioclase crystals. The augite is treated as an equant unfaceted crystal, whereas the plagioclase is faceted, and our treatment applies generally to such combinations. The resulting three-grain junctions in natural rock samples are commonly of two distinct types. In one, the two augite-plagioclase interfaces grow towards each other with opposite curvatures, whereas in the second the curvatures of the interfaces have the same sign, giving a distinctive morphology which we have called an eagle’s beak. The present paper provides a theoretical framework to provide explanations for both of these morphologies, based on the kinetics of interfacial growth.
{"title":"Interfacial growth morphologies in dense eutectic crystal mushes","authors":"A C Fowler, Marian B Holness","doi":"10.1093/imamat/hxad033","DOIUrl":"https://doi.org/10.1093/imamat/hxad033","url":null,"abstract":"We consider the interfacial growth morphologies of crystals growing in contact in crystal mushes, with a specific application to those formed by the solidification of basaltic magma. We focus on the particular case of an augite (pyroxene) crystal growing between two plagioclase crystals. The augite is treated as an equant unfaceted crystal, whereas the plagioclase is faceted, and our treatment applies generally to such combinations. The resulting three-grain junctions in natural rock samples are commonly of two distinct types. In one, the two augite-plagioclase interfaces grow towards each other with opposite curvatures, whereas in the second the curvatures of the interfaces have the same sign, giving a distinctive morphology which we have called an eagle’s beak. The present paper provides a theoretical framework to provide explanations for both of these morphologies, based on the kinetics of interfacial growth.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2023-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138538044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract One of the reasons why many neural networks are capable of replicating complicated tasks or functions is their universal approximation property. Though the past few decades have seen tremendous advances in theories of neural networks, a single constructive and elementary framework for neural network universality remains unavailable. This paper is an effort to provide a unified and constructive framework for the universality of a large class of activation functions including most of the existing ones. At the heart of the framework is the concept of neural network approximate identity (nAI). The main result is: any nAI activation function is universal in the space of continuous functions on compacta. It turns out that most of the existing activation functions are nAI, and thus universal. The framework induces several advantages over the contemporary counterparts. First, it is constructive with elementary means from functional analysis, probability theory, and numerical analysis. Second, it is one of the first unified and constructive attempts that is valid for most of the existing activation functions. Third, it provides new proofs for most activation functions. Fourth, for a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with a predetermined number of neurons and the values of weights/biases. Fifth, the framework allows us to abstractly present the first universal approximation with a favorable non-asymptotic rate. Sixth, our framework also provides insights into the developments, and hence providing constructive derivations, of some of the existing approaches.
{"title":"A unified and constructive framework for the universality of neural networks","authors":"Tan Bui-Thanh","doi":"10.1093/imamat/hxad032","DOIUrl":"https://doi.org/10.1093/imamat/hxad032","url":null,"abstract":"Abstract One of the reasons why many neural networks are capable of replicating complicated tasks or functions is their universal approximation property. Though the past few decades have seen tremendous advances in theories of neural networks, a single constructive and elementary framework for neural network universality remains unavailable. This paper is an effort to provide a unified and constructive framework for the universality of a large class of activation functions including most of the existing ones. At the heart of the framework is the concept of neural network approximate identity (nAI). The main result is: any nAI activation function is universal in the space of continuous functions on compacta. It turns out that most of the existing activation functions are nAI, and thus universal. The framework induces several advantages over the contemporary counterparts. First, it is constructive with elementary means from functional analysis, probability theory, and numerical analysis. Second, it is one of the first unified and constructive attempts that is valid for most of the existing activation functions. Third, it provides new proofs for most activation functions. Fourth, for a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with a predetermined number of neurons and the values of weights/biases. Fifth, the framework allows us to abstractly present the first universal approximation with a favorable non-asymptotic rate. Sixth, our framework also provides insights into the developments, and hence providing constructive derivations, of some of the existing approaches.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135086306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We model nematic liquid crystal configurations inside three-dimensional prisms, with a polygonal cross-section and Dirichlet boundary conditions on all prism surfaces. We work in a reduced Landau-de Gennes framework, and the Dirichlet conditions on the top and bottom surfaces are special in the sense, that they are critical points of the reduced Landau-de Gennes energy on the polygonal cross-section. The choice of the boundary conditions allows us to make a direct correspondence between the three-dimensional Landau-de Gennes critical points and pathways on the two-dimensional Landau-de Gennes solution landscape on the polygonal cross-section. We explore this concept by means of asymptotic analysis and numerical examples, with emphasis on a cuboid and a hexagonal prism, focusing on three-dimensional multistability tailored by two-dimensional solution landscapes.
