Yoichiro Mori

Han Zhou, Yuan-Nan Young, Yoichiro Mori

In this work, we present a mathematical and computational framework to model the dynamics of open lipid bilayer membranes interacting with ambient Stokes flow. The model explicitly couples the three-dimensional viscous fluid, the two-dimensional membrane surface, and its one-dimensional free edge. We develop an axisymmetric hybrid BEM-FEM method that solves the problem with an effective one-dimensional formulation. A key component is a local mesh refinement strategy designed to accurately resolve singularities and boundary layers originating at the membrane edge. Several numerical examples are provided to showcase its ability to capture intricate edge dynamics and multiscale fluid-membrane coupling.

Charles L. Epstein, Yoichiro Mori, Han Zhou

We study a bulk-surface coupled Laplace system involving an embedded open boundary. The problem is reformulated as an integro-differential equation using boundary integral representations, for which we establish existence and uniqueness of the solution. A Wiener-Hopf technique is employed to study the solution regularity and derive asymptotic expressions for the edge singularity. Building on these results, we develop a finite element method that incorporates the singularity structure and provide a rigorous error analysis. Numerical experiments confirm the theoretical convergence rates.

Joshua A. McGinnis, Xinbo Li, Yoichiro Mori

Physical learning methods train physical networks to perform computational tasks using only local update rules, exploiting the physics of the system to handle the global transfer of information. We provide the first local convergence analysis of three such methods -- Equilibrium Propagation (EP), Coupled Learning (CL), and a new method we call Adjoint Coupled Learning (AL) -- for linear circuits, in the limit of small-nudging for both discrete and continuous time. EP and AL perform gradient descent on a natural loss function, while CL follows modified dynamics with an additional cubic correction. Assuming the existence of a solution, we identify a coercivity condition, expressed as a rank condition on a matrix built from the network's incidence structure, under which the training loss decays exponentially and the parameters converge to the solution manifold. We show that coercivity can fail by exhibiting a kite circuit in which a symmetry causes the coercivity constant to degenerate on the solution manifold, but prove using Sard's theorem that such degeneracies are non-generic: coercivity holds at every point of the solution manifold for almost every choice of desired output.

Joshua A. McGinnis, Adam G. Kline, Yoichiro Mori

In this paper we show that the physical learning methods known as coupled learning (CL) and equilibrium propagation (EP) conserve a mass-like quantity in the trainable parameters in the continuous-time, small-nudging limit. We prove that this conservation holds in a broad range of physically relevant settings. We then show that the conservation law constrains the training dynamics in a way that makes convergence reliable in important settings for linear circuits. We conclude by discussing some practical implications of this conservation law.