| 1. |
- Gerken, Jan, 1991, et al.
(författare)
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Geometric deep learning and equivariant neural networks
- 2023
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Ingår i: Artificial Intelligence Review. - : Springer Nature. - 1573-7462 .- 0269-2821. ; 56:12, s. 14605-14662
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Tidskriftsartikel (refereegranskat)abstract
- We survey the mathematical foundations of geometric deep learning, focusing on group equivariant and gauge equivariant neural networks. We develop gauge equivariant convolutional neural networks on arbitrary manifolds M using principal bundles with structure group K and equivariant maps between sections of associated vector bundles. We also discuss group equivariant neural networks for homogeneous spaces M= G/ K , which are instead equivariant with respect to the global symmetry G on M . Group equivariant layers can be interpreted as intertwiners between induced representations of G, and we show their relation to gauge equivariant convolutional layers. We analyze several applications of this formalism, including semantic segmentation and object detection networks. We also discuss the case of spherical networks in great detail, corresponding to the case M= S2= SO (3) / SO (2) . Here we emphasize the use of Fourier analysis involving Wigner matrices, spherical harmonics and Clebsch–Gordan coefficients for G= SO (3) , illustrating the power of representation theory for deep learning.
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| 2. |
- Gyllingberg, Linnéa, et al.
(författare)
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Finding analytical approximations for discrete, stochastic, individual-based models of ecology
- 2023
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Ingår i: Mathematical Biosciences. - : Elsevier. - 0025-5564 .- 1879-3134. ; 365
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Tidskriftsartikel (refereegranskat)abstract
- Discrete time, spatially extended models play an important role in ecology, modelling population dynamics of species ranging from micro-organisms to birds. An important question is how ’bottom up’, individual-based models can be approximated by ’top down’ models of dynamics. Here, we study a class of spatially explicit individual-based models with contest competition: where species compete for space in local cells and then disperse to nearby cells. We start by describing simulations of the model, which exhibit large-scale discrete oscillations and characterize these oscillations by measuring spatial correlations. We then develop two new approximate descriptions of the resulting spatial population dynamics. The first is based on local interactions of the individuals and allows us to give a difference equation approximation of the system over small dispersal distances. The second approximates the long-range interactions of the individual-based model. These approximations capture demographic stochasticity from the individual-based model and show that dispersal stabilizes population dynamics. We calculate extinction probability for the individual-based model and show convergence between the local approximation and the non-spatial global approximation of the individual-based model as dispersal distance and population size simultaneously tend to infinity. Our results provide new approximate analytical descriptions of a complex bottom-up model and deepen understanding of spatial population dynamics.
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| 3. |
- Cohen, David, et al.
(författare)
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Drift-preserving numerical integrators for stochastic Poisson systems
- 2021
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Ingår i: International Journal of Computer Mathematics. - : Informa UK Limited. - 0020-7160 .- 1029-0265. ; 99:1
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Tidskriftsartikel (refereegranskat)abstract
- We perform a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems. For the considered additive noise perturbation of such systems, we show the long-time behaviour of the energy and quadratic Casimirs for the exact solution. We then propose and analyse a drift-preserving splitting scheme for such problems with the following properties: exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, weak order of convergence 2. These properties are illustrated with numerical experiments.
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| 4. |
- Hansbo, Peter, et al.
(författare)
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A simple nonconforming tetrahedral element for the Stokes equations
- 2022
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier. - 0045-7825 .- 1879-2138. ; 400
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we apply a nonconforming rotated bilinear tetrahedral element to the Stokes problem in R3. We show that the element is stable in combination with a piecewise linear, continuous, approximation of the pressure. This gives an approximation similar to the well known continuous P2–P1 Taylor–Hood element, but with fewer degrees of freedom. The element is a stable non-conforming low order element which fulfils Korn's inequality, leading to stability also in the case where the Stokes equations are written on stress form for use in the case of free surface flow.
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| 5. |
- Backlund, Ulf, et al.
(författare)
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Semi-Bloch Functions in Several Complex Variables
- 2016
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Ingår i: Journal of Geometric Analysis. - : Springer Science and Business Media LLC. - 1050-6926 .- 1559-002X. ; 26:1, s. 463-473
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Tidskriftsartikel (refereegranskat)abstract
- Let M be an n-dimensional complex manifold. A holomorphic function f : M -> C is said to be semi-Bloch if for every lambda is an element of C the function g(lambda) = exp(lambda f(z)) is normal on M. We characterize semi-Bloch functions on infinitesimally Kobayashi non-degenerate M in geometric as well as analytic terms. Moreover, we show that on such manifolds, semi-Bloch functions are normal.
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| 6. |
- Bökman, Georg, 1994, et al.