{"title":"A Reduced Landau-de Gennes Study for Nematic Equilibria in Three-Dimensional Prisms","authors":"Yucen Han, Baoming Shi, Lei Zhang, Apala Majumdar","doi":"10.1093/imamat/hxad031","DOIUrl":"https://doi.org/10.1093/imamat/hxad031","url":null,"abstract":"Abstract We model nematic liquid crystal configurations inside three-dimensional prisms, with a polygonal cross-section and Dirichlet boundary conditions on all prism surfaces. We work in a reduced Landau-de Gennes framework, and the Dirichlet conditions on the top and bottom surfaces are special in the sense, that they are critical points of the reduced Landau-de Gennes energy on the polygonal cross-section. The choice of the boundary conditions allows us to make a direct correspondence between the three-dimensional Landau-de Gennes critical points and pathways on the two-dimensional Landau-de Gennes solution landscape on the polygonal cross-section. We explore this concept by means of asymptotic analysis and numerical examples, with emphasis on a cuboid and a hexagonal prism, focusing on three-dimensional multistability tailored by two-dimensional solution landscapes.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135430039","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Y Peña Pérez, J Sánchez Ortíz, F J Ariza Hernández, M P Árciga Alejandre
Abstract In this paper, we study a Dirichlet problem for a fractional heat equation, with spacial fractional derivative in the sense of Riemann–Liouville on a finite interval. The main ideas of Fokas method is employed, where the Lax pairs are used to obtain an integral representation of solutions.
{"title":"Initial-boundary value problem for a fractional heat equation on an interval","authors":"Y Peña Pérez, J Sánchez Ortíz, F J Ariza Hernández, M P Árciga Alejandre","doi":"10.1093/imamat/hxad029","DOIUrl":"https://doi.org/10.1093/imamat/hxad029","url":null,"abstract":"Abstract In this paper, we study a Dirichlet problem for a fractional heat equation, with spacial fractional derivative in the sense of Riemann–Liouville on a finite interval. The main ideas of Fokas method is employed, where the Lax pairs are used to obtain an integral representation of solutions.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135728101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract This paper is concerned with an approach based on the topological sensitivity notion to solve a geometric inverse problem for a linear wave equation. The considered inverse problem is motivated by elastography. More precisely, the modeling of our application system has been aimed toward the detection of a breast tumor, in particular, and to enable the calculation of the tumor size, location, and type. We start our analysis by rephrasing the considered inverse problem as an optimization one minimizing an energy cost functional. We establish an estimation describing the asymptotic behavior of the wave equation solution with respect to the presence of a small tumor in the breast which plays an important role in the derivation of a topological asymptotic formula for the considered cost function. Based on the derived theoretical results, we have developed a numerical algorithm for solving our inverse problem, which requires only one iteration. Some numerical experiments are presented to point out the efficiency and accuracy of the proposed approach.
{"title":"Application of the topological sensitivity method to the detection of Breast cancer","authors":"Sabeur Mansouri, Mohamed BenSalah","doi":"10.1093/imamat/hxad028","DOIUrl":"https://doi.org/10.1093/imamat/hxad028","url":null,"abstract":"Abstract This paper is concerned with an approach based on the topological sensitivity notion to solve a geometric inverse problem for a linear wave equation. The considered inverse problem is motivated by elastography. More precisely, the modeling of our application system has been aimed toward the detection of a breast tumor, in particular, and to enable the calculation of the tumor size, location, and type. We start our analysis by rephrasing the considered inverse problem as an optimization one minimizing an energy cost functional. We establish an estimation describing the asymptotic behavior of the wave equation solution with respect to the presence of a small tumor in the breast which plays an important role in the derivation of a topological asymptotic formula for the considered cost function. Based on the derived theoretical results, we have developed a numerical algorithm for solving our inverse problem, which requires only one iteration. Some numerical experiments are presented to point out the efficiency and accuracy of the proposed approach.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135367248","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ivan Yu. Tyukin, Desmond J. Higham, Eliyas Woldegeorgis, Alexander N. Gorban
Abstract We develop and study new adversarial perturbations that enable an attacker to gain control over decisions in generic Artificial Intelligence (AI) systems including deep learning neural networks. In contrast to adversarial data modification, the attack mechanism we consider here involves alterations to the AI system itself. Such a stealth attack could be conducted by a mischievous, corrupt or disgruntled member of a software development team. It could also be made by those wishing to exploit a “democratization of AI” agenda, where network architectures and trained parameter sets are shared publicly. We develop a range of new implementable attack strategies with accompanying analysis, showing that with high probability a stealth attack can be made transparent, in the sense that system performance is unchanged on a fixed validation set which is unknown to the attacker, while evoking any desired output on a trigger input of interest. The attacker only needs to have estimates of the size of the validation set and the spread of the AI’s relevant latent space. In the case of deep learning neural networks, we show that a one neuron attack is possible—a modification to the weights and bias associated with a single neuron—revealing a vulnerability arising from over-parameterization. We illustrate these concepts using state of the art architectures on two standard image data sets. Guided by the theory and computational results, we also propose strategies to guard against stealth attacks.
