Geometric Partial Differential Equations Research Articles

Geometric neural operators (gnps) for data-driven deep learning in non-euclidean settings

Abstract We introduce Geometric Neural Operators (GNPs) for data-driven deep learning of geometric features for tasks in non-euclidean settings. We present a formulation for accounting for geometric contributions along with practical neural network architectures and factorizations for training. We then demonstrate how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures of surfaces, (ii) to approximate solutions of geometric partial differential equations on manifolds, and (iii) to solve Bayesian inverse problems for identifying manifold shapes. These results show a few ways GNPs can be used for incorporating the roles of geometry in the data-driven learning of operators.

A unified structure-preserving parametric finite element method for anisotropic surface diffusion

We propose and analyze a unified structure-preserving parametric finite element method (SP-PFEM) for the anisotropic surface diffusion of curves in two dimensions ( d = 2 ) (d=2) and surfaces in three dimensions ( d = 3 ) (d=3) with an arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) , where n ∈ S d − 1 \boldsymbol {n}\in \mathbb {S}^{d-1} represents the outward unit vector. By introducing a novel unified surface energy matrix G k ( n ) \boldsymbol {G}_k(\boldsymbol {n}) depending on γ ( n ) \gamma (\boldsymbol {n}) , the Cahn-Hoffman ξ \boldsymbol {\xi } -vector and a stabilizing function k ( n ) : S d − 1 → R k(\boldsymbol {n}):\ \mathbb {S}^{d-1}\to {\mathbb R} , we obtain a unified and conservative variational formulation for the anisotropic surface diffusion via different surface differential operators, including the surface gradient operator, the surface divergence operator and the surface Laplace-Beltrami operator. A SP-PFEM discretization is presented for the variational problem. In order to establish the unconditional energy stability of the proposed SP-PFEM under a very mild condition on γ ( n ) \gamma (\boldsymbol {n}) , we propose a new framework via local energy estimate for proving energy stability/structure-preserving properties of the parametric finite element method for the anisotropic surface diffusion. This framework sheds light on how to prove unconditional energy stability of other numerical methods for geometric partial differential equations. Extensive numerical results are reported to demonstrate the efficiency and accuracy as well as structure-preserving properties of the proposed SP-PFEM for the anisotropic surface diffusion with arbitrary anisotropic surface energy density γ ( n ) \gamma (\boldsymbol {n}) arising from different applications.

Convergence of a stabilized parametric finite element method of the Barrett–Garcke–Nürnberg type for curve shortening flow

The parametric finite element methods of the Barrett–Garcke–Nürnberg (BGN) type have been successful in preventing mesh distortion/ degeneration in approximating the evolution of surfaces under various geometric flows, including mean curvature flow, Willmore flow, Helfrich flow, surface diffusion, and so on. However, the rigorous justification of convergence of the BGN-type methods and the characeterization of the particle trajectories produced by these methods still remain open since this class of methods was proposed in 2007. The main difficulty lies in the stability of the artificial tangential velocity implicitly determined by the BGN methods. In this paper, we give the first proof of convergence of a stabilized BGN method for curve shortening flow, with optimal-order convergence in L 2 L^2 norm for finite elements of degree k ≥ 2 k \geq 2 under the stepsize condition τ ≤ c h k + 1 \tau \leq c h^{k+1} (for any fixed constant c c ). Moreover, we give the first rigorous characterization of the particle trajectories produced by the BGN-type methods for one-dimensional curves, i.e., we prove that the particle trajectories produced by the stabilized BGN methods converge to the particle trajectories determined by a system of geometric partial differential equations which differs from the standard curve shortening flow by a tangential motion. The characterization of the particle trajectories also rigorously explains, for one-dimensional curves, why the BGN-type methods could maintain the quality of the underlying evolving mesh.

Interfaces, Free Boundaries and Geometric Partial Differential Equations

Partial differential equations arising in the context of interfaces and free boundaries encompass a flourishing area of research. The workshop focused on new developments and emerging new themes. At the same time also new interesting results on more traditional areas like, e.g. regularity theory and classical numerical approaches have been addressed. By convening experts from various disciplines related to modeling, analysis, and numerical methods concerning interfaces and free boundaries, the workshop facilitated progress on longstanding open questions and paved the way for novel research directions.

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. The lack of coercivity of Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomes highly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.

