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On the effect of phase transition on the manifold dimensionality: application to the Ising model. (English) Zbl 06945830
Summary: Fields can be represented in a discrete manner from their values at some locations, the nodes when considering finite element descriptions. Thus, each discrete scalar solution can be considered as a point in \(\mathbb{R}^N\) (\(N\) being the number of nodes used for approximating the scalar field). Most manifold learning techniques (linear and nonlinear) are based on the fact that those solutions define a slow manifold of dimension \(n \ll N\) embedded in the space \(\mathbb{R}^N\). This paper explores such a behavior in systems exhibiting phase transitions in order to analyze the evolution of the local dimensionality \(n\) when the system moves from one side of the critical behavior to the other. For that purpose we consider the Ising model.
MSC:
74 Mechanics of deformable solids
68 Computer science
35 Partial differential equations
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References:
[1] 10.2514/1.35374
[2] 10.1007/s00220-006-0143-9 · Zbl 1117.82006
[3] 10.1007/s10955-008-9603-2 · Zbl 1161.82313
[4] 10.1007/s10955-008-9677-x · Zbl 1167.82009
[5] ; Fisher, Lectures in theoretical physics, 1, (1965)
[6] 10.1109/T-C.1971.223208 · Zbl 0216.50201
[7] 10.1007/s10439-010-9976-7
[8] 10.1007/BF02980577
[9] 10.5566/ias.v27.p183-192 · Zbl 1247.94008
[10] 10.1162/neco.1997.9.7.1493
[11] 10.1007/s11837-011-0037-y
[12] 10.1007/978-0-387-39351-3 · Zbl 1128.68024
[13] 10.1007/s11831-016-9172-5 · Zbl 1390.74196
[14] 10.1063/1.1699114
[15] 10.1103/RevModPhys.25.353 · Zbl 0053.18601
[16] 10.1007/s00407-004-0088-3 · Zbl 1068.82001
[17] 10.1103/PhysRev.65.117 · Zbl 0060.46001
[18] ; Polito, Advances in Neural Information Processing Systems 14, 1255, (2001)
[19] 10.1126/science.290.5500.2323
[20] 10.1162/089976698300017467
[21] ; Schölkopf, Advances in kernel methods, 327, (1999)
[22] 10.1126/science.290.5500.2319
[23] 10.1016/j.matdes.2017.10.023
[24] 10.1016/j.compmedimag.2014.06.001
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