Related papers: A gradient flow formulation for the stochastic Ama…
We provide gradient flow interpretations for the continuous-time continuous-state Hopfield neural network (HNN). The ordinary and stochastic differential equations associated with the HNN were introduced in the literature as analog…
This is a sequel to our paper `On the kernel learning problem'. We identify a canonical choice of Riemannian gradient flow, to find the stationary points in the kernel learning problem. In the presence of Gaussian noise variables, this flow…
Sampling a target probability distribution with an unknown normalization constant is a fundamental challenge in computational science and engineering. Recent work shows that algorithms derived by considering gradient flows in the space of…
We study both the local and global existence of a gradient flow of the Sinai-Ruelle-Bowen entropy functional on a Hilbert manifold of expanding maps of a circle equipped with a Sobolev norm in the tangent space of the manifold. We show…
We study the power spectrum of a space-time dependent neural field which describes the average membrane potential of neurons in a single layer. This neural field is modelled by a dissipative integro-differential equation, the so-called…
We investigate two-dimensional neural fields as a model of the dynamics of macroscopic activations in a cortex-like neural system. While the one-dimensional case has been treated comprehensively by Amari 30 years ago, two-dimensional neural…
This paper is devoted to the investigation of gradient flows in asymmetric metric spaces (for example, irreversible Finsler manifolds and Minkowski normed spaces) by means of discrete approximation. We study basic properties of curves and…
Linear networks provide valuable insights into the workings of neural networks in general. This paper identifies conditions under which the gradient flow provably trains a linear network, in spite of the non-strict saddle points present in…
Consider the stochastic evolution equation in a separable Hilbert space with a nice multiplicative noise and a locally Dini continuous drift. We prove that for any initial data the equation has a unique (possibly explosive) mild solution.…
Perturbative calculations of gradient flow observables are technically challenging. Current results are limited to a few quantities and, in general, to low perturbative orders. Numerical stochastic perturbation theory is a potentially…
We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by…
We derive the system of differential equations for the gradient flow characterizing the training process of linear in-context learning in full generality. Next, we explore the geometric structure of the gradient flows in two instances,…
We investigate a variational approach to nonpotential perturbations of gradient flows of nonconvex energies in Hilbert spaces. We prove existence of solutions to elliptic-in-time regularizations of gradient flows by combining the…
Wilson-Cowan and Amari-type models capture nonlinear neural population dynamics, providing a fundamental framework for modeling how sensory and other exogenous inputs shape activity in neural tissue. We study the controllability properties…
Sampling a probability distribution with an unknown normalization constant is a fundamental problem in computational science and engineering. This task may be cast as an optimization problem over all probability measures, and an initial…
In dually flat manifolds, there is a deep connection between gradient flows and pregeodesics. This was one of the many important contributions of Amari to information geometry. In this paper, we extend the study of this relationship to…
We derive curvature flows in the Heisenberg group by formal asymptotic expansion of a nonlocal mean-field equation under the anisotropic rescaling of the Heisenberg group. This is motivated by the aim of connecting mechanisms at a…
The dynamics of gradient and Hamiltonian flows with particular application to flows on adjoint orbits of a Lie group and the extension of this setting to flows on a loop group are discussed. Different types of gradient flows that arise from…
The brain as an astonishingly remarkable device has been studied from various angles. It is now well known that neurons are the seat of all activities of the brain function. The dynamical properties pertaining to a single neuron and a…
Field inversion and machine learning are implemented in this study to describe three-dimensional separation flow around an axisymmetric hill and augment the Spart-Allmaras model. The discrete adjoint method is used to solve the field…