Preface for the special issue on “Geometric Partial Differential Equations and Applications”

Preface for the special issue on “Geometric Partial Differential Equations and Applications”

Stability for Serrin’s Problem and Alexandroff’s Theorem in Warped Product Manifolds

Abstract We prove quantitative versions for several results from geometric partial differential equations. Firstly, we obtain a double stability theorem for Serrin’s overdetermined problem in spaceforms. Secondly, we prove stability theorems for Brendle’s Heintze–Karcher inequality respectively constant mean curvature classification in a class of warped product spaces. The key tool is the first author’s recent development of stability for level sets of a function under smallness of the traceless Hessian thereof.

Gradient estimates for a general type of nonlinear parabolic equations under geometric conditions and related problems

In this paper, we establish gradient estimates for the positive bounded solutions to a general type of nonlinear parabolic equation concerning the weighted Laplacian ∂∂t−a(x,t)−Δfu(x,t)=F(u(x,t))on a smooth metric measure space with the metric evolving under the (k,∞)-super Perelman–Ricci flow and the Yamabe flow. Applications of our results include Liouville type results and gradient estimates for some important geometric partial differential equations such as the equations involving gradient Ricci solitons and the Einstein-scalar field Lichnerowicz type equations. Our results generalize and improve many previous works.

Force Estimation during Cell Migration Using Mathematical Modelling.

Cell migration is essential for physiological, pathological and biomedical processes such as, in embryogenesis, wound healing, immune response, cancer metastasis, tumour invasion and inflammation. In light of this, quantifying mechanical properties during the process of cell migration is of great interest in experimental sciences, yet few theoretical approaches in this direction have been studied. In this work, we propose a theoretical and computational approach based on the optimal control of geometric partial differential equations to estimate cell membrane forces associated with cell polarisation during migration. Specifically, cell membrane forces are inferred or estimated by fitting a mathematical model to a sequence of images, allowing us to capture dynamics of the cell migration. Our approach offers a robust and accurate framework to compute geometric mechanical membrane forces associated with cell polarisation during migration and also yields geometric information of independent interest, we illustrate one such example that involves quantifying cell proliferation levels which are associated with cell division, cell fusion or cell death.

Sobolev spaces and $$\nabla $$-differential operators on manifolds I: basic properties and weighted spaces

We study covariant Sobolev spaces and $$\nabla $$ -differential operators with coefficients in general Hermitian vector bundles on Riemannian manifolds, stressing a coordinate-free approach that uses connections (which are typically denoted $$\nabla $$ ). These concepts arise naturally from geometric partial differential equations, including some that are formulated on plain Euclidean domains, for instance, from problems formulated on the boundary of smooth domains or in relation to the weighted Sobolev spaces used to study PDEs on polyhedral domains. We prove several basic properties of the covariant Sobolev spaces and of the $$\nabla $$ -differential operators on general manifolds. For instance, we prove mapping properties for our differential operators and the independence of the covariant Sobolev spaces on the choices of the connection $$\nabla $$ , as long as the new connection is obtained using a totally bounded perturbation. We also introduce the Fréchet finiteness condition (FFC) for totally bounded vector fields, which is satisfied, for instance, by open subsets of manifolds with bounded geometry. When (FFC) is satisfied, we provide several equivalent definitions of our covariant Sobolev spaces and of our $$\nabla $$ -differential operators. We also introduce and study the notion of a $$\nabla $$ -bidifferential operator (a bilinear version of differential operators), obtaining results similar to those obtained for $$\nabla $$ -differential operators. Bilinear differential operators are necessary for a global, geometric discussion of variational problems. We tried to write the paper so that it is accessible to a large audience.

Moduli theory, stability of fibrations and optimal symplectic connections

K-polystability is, on the one hand, conjecturally equivalent to the existence of certain canonical K\"ahler metrics on polarised varieties, and, on the other hand, conjecturally gives the correct notion to form moduli. We introduce a notion of stability for families of K-polystable varieties, extending the classical notion of slope stability of a bundle, viewed as a family of K-polystable varieties via the associated projectivisation. We conjecture that this is the correct condition for forming moduli of fibrations. Our main result relates this stability condition to K\"ahler geometry: we prove that the existence of an optimal symplectic connection implies semistability of the fibration. An optimal symplectic connection is a choice of fibrewise constant scalar curvature K\"ahler metric, satisfying a certain geometric partial differential equation. We conjecture that the existence of such a connection is equivalent to polystability of the fibration. We prove a finite dimensional analogue of this conjecture, by describing a GIT problem for fibrations embedded in a fixed projective space, and showing that GIT polystability is equivalent to the existence of a zero of a certain moment map.

