Communications on Pure and Applied Mathematics最新文献

This paper studies the infinite-width limit of deep linear neural networks (NNs) initialized with random parameters. We obtain that, when the number of parameters diverges, the training dynamics converge (in a precise sense) to the dynamics obtained from a gradient descent on an infinitely wide deterministic linear NN. Moreover, even if the weights remain random, we get their precise law along the training dynamics, and prove a quantitative convergence result of the linear predictor in terms of the number of parameters. We finally study the continuous-time limit obtained for infinitely wide linear NNs and show that the linear predictors of the NN converge at an exponential rate to the minimal $�{\ell }_2$-norm minimizer of the risk.

We study the Calogero–Moser derivative nonlinear Schrödinger NLS equation

A classical problem in fluid dynamics concerns the interaction of multiple vortex rings sharing a common axis of symmetry in an incompressible, inviscid three-dimensional fluid. In 1858, Helmholtz observed that a pair of similar thin, coaxial vortex rings may pass through each other repeatedly due to the induced flow of the rings acting on each other. This celebrated configuration, known as leapfrogging, has not yet been rigorously established. We provide a mathematical justification for this phenomenon by constructing a smooth solution of the 3D Euler equations  exhibiting this motion pattern.

We study uniformly random lozenge tilings of general simply connected polygons. Under a technical assumption that is presumably generic with respect to polygon shapes, we show that the local statistics around a cusp point of the arctic curve converge to the Pearcey process. This verifies the widely predicted universality of edge statistics in the cusp case. Together with the smooth and tangent cases proved by Aggarwal-Huang and Aggarwal-Gorin, these are believed to be the three types of edge statistics that can arise in a generic polygon. Our proof is via a local coupling of the random tiling with nonintersecting Bernoulli random walks (NBRW). To leverage this coupling, we establish an optimal concentration estimate for the tiling height function around the cusp. As another step and also a result of potential independent interest, we show that the local statistics of NBRW around a cusp converge to the Pearcey process when the initial configuration consists of two parts with proper density growth, via careful asymptotic analysis of the determinantal formulas.

We introduce a simplified model for wave turbulence theory—the Wick nonlinear Schrödinger equation, of which the main feature is the absence of all self-interactions in the correlation expansions of its solutions. For this model, we derive several wave kinetic equations that govern the effective statistical behavior of its solutions in various regimes. In the homogeneous setting, where the initial correlation is translation invariant, we obtain a wave kinetic equation similar to the one predicted by the formal theory. In the inhomogeneous setting, we obtain a wave kinetic equation that describes the statistical behavior of the wavepackets of the solutions, accounting for both the transport of wavepackets and collisions among them. Another wave kinetic equation, which seems new in the literature, also appears in a certain scaling regime of this setting and provides a more refined collision picture.

We study the Langevin dynamics for spherical $�p$-spin models, focusing on the short time regime described by the Cugliandolo–Kurchan equations. Confirming a prediction of Cugliandolo and Kurchan, we show the asymptotic energy achieved is exactly $��E_{\infty }��\left(�p�\right)��=�2�\sqrt{�\frac{�p�-�1�}{p}�}�$ in the low temperature limit. The upper bound uses hardness results for Lipschitz optimization algorithms and applies for all temperatures. For the lower bound, we prove the dynamics reaches and stays above the lowest energy of any approximate local maximum. In fact the latter behavior holds for any Hamiltonian obeying natural smoothness estimates, even with disorder-dependent initialization and on exponential time-scales.

Given any (not necessarily connected) combinatorial finite graph and any compact smooth 6-manifold $�M^6$ with the third Betti number $��b_3�\ne �0�$, we construct a calibrated 3-dimensional homologically area minimizing surface on $�M$ equipped in a smooth metric $�g$, so that the singular set of the surface is precisely an embedding of this finite graph. Moreover, the calibration form near the singular set is a smoothly $��G�L�(�6�,�R�)�$ twisted special Lagrangian form. The constructions are based on some unpublished ideas of Professor Camillo De Lellis and Professor Robert Bryant.

For the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta-convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta-convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

We prove that every non-degenerate minimal submanifold of codimension two can be obtained as the energy concentration set of a family of critical maps for the (rescaled) Ginzburg–Landau functional. The proof is purely variational, and follows the strategy laid out by Jerrard and Sternberg, extending a recent result for geodesics by Colinet–Jerrard–Sternberg. The same proof applies also to the $��U�\left(�1�\right)�$-Yang–Mills–Higgs and to the Allen–Cahn–Hilliard energies. While for the latter energies gluing methods are also effective, in general dimension our proof is by now the only available one in the Ginzburg–Landau setting, where the weaker energy concentration is the main technical difficulty.