Simon M Finney, Matthew G Hennessy, Andreas Münch, Sarah L Waters
We study an elastic particle translating axially along the centre-line of a rigid cylindrical tube filled with a Newtonian viscous fluid. The flow is pressure-driven and an axial body force is applied to the particle. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small elastic strains in the particle. In this case, there is a one-way decoupling of the fluid-structure interaction problem. The leading-order fluid problem is shown to be pressure-driven Stokes flow past a rigid sphere, and is solved using the semi-analytical method of reflections. The traction exerted by the fluid on the particle can be computed and used to formulate a pure solid-mechanics problem for the deformation of the particle, which can be solved analytically. This framework is used to investigate the role of the background flow, an axial body force, and the tube wall on the particle’s leading-order translational velocity, resulting deformation, and induced solid stress. By considering the first-order fluid problem the next-order correction to the translational velocity of the particle is shown to be zero. Depending on the magnitude of the ratio of applied body force to viscous forces, the particle can either have a bullet-like shape, an anti-bullet shape, or retain its original spherical shape. A non-linear arbitrary Lagrangian-Eulerian finite element implementation is used, in conjunction with various existing results from the literature, to validate the method of reflections solutions and interrogate their range of validity.
{"title":"The impact of confinement on the deformation of an elastic particle under axisymmetric tube flow","authors":"Simon M Finney, Matthew G Hennessy, Andreas Münch, Sarah L Waters","doi":"10.1093/imamat/hxae022","DOIUrl":"https://doi.org/10.1093/imamat/hxae022","url":null,"abstract":"We study an elastic particle translating axially along the centre-line of a rigid cylindrical tube filled with a Newtonian viscous fluid. The flow is pressure-driven and an axial body force is applied to the particle. We consider the regime in which the ratio of typical viscous fluid stress to elastic stiffness is small, leading to small elastic strains in the particle. In this case, there is a one-way decoupling of the fluid-structure interaction problem. The leading-order fluid problem is shown to be pressure-driven Stokes flow past a rigid sphere, and is solved using the semi-analytical method of reflections. The traction exerted by the fluid on the particle can be computed and used to formulate a pure solid-mechanics problem for the deformation of the particle, which can be solved analytically. This framework is used to investigate the role of the background flow, an axial body force, and the tube wall on the particle’s leading-order translational velocity, resulting deformation, and induced solid stress. By considering the first-order fluid problem the next-order correction to the translational velocity of the particle is shown to be zero. Depending on the magnitude of the ratio of applied body force to viscous forces, the particle can either have a bullet-like shape, an anti-bullet shape, or retain its original spherical shape. A non-linear arbitrary Lagrangian-Eulerian finite element implementation is used, in conjunction with various existing results from the literature, to validate the method of reflections solutions and interrogate their range of validity.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142251247","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}
Takeo Kamizawa, Andrzej Jamiołkowski, Takashi Matsuoka, Noboru Watanabe
In this paper we will discuss the p-reducibility/irreducibility of positive polynomials, and we will give some sufficient conditions for quintic polynomials to be p-reducible/irreducible. This research is closely related to some problems in bio-chemistry, especially to the cooperativity in bio-systems. We will study some applications of our results to these problems.
{"title":"On the P-Irreducibility of Quintic Positive Polynomials","authors":"Takeo Kamizawa, Andrzej Jamiołkowski, Takashi Matsuoka, Noboru Watanabe","doi":"10.1093/imamat/hxae021","DOIUrl":"https://doi.org/10.1093/imamat/hxae021","url":null,"abstract":"In this paper we will discuss the p-reducibility/irreducibility of positive polynomials, and we will give some sufficient conditions for quintic polynomials to be p-reducible/irreducible. This research is closely related to some problems in bio-chemistry, especially to the cooperativity in bio-systems. We will study some applications of our results to these problems.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142185450","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}
We give a new Omega Calculus (a.k.a MacMahon’s Partition Analysis) based integral-free representation for the solution of a non-autonomous and non-homogeneous evolution equation. Our new representation generalizes some of the main results of the recent work of Francisco Neto (2024, A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus. Ann. Inst. Henri Poincaré D, 11, 383) and Bassom et al. (2023, An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation. Appl. Math. Lett., 139, 108537) and show that we can indeed compute the coefficients of the Maclaurin series solution associated with the evolution equation starting with the Peano-Baker series. Furthermore, we discuss in the context of our framework the inverse problem for homogeneous evolution equations in a Hilbert space answering a question left open by Bassom et al. in this case; that is, assuming the solution of the homogeneous evolution equation is a known analytic function the problem concerns the determination of the associated generator of the dynamics. Finally, in order to illustrate the versatility of our approach we explicitly determine the Maclaurin series solution related to the power series method in the context of the vibration problems for the non-uniform (tapered) Euler-Bernoulli beam and thus we explicitly solve the recursion relations considered by Adair and Jaeger (2018, A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams. J. Vib. Control, 24, 3855-3864).