{"title":"The feasibility and inevitability of stealth attacks","authors":"Ivan Yu. Tyukin, Desmond J. Higham, Eliyas Woldegeorgis, Alexander N. Gorban","doi":"10.1093/imamat/hxad027","DOIUrl":"https://doi.org/10.1093/imamat/hxad027","url":null,"abstract":"Abstract We develop and study new adversarial perturbations that enable an attacker to gain control over decisions in generic Artificial Intelligence (AI) systems including deep learning neural networks. In contrast to adversarial data modification, the attack mechanism we consider here involves alterations to the AI system itself. Such a stealth attack could be conducted by a mischievous, corrupt or disgruntled member of a software development team. It could also be made by those wishing to exploit a “democratization of AI” agenda, where network architectures and trained parameter sets are shared publicly. We develop a range of new implementable attack strategies with accompanying analysis, showing that with high probability a stealth attack can be made transparent, in the sense that system performance is unchanged on a fixed validation set which is unknown to the attacker, while evoking any desired output on a trigger input of interest. The attacker only needs to have estimates of the size of the validation set and the spread of the AI’s relevant latent space. In the case of deep learning neural networks, we show that a one neuron attack is possible—a modification to the weights and bias associated with a single neuron—revealing a vulnerability arising from over-parameterization. We illustrate these concepts using state of the art architectures on two standard image data sets. Guided by the theory and computational results, we also propose strategies to guard against stealth attacks.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135666553","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Larkspur Brudvik-Lindner, Dimitrios Mitsotakis, Athanasios E Tzavaras
Abstract We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn’t possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time.
{"title":"Oscillatory and regularized shock waves for a dissipative Peregrine-Boussinesq system","authors":"Larkspur Brudvik-Lindner, Dimitrios Mitsotakis, Athanasios E Tzavaras","doi":"10.1093/imamat/hxad030","DOIUrl":"https://doi.org/10.1093/imamat/hxad030","url":null,"abstract":"Abstract We consider a dissipative, dispersive system of the Boussinesq type, which describes wave phenomena in scenarios where dissipation plays a significant role. Examples include undular bores in rivers or oceans, where turbulence-induced dissipation significantly influences their behavior. In this study, we demonstrate that the proposed system admits traveling wave solutions known as diffusive-dispersive shock waves. These solutions can be categorized as oscillatory and regularized shock waves, depending on the interplay between dispersion and dissipation effects. By comparing numerically computed solutions with laboratory data, we observe that the proposed model accurately captures the behavior of undular bores over a broad range of phase speeds. Traditionally, undular bores have been approximated using the original Peregrine system, which, even though it doesn’t possess these as traveling wave solutions, tends to offer accurate approximations within suitable time scales. To shed light on this phenomenon, we demonstrate that the discrepancy between the solutions of the dissipative Peregrine system and the non-dissipative counterpart is proportional to the product of the dissipation coefficient and the observation time.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135728896","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Correction to: On the use of asymptotically motivated gauge functions to obtain convergent series solutions to nonlinear ODEs","authors":"","doi":"10.1093/imamat/hxad026","DOIUrl":"https://doi.org/10.1093/imamat/hxad026","url":null,"abstract":"","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136378025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.
{"title":"Random attractors for a stochastic nonlocal delayed reaction-diffusion equation on a semi-infinite interval","authors":"Wenjie Hu, Quanxin Zhu, Tomás Caraballo","doi":"10.1093/imamat/hxad025","DOIUrl":"https://doi.org/10.1093/imamat/hxad025","url":null,"abstract":"Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135148643","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract We investigate the problem of learning a water quality model (BOD-DO model) from given data. Assuming that all parameters in the model are constants, we reformulate the problem as a system of linear equations for the unknown terms. Since in practice the system is often under-determined or over-determined and the observed data are noisy, we use an $l^{1}$-weighted regularization method to find a stable approximate solution. Then, Nesterov’s algorithm is used to solve the regularized problem. Learning models with variable coefficients are also discussed. Numerical examples show that our approach works well with noisy data and has the ability to learn the BOD-DO model.
{"title":"Learning river water quality models by <i>l1</i>-weighted regularization","authors":"Dinh Nho Hào, Duong Xuan Hiep, Pham Quy Muoi","doi":"10.1093/imamat/hxad023","DOIUrl":"https://doi.org/10.1093/imamat/hxad023","url":null,"abstract":"Abstract We investigate the problem of learning a water quality model (BOD-DO model) from given data. Assuming that all parameters in the model are constants, we reformulate the problem as a system of linear equations for the unknown terms. Since in practice the system is often under-determined or over-determined and the observed data are noisy, we use an $l^{1}$-weighted regularization method to find a stable approximate solution. Then, Nesterov’s algorithm is used to solve the regularized problem. Learning models with variable coefficients are also discussed. Numerical examples show that our approach works well with noisy data and has the ability to learn the BOD-DO model.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134950164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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