Nonlinear vibration analysis of cracked pipe conveying fluid under primary and superharmonic resonances

The nonlinear vibration response of the pipe conveying fluid with a breathing transverse crack, under primary and secondary excitations is investigated. Attention is concentrated on the superharmonic resonance which is the most sensitive phenomena for small crack detection. The crack is modeled with a transverse partial cut of the pipe wall thickness. The equivalent bending stiffness of the cracked pipe is derived based on an energy method and the breathing crack model is used to simulate the transition state between the fully open and the fully closed states. Based on Von Karman’s nonlinear geometric assumption and Euler–Bernoulli beam theory, the nonlinear geometric partial differential equations due to stretching effect, have been derived. The fractional viscoelastic model and the plug flow assumptions are used to model the damping and fluid flow. The coupled nonlinear partial differential equations are reduced into nonlinear ordinary differential equations by the Galerkin method. The method of multiple scales is adopted to analyze steady-state solutions for the primary and superharmonic resonances. A parametric sensitivity analysis is carried out and the effects of different parameters, namely the geometry and location of the crack, nonlinear geometric parameter, fluid flow velocity, mass ratio, fractional derivative order and retardation time on the frequency-response solution are examined. In general, this paper shows that the superharmonic response amplitude measurement is useful tool for damage detection of pipes conveying fluid in the early stages.

Uniqueness of optimal symplectic connections

Abstract Consider a holomorphic submersion between compact Kähler manifolds, such that each fibre admits a constantscalar curvature Kähler metric. When the fibres admit continuous automorphisms, a choice of fibrewise constant scalarcurvature Kähler metric is not unique. An optimal symplectic connection is a choice of fibrewise constant scalar curvature Kähler metric satisfying a geometric partial differential equation. The condition generalises the Hermite-Einstein condition for a holomorphic vector bundle through the induced fibrewise Fubini-Study metric on the associated projectivisation. We prove various foundational analytic results concerning optimal symplectic connections. Our main result proves that optimal symplectic connections are unique, up to the action of the automorphism group of the submersion, when they exist. Thus optimal symplectic connections are canonical relatively Kähler metrics when they exist. In addition, we show that the existence of an optimal symplectic connection forces the automorphism group of the submersion to be reductive and that an optimal symplectic connection is automatically invariant under a maximal compact subgroup of this automorphism group. We also prove that when a submersion admits an optimal symplectic connection, it achieves the absolute minimum of a natural log norm functional, which we define.

Optimal Symplectic Connections on Holomorphic Submersions

The main result of this paper gives a new construction of extremal Kähler metrics on the total space of certain holomorphic submersions, giving a vast generalisation and unification of results of Hong, Fine and others. The principal new ingredient is a novel geometric partial differential equation on such fibrations, which we call the optimal symplectic connection equation.We begin with a smooth fibration for which all fibres admit a constant scalar curvature Kähler metric. When the fibres admit automorphisms, such metrics are not unique in general, but rather are unique up to the action of the automorphism group of each fibre. We define an equation which, at least conjecturally, determines a canonical choice of constant scalar curvature Kähler metric on each fibre. When the fibration is a projective bundle, this equation specialises to asking that the hermitian metric determining the fibrewise Fubini‐Study metric is Hermite‐Einstein.Assuming the existence of an optimal symplectic connection and the existence of an appropriate twisted extremal metric on the base of the fibration, we show that the total space of the fibration itself admits an extremal metric for certain polarisations making the fibres small. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Quenched Mass Transport of Particles Toward a Target

We consider the stochastic target problem of finding the collection of initial laws of a mean-field stochastic differential equation such that we can control its evolution to ensure that it reaches a prescribed set of terminal probability distributions, at a fixed time horizon. Here, laws are considered conditionally to the path of the Brownian motion that drives the system. This kind of problems is motivated by limiting behavior of interacting particles systems with applications in, for example, agricultural crop management. We establish a version of the geometric dynamic programming principle for the associated reachability sets and prove that the corresponding value function is a viscosity solution of a geometric partial differential equation. This provides a characterization of the initial masses that can be almost surely transported toward a given target, along the paths of a stochastic differential equation. Our results extend those of Soner and Touzi, Journal of the European Mathematical Society (2002) to our setting.

Surface, Bulk, and Geometric Partial Differential Equations: Interfacial, stochastic, non-local and discrete structures

Partial differential equations in complex domains with free boundaries and interfaces continue to be flourishing research areas at the interfaces between PDE theory, differential geometry, numerical analysis and applications. Main themes of the workshop have been PDEs on evolving domains, phase field approaches, interactions of bulk and surface PDEs, curvature driven evolution equations. Applications particular from biology, such as cell and cancer modelling and fluid as well solid mechanics have been subjects of the conference.