我们给出了一种新的基于欧米茄微积分（又称麦克马洪分区分析法）的无积分表示法，用于求解非自治和非均质演化方程。我们的新表示法概括了弗朗西斯科-内托（Francisco Neto，2024，A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus.Ann.Henri Poincaré D, 11, 383）和 Bassom 等人（2023，经典非自治抽象演化方程的显式麦克劳林级数解。Appl.Lett., 139, 108537），并证明我们确实可以从皮诺-贝克级数开始计算与演化方程相关的麦克劳林级数解的系数。此外，我们还讨论了希尔伯特空间中同质演化方程的逆问题，回答了巴索姆等人在这种情况下提出的一个问题，即假设同质演化方程的解是一个已知的解析函数，那么问题就涉及如何确定相关的动力学发电机。最后，为了说明我们方法的多功能性，我们在非均匀（锥形）欧拉-伯努利梁的振动问题中明确确定了与幂级数方法相关的 Maclaurin 级数解，从而明确解决了 Adair 和 Jaeger（2018，A power series solution for rotating nonuniform Euler-Bernoulli cantilever beams.J. Vib.Control, 24, 3855-3864）。
{"title":"An explicit Maclaurin series solution to non-autonomous and non-homogeneous evolution equation, Omega Calculus, and associated applications","authors":"Antônio Francisco Neto","doi":"10.1093/imamat/hxae020","DOIUrl":"https://doi.org/10.1093/imamat/hxae020","url":null,"abstract":"We give a new Omega Calculus (a.k.a MacMahon’s Partition Analysis) based integral-free representation for the solution of a non-autonomous and non-homogeneous evolution equation. Our new representation generalizes some of the main results of the recent work of Francisco Neto (2024, A basis- and integral-free representation of time-dependent perturbation theory via the Omega matrix calculus. Ann. Inst. Henri Poincaré D, 11, 383) and Bassom et al. (2023, An explicit Maclaurin series solution to a classic non-autonomous abstract evolution equation. Appl. Math. Lett., 139, 108537) and show that we can indeed compute the coefficients of the Maclaurin series solution associated with the evolution equation starting with the Peano-Baker series. Furthermore, we discuss in the context of our framework the inverse problem for homogeneous evolution equations in a Hilbert space answering a question left open by Bassom et al. in this case; that is, assuming the solution of the homogeneous evolution equation is a known analytic function the problem concerns the determination of the associated generator of the dynamics. Finally, in order to illustrate the versatility of our approach we explicitly determine the Maclaurin series solution related to the power series method in the context of the vibration problems for the non-uniform (tapered) Euler-Bernoulli beam and thus we explicitly solve the recursion relations considered by Adair and Jaeger (2018, A power series solution for rotating nonuniform Euler–Bernoulli cantilever beams. J. Vib. Control, 24, 3855-3864).","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141614614","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}
One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin-Helmholtz (KH) instability. It predicts short-wave instabilities that in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity) render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid-liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid-liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid-liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilisation of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.
{"title":"Stability analysis of viscous multi-layer shear flows with interfacial slip","authors":"A. Katsiavria, Demetrios T Papageorgiou","doi":"10.1093/imamat/hxae012","DOIUrl":"https://doi.org/10.1093/imamat/hxae012","url":null,"abstract":"\u0000 One of the most fundamental interfacial instabilities in ideal, immiscible, incompressible multifluid flows is the celebrated Kelvin-Helmholtz (KH) instability. It predicts short-wave instabilities that in the absence of other mollifying physical mechanisms (e.g. surface tension, viscosity) render the nonlinear problem ill-posed and lead to finite-time singularities. The crucial driving mechanism is the jump in tangential velocity across the liquid-liquid interface, i.e. interfacial slip, that can occur since viscosity is absent. The purpose of the present work is to analyse analogous instabilities for viscous flows at small or moderate Reynolds numbers as opposed to the infinite Reynolds numbers that underpin KH instabilities. The problem is physically motivated by both experiments and simulations. The fundamental model considered consists of two superposed viscous, incompressible, immiscible fluid layers sheared in a plane Couette flow configuration, with slip present at the deforming liquid-liquid interface. The origin of slip in viscous flows has been observed in experiments and molecular dynamics simulations, and can be modelled by employing a Navier-slip boundary condition at the liquid-liquid interface. The emerging novel instabilities are studied in detail here. The linear stability of the system is addressed asymptotically for long- and short-waves, and for arbitrary wavenumbers using a combination of analytical and numerical calculations. Slip is found to be capable of destabilising perturbations of all wavelengths. In regimes where the flow is stable to perturbations of all wavelengths in the absence of slip, its presence can induce a Turing-type instability by destabilisation of a small band of finite wavenumber perturbations. In the case where the underlying layer is asymptotically thin, the results are found to agree with the linear properties of a weakly non-linear asymptotic model that is also derived here. The weakly nonlinear model extends previous work by the authors that had a thin overlying layer that produces a different evolution equation.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141348499","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}
The problem of propagation of nonlinear waves in a 1D bar is studied, wherein the linearized strain tensor is considered as a function of the Cauchy stress tensor. Specifically, two constitutive equations for non-Green elastic solids are investigated, introducing a novel numerical iterative method capable to obtain approximate solutions of one nonlinear constitutive equation for rock, and one constitutive equation that shows a strain limiting behaviour. The numerical results are compared with exact solutions for the case of a linearized elastic solid.