(författare)
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ZZ-Net: A Universal Rotation Equivariant Architecture for 2D Point Clouds
- 2022
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Ingår i: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition. - : IEEE Computer Society. - 1063-6919. ; 2022-June, s. 10966-10975
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Konferensbidrag (refereegranskat)abstract
- In this paper, we are concerned with rotation equivariance on 2D point cloud data. We describe a particular set of functions able to approximate any continuous rotation equivariant and permutation invariant function. Based on this result, we propose a novel neural network architecture for processing 2D point clouds and we prove its universality for approximating functions exhibiting these symmetries. We also show how to extend the architecture to accept a set of 2D-2D correspondences as indata, while maintaining similar equivariance properties. Experiments are presented on the estimation of essential matrices in stereo vision.
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| 7. |
- Salmi, Tapio, et al.
(författare)
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A Robust Method for the Estimation of Kinetic Parameters for Systems Including Slow and Rapid Reactions - From Differential-Algebraic Model to Differential Model
- 2020
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Ingår i: Processes. - : MDPI. - 2227-9717. ; 8:12
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Tidskriftsartikel (refereegranskat)abstract
- Reliable estimation of kinetic parameters in chemical systems comprising both slow and rapid reaction steps and rapidly reacting intermediate species is a difficult differential-algebraic problem. Consequently, any conventional approach easily leads to serious convergence and stability problems during the parameter estimation. A robust method is proposed to surmount this dilemma: the system of ordinary differential equations and nonlinear algebraic equations is converted to ordinary differential equations, which are solved in-situ during the parameter estimation. The approach was illustrated with two generic examples and an example from green chemistry: synthesis of dimethyl carbonate from carbon dioxide and methanol.
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| 8. |
- Behrstock, Jason, et al.
(författare)
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Square percolation and the threshold for quadratic divergence in random right-angled Coxeter groups
- 2022
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Ingår i: Random structures & algorithms (Print). - : John Wiley & Sons. - 1042-9832 .- 1098-2418. ; 60:4, s. 594-630
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Tidskriftsartikel (refereegranskat)abstract
- Given a graph (Formula presented.), its auxiliary square-graph (Formula presented.) is the graph whose vertices are the non-edges of (Formula presented.) and whose edges are the pairs of non-edges which induce a square (i.e., a 4-cycle) in (Formula presented.). We determine the threshold edge-probability (Formula presented.) at which the Erdős–Rényi random graph (Formula presented.) begins to asymptotically almost surely (a.a.s.) have a square-graph with a connected component whose squares together cover all the vertices of (Formula presented.). We show (Formula presented.), a polylogarithmic improvement on earlier bounds on (Formula presented.) due to Hagen and the authors. As a corollary, we determine the threshold (Formula presented.) at which the random right-angled Coxeter group (Formula presented.) a.a.s. becomes strongly algebraically thick of order 1 and has quadratic divergence.
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| 9. |
- Danielsson, Joel Larsson, et al.
(författare)
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Polarised random κ-SAT
- 2023
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Ingår i: Combinatorics, probability & computing. - : Cambridge University Press. - 0963-5483 .- 1469-2163. ; 32:6, s. 885-899
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Tidskriftsartikel (refereegranskat)abstract
- In this paper we study a variation of the random κ-SAT problem, called polarised random κ-SAT, which contains both the classical random κ-SAT model and the random version of monotone κ-SAT another well-known NP-complete version of SAT. In this model there is a polarisation parameter p, and in half of the clauses each variable occurs negated with probability p and pure otherwise, while in the other half the probabilities are interchanged. For p = 1/2 we get the classical random κ-SAT model, and at the other extreme we have the fully polarised model where p = 0, or 1. Here there are only two types of clauses: clauses where all κ variables occur pure, and clauses where all κ variables occur negated. That is, for p = 0, and p=1, we get an instance of random monotone κ-SAT. We show that the threshold of satisfiability does not decrease as p moves away from 1/2 and thus that the satisfiability threshold for polarised random κ-SAT with p ≠ 1/2 is an upper bound on the threshold for random κ-SAT. Hence the satisfiability threshold for random monotone κ-SAT is at least as large as for random κ-SAT, and we conjecture that asymptotically, for a fixed κ, the two thresholds coincide.
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| 10. |
- Day, A. Nicholas, et al.
(författare)
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Upper density of monochromatic paths in edge-coloured infinite complete graphs and bipartite graphs
- 2023
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Ingår i: European journal of combinatorics (Print). - : Elsevier. - 0195-6698 .- 1095-9971. ; 110
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Tidskriftsartikel (refereegranskat)abstract
- The upper density of an infinite graph G with V(G)⊆N is defined as d¯(G)=lim supn→∞|V(G)∩{1,…,n}|/n. Let KN be the infinite complete graph with vertex set N. Corsten, DeBiasio, Lamaison and Lang showed that in every 2-edge-colouring of KN, there exists a monochromatic path with upper density at least (12+8)/17, which is best possible. In this paper, we extend this result to k-edge-colouring of KN for k≥3. We conjecture that every k-edge-coloured KN contains a monochromatic path with upper density at least 1/(k−1), which is best possible (when k−1 is a prime power). We prove that this is true when k=3 and asymptotically when k=4. Furthermore, we show that this problem can be deduced from its bipartite variant, which is of independent interest.
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