Training overdamped dynamics

TL;DR

This work develops a framework for manipulating overdamped dynamics through local, physically motivated update rules inspired by ideas from physical learning and directed aging, and derives approximate directed-aging and equilibrium-propagation rules tailored to dissipative systems.

Abstract

In regimes where inertia is negligible, the temporal evolution is governed by overdamped dynamics. This limit is particularly relevant in soft-matter contexts, such as polymers, colloidal suspensions, and processes occurring at the cellular scale. Being able to manipulate the dynamics of such many-particle systems would enable control over rate-dependent elastic responses, time-dependent material properties, relaxation processes, and perhaps the hydrodynamics of suspensions. In this work, we develop a framework for manipulating overdamped dynamics through local, physically motivated update rules. Our approach is inspired by ideas from physical learning and directed aging, in which microscopic parameters adapt autonomously to endow a material with a desired function. Using the Rayleighian formulation, whose minimization reproduces the overdamped equations of motion, we derive approximate directed-aging and equilibrium-propagation rules tailored to dissipative systems. To demonstrate these ideas, we study a disordered Maxwell material that behaves elastically at short times but flows at long times. By locally modifying the viscous damping, we show that one can tune the viscous Poisson's ratio and shape local mechanical responses. These results illustrate how materials can be trained to exhibit targeted rate-dependent elastic and viscous behaviors.

Figures (5)

• Figure 1: Training with directed aging to control the Poisson's ratio under compression. (a) The Poisson's ratio as a function of strain for different actuation rates. (b) The Poisson's ratio at the training strain as a function of actuation rate, $\dot{\epsilon}_m$. The results are obtained for different training rates. Parameters: $N=512$, $\dot{\epsilon}_{Age}\gamma_{i}\left(0\right)/k=8\cdot10^{-4}$, $\epsilon_{Age}=0.02$ and number of training cycles is $10^5$. In (a): $r_{Age}=10^{5}$.
• Figure 2: Training with directed aging to control the Poisson's ratio under expansion. (a) The Poisson's ratio as a function of strain (expansion) for different actuation rates. (b) The Poisson's ratio at the training strain as a function of actuation rate. The results are obtained for different training rates. Parameters: $N=512$, $\dot{\epsilon}_{Age}\gamma_{i}\left(0\right)/k=4\cdot10^{-3}$, $\epsilon_{Age}=0.02$ and number of training cycles is $10^5$.
• Figure 3: Training local response with directed aging. (a) An illustration of the network with four sources and a single target. (b) The strain on the target as a function of the strain on the source. (c) The average ratio of the strain of the target to the source, as a function of actuation rate for different number of sources. (d) The distribution of the inverse dissipation parameters appears to be power-law, similarly to the results for the Poisson's ratio. Parameters: $N=256$, $\dot{\epsilon}_{Age}\gamma_{i}\left(0\right)/k=2\cdot10^{-3}$, $\epsilon_{Age}=0.02$ and number of training cycles is $10^5$. In (b) $N_{S}=10$.
• Figure 4: Manipulating simultaneously both the elastic and viscous response using directed aging. (a) The Poisson's ratio as a function of strain for different actuation rates. (b) The Poisson's ratio at the training strain as a function of actuation rate. The results are obtained for different training rates. Simulation parameters: $r_{Age}=5\cdot10^{5},$$\epsilon_{Age}=-0.02$, $N=512$, $\dot{\epsilon}_{train}=5$, $\dot{\epsilon}_{train\,k}r_{k}=0.01,$$\gamma_{0}=1$, $k_{0}=1$ and number of training cycles is $10^5$.
• Figure 5: Using equilibrium propagation to train localized response. (a) The error as a function of the number of iterations. (b) The strain on the target as a function of the strain on the source. Blue: We nudge the system with a force that is proportional to the difference in the velocity and the desired velocity $\left(\dot{\epsilon}_{T}-\dot{\epsilon}_{D}\right)$. Red: The nudging depends on the distance between the nodes of the target and their desired value. The dashed line corresponds to the desired strain. Here, $N=256$, actuation time normalized by dissipation coefficient $\frac{\epsilon_{Age}k}{\dot{\epsilon}_{Age}\gamma}=0.04$, $\epsilon_{Age}=0.02$, $N_{S}=5$ and $\beta=0.001$.