{"title":"An iteration method to study nonlinear wave propagation for a non-Green elastic 1D bar","authors":"R. Bustamante, P. Arrue, O. Orellana, R. Meneses","doi":"10.1093/imamat/hxae017","DOIUrl":"https://doi.org/10.1093/imamat/hxae017","url":null,"abstract":"\u0000 The problem of propagation of nonlinear waves in a 1D bar is studied, wherein the linearized strain tensor is considered as a function of the Cauchy stress tensor. Specifically, two constitutive equations for non-Green elastic solids are investigated, introducing a novel numerical iterative method capable to obtain approximate solutions of one nonlinear constitutive equation for rock, and one constitutive equation that shows a strain limiting behaviour. The numerical results are compared with exact solutions for the case of a linearized elastic solid.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141358337","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}
This paper considers a density-suppressed motility model with a strong Allee effect under the homogeneous Neumman boundary condition. We first establish the global existence of bounded classical solutions to a parabolic-parabolic system over a $N $-dimensional $mathbf{(Nle 3)}$ bounded domain $varOmega $, as well as the global existence of bounded classical solutions to a parabolic-elliptic system over the multidimensional bounded domain $varOmega $ with smooth boundary. We then investigate the linear stability at the positive equilibria for the full parabolic case and parabolic-elliptic case respectively, and find the influence of Allee effect on the local stability of the equilibria. By treating the Allee effect as a bifurcation parameter, we focus on the one-dimensional stationary problem and obtain the existence of non-constant positive steady states, which corresponds to small perturbations from the constant equilibrium $(1,1)$. Furthermore, we present some properties through theoretical analysis on pitchfork type and turning direction of the local bifurcations. The stability results provide a stable wave mode selection mechanism for the model considered in this paper. Finally, numerical simulations are performed to demonstrate our theoretical results.
{"title":"Global existence and steady states of the density-suppressed motility model with strong Allee effect","authors":"Cui Song, Zhicheng Wang, Zhaosheng Feng","doi":"10.1093/imamat/hxae013","DOIUrl":"https://doi.org/10.1093/imamat/hxae013","url":null,"abstract":"\u0000 This paper considers a density-suppressed motility model with a strong Allee effect under the homogeneous Neumman boundary condition. We first establish the global existence of bounded classical solutions to a parabolic-parabolic system over a $N $-dimensional $mathbf{(Nle 3)}$ bounded domain $varOmega $, as well as the global existence of bounded classical solutions to a parabolic-elliptic system over the multidimensional bounded domain $varOmega $ with smooth boundary. We then investigate the linear stability at the positive equilibria for the full parabolic case and parabolic-elliptic case respectively, and find the influence of Allee effect on the local stability of the equilibria. By treating the Allee effect as a bifurcation parameter, we focus on the one-dimensional stationary problem and obtain the existence of non-constant positive steady states, which corresponds to small perturbations from the constant equilibrium $(1,1)$. Furthermore, we present some properties through theoretical analysis on pitchfork type and turning direction of the local bifurcations. The stability results provide a stable wave mode selection mechanism for the model considered in this paper. Finally, numerical simulations are performed to demonstrate our theoretical results.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141386700","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}
Recent experimental work has highlighted that electrolyte-driven diffusiophoresis is likely to be the most significant phoretic motion in a mixture of silica nanoparticles and relatively large latex particles, which are commonly used in coatings. In this present work, this diffusiophoretic effect, powered by gradients in the nanoparticles and their stabilising cations, is modelled in drying films. A continuum hydrodynamic model is derived, and the resulting partial differential equations solved numerically. An asymptotic solution is found for high evaporation rate. It is found that the final film structure is governed by the relative magnitudes of the diffusive and diffusiophoretic terms. Two methods are discovered to control the resulting stratification: (i) setting the surface charge on the particles, and (ii) setting the background salt concentration. Either of these can be used to select either small- or large-on-top stratification or a homogenous film. The diffusiophoretic term promotes small-on-top stratification, and so may account for experimental observations of accumulated small particles at the top surface of dried films.