Surface Crouzeix–Raviart element for the Laplace–Beltrami equation

This paper is concerned with the nonconforming finite element discretization of geometric partial differential equations. In specific, we construct a surface Crouzeix-Raviart element on the linear approximated surface, analogous to a flat surface. The optimal error estimations are established even though the presentation of the geometric error. By taking the intrinsic viewpoint of manifolds, we introduce a new superconvergent gradient recovery method for the surface Crouzeix-Raviart element using only the information of discretization surface. The potential of serving as an asymptotically exact {\it a posteriori} error estimator is also exploited. A series of benchmark numerical examples are presented to validate the theoretical results and numerically demonstrate the superconvergence of the gradient recovery method.

Primary and secondary resonances in pipes conveying fluid with the fractional viscoelastic model

Nonlinear forced vibrations of a fractional viscoelastic pipe conveying fluid exposed to the time-dependent excitations is investigated in the present work. Attention is focused in particular on the primary and secondary resonances with the Kelvin–Voigt fractional order constitutive relationship model. The nonlinear geometric partial differential equations due to stretching effect have been expressed by assumptions with Von Karman’s strain-displacement relation and Euler–Bernoulli beam theory. Viscoelastic fractional model for damping and stiffness, and also plug flow model for fluid flow are considered to derive the equation of motion. Based on the Galerkin truncation, the coupled Fluid-Solid interaction nonlinear equation transferred to ordinary differential equations. The method of multiple scales is adopted to analyze steady-state solutions for the primary, superharmonic, and subharmonic resonances. Finally, the detailed parametric studies on the nonlinear dynamic behavior are discussed. Results delineate that the fractional derivative order and the retardation time have significant effects on the oscillation exhibited for different values of flow velocity.

The dynamics of geometric PDEs: Surface evolution equations and a comparison with their small gradient approximations.

Apart from three-dimensional continuum and discrete models, the evolution of surfaces is usually described by spatially two-dimensional partial differential equations (PDEs). These models are often derived from or at least motivated by small gradient approximations, but the studied surfaces do not fulfill this requirement in all cases. We will investigate how to overcome the small gradient approximation by using geometric PDEs. Therefore, we will introduce a method to simulate the evolution of surfaces with respect to local geometric properties. In contrast to traditional PDEs, this method does not depend on the parametrization of the surface. It will not only allow us to simulate surface evolution on flat geometries but also on more complex shaped objects. For small gradients, the studies of simple model equations show similar results compared to the related PDEs. For large gradients the results differ fundamentally. Hence, the small gradient approximation should only be applied in cases where large gradients does not appear. Specifically, we exemplify this using various equations including the (damped) Kuramoto-Sivashinsky equation, which is used as a minimal model for low-energetic erosion and deposition processes, and its geometric PDE counterpart.

Isogeometric analysis for phase-field models of geometric PDEs and high-order PDEs on stationary and evolving surfaces

In this paper, we present an isogeometric analysis (IGA) for phase-field models of three different yet closely related classes of partial differential equations (PDEs): geometric PDEs, high-order PDEs on stationary surfaces, and high-order PDEs on evolving surfaces; the latter can be a coupling of the former two classes. In the context of geometric PDEs, we consider mean curvature flow and Willmore flow problems and their corresponding phase-field approximations which yield second-order and fourth-order nonlinear parabolic PDEs. Through some numerical examples, we study the convergence behavior of isogeometric analysis for these equations using the method of manufactured solutions. Moreover, we study numerically the convergence of these phase-field approximations to the sharp interface solutions. As for the high-order PDEs on stationary surfaces, we consider a model problem which is the Cahn–Hilliard equation on a unit sphere, where the surface is modeled using a diffuse-interface approach. Finally, as a model problem for high-order PDEs on evolving surfaces, we consider a phase-field model of a deforming multicomponent vesicle which couples the vesicle shape changes with the phase separation process on the vesicle surface. The model consists of two fourth-order nonlinear PDEs which their direct finite element formulation in a Galerkin framework necessitates smooth basis functions with at least global C1 continuity; a condition that can be easily satisfied using spline bases in IGA. We solve the coupled equations both in two dimensions, where the vesicle is a curve, and in three dimensions, where the vesicle is a surface. The simulation results agree with the numerical and experimental results from the literature.