{"title":"Stratification in drying films: Diffusiophoresis driven by nanoparticles and their counterions","authors":"Clare R. Rees-Zimmerman, Alexander F Routh","doi":"10.1093/imamat/hxae015","DOIUrl":"https://doi.org/10.1093/imamat/hxae015","url":null,"abstract":"\u0000 Recent experimental work has highlighted that electrolyte-driven diffusiophoresis is likely to be the most significant phoretic motion in a mixture of silica nanoparticles and relatively large latex particles, which are commonly used in coatings. In this present work, this diffusiophoretic effect, powered by gradients in the nanoparticles and their stabilising cations, is modelled in drying films. A continuum hydrodynamic model is derived, and the resulting partial differential equations solved numerically. An asymptotic solution is found for high evaporation rate. It is found that the final film structure is governed by the relative magnitudes of the diffusive and diffusiophoretic terms. Two methods are discovered to control the resulting stratification: (i) setting the surface charge on the particles, and (ii) setting the background salt concentration. Either of these can be used to select either small- or large-on-top stratification or a homogenous film. The diffusiophoretic term promotes small-on-top stratification, and so may account for experimental observations of accumulated small particles at the top surface of dried films.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141383990","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}
A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function $T$. The field $T$ is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outward is formulated both in terms of the Schwarz function of the boundary and a Polubarinova-Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova-Galin equation is used to compute the evolution of the boundary having other initial shapes.
{"title":"Exact and numerical solutions of a free boundary problem with a reciprocal growth law","authors":"N. R. McDonald, Samuel J Harris","doi":"10.1093/imamat/hxae014","DOIUrl":"https://doi.org/10.1093/imamat/hxae014","url":null,"abstract":"\u0000 A two-dimensional free boundary problem is formulated in which the normal velocity of the boundary is proportional to the inverse of the gradient of a harmonic function $T$. The field $T$ is defined in a simply connected region which includes the point at infinity where it has a logarithmic singularity. The growth problem in which the boundary expands outward is formulated both in terms of the Schwarz function of the boundary and a Polubarinova-Galin equation for the conformal map of the region from the exterior of the unit disk. An expanding free boundary is shown to be stable and explicit solutions for growing ellipses and a class of polynomial lemniscates are derived. Numerical solution of the Polubarinova-Galin equation is used to compute the evolution of the boundary having other initial shapes.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141267584","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}
Tamara G Grossmann, Urszula Julia Komorowska, Jonas Latz, Carola-Bibane Schönlieb
Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen–Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.
{"title":"Can physics-informed neural networks beat the finite element method?","authors":"Tamara G Grossmann, Urszula Julia Komorowska, Jonas Latz, Carola-Bibane Schönlieb","doi":"10.1093/imamat/hxae011","DOIUrl":"https://doi.org/10.1093/imamat/hxae011","url":null,"abstract":"Partial differential equations play a fundamental role in the mathematical modelling of many processes and systems in physical, biological and other sciences. To simulate such processes and systems, the solutions of PDEs often need to be approximated numerically. The finite element method, for instance, is a usual standard methodology to do so. The recent success of deep neural networks at various approximation tasks has motivated their use in the numerical solution of PDEs. These so-called physics-informed neural networks and their variants have shown to be able to successfully approximate a large range of partial differential equations. So far, physics-informed neural networks and the finite element method have mainly been studied in isolation of each other. In this work, we compare the methodologies in a systematic computational study. Indeed, we employ both methods to numerically solve various linear and nonlinear partial differential equations: Poisson in 1D, 2D, and 3D, Allen–Cahn in 1D, semilinear Schrödinger in 1D and 2D. We then compare computational costs and approximation accuracies. In terms of solution time and accuracy, physics-informed neural networks have not been able to outperform the finite element method in our study. In some experiments, they were faster at evaluating the solved PDE.","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141145861","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":"Forward to the special issue on “The Applied Mathematics of Machine Learning”","authors":"","doi":"10.1093/imamat/hxae010","DOIUrl":"https://doi.org/10.1093/imamat/hxae010","url":null,"abstract":"","PeriodicalId":56297,"journal":{"name":"IMA Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":1.2,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141122657","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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