Multistable Physical Neural Networks

TL;DR

The paper addresses realizing computation and memory in mechanical hardware by embedding bistable chambers into physical neural networks (PNNs). It develops a bistable flow-network model with $Q_{ij} = C_{ij}(p_j - p_i)$, $C_{ij} = 1/R_{ij}$ and a common bistable pressure–volume relation $p_i = f(v_i)$, and analyzes equilibrium and stability via a graph Laplacian $W$ and an effective potential. Two training paradigms are proposed: global supervised learning to design topology and conductances, and local physical supervised learning that adjusts tube conductances via local signals, enabling memory-enabled, multi-task computation. Numerical demonstrations on lattices of bistable nodes show the network can store memory and perform pattern-writing and task-oriented computations with a single input in soft-actuation contexts. The work demonstrates a route to computational matter, with potential impact on smart metamaterials, soft robotics, and microfluidic devices, representing a step toward computational matter.

Abstract

Artificial neural networks (ANNs), which are inspired by the brain, are a central pillar in the ongoing breakthrough in artificial intelligence. In recent years, researchers have examined mechanical implementations of ANNs, denoted as Physical Neural Networks (PNNs). PNNs offer the opportunity to view common materials and physical phenomena as networks, and to associate computational power with them. In this work, we incorporated mechanical bistability into PNNs, enabling memory and a direct link between computation and physical action. To achieve this, we consider an interconnected network of bistable liquid-filled chambers. We first map all possible equilibrium configurations or steady states, and then examine their stability. Building on these maps, both global and local algorithms for training multistable PNNs are implemented. These algorithms enable us to systematically examine the network's capability to achieve stable output states and thus the network's ability to perform computational tasks. By incorporating PNNs and multistability, we can design structures that mechanically perform tasks typically associated with electronic neural networks, while directly obtaining physical actuation. The insights gained from our study pave the way for the implementation of intelligent structures in smart tech, metamaterials, medical devices, soft robotics, and other fields.

Figures (7)

• Figure 1: Illustration of a hierarchical metamaterial structure composed of bistable nodes interconnected by rigid tubes, showcasing the concept of design of the physical neural network in advanced metamaterial framework (image generated by OpenAI's DALL-E). Each node within this network is characterized by a distinct non-monotonic pressure-volume relationship, leading to two practical and stable equilibrium phases, symbolized as $v_{b=0}$ and $v_{b=1}$, or more simply, '0' and '1'. Illustrated by the blue curve is the node's elastic potential, which clearly delineates two stable states separated by an unstable state, associated with the spinodal branch.
• Figure 2: A theoretical investigation of a grounded flow network with four bistable nodes dictated by external pressure $p_{BC}$. The viscous resistances are denoted by $R_{1,2,3,4}$. (a) Typical pressure-volume curve of a bistable element. The domain of volumes in binary states '0' and '1' is $v\in v_{b=0}$ and $v\in v_{b=1}$, respectively. A pressure higher than $p_{max}$ or lower than $p_{min}$ is denoted by $p_{u}$ and $p_{d}$, respectively, and the volumes are defined in a one-value manner. For a pressure within the bistable domain, there are three different options for volume, marked by orange dots. (b) $\{v_{1};v_{2}\}$ space to describe the dynamic solutions of the system for the case where $R_{1}/R_{3}=R_{2}/R_{4}$. The black lines are solutions of the steady state equations. The intersection between these lines describes the fixed points of the system (green-stable, red-unstable). The gray arrows describe the dynamic solution field, namely, $\{v_{1}(t),v_{2}(t)\}$. Several solution trajectories are presented for different initial conditions (marked with empty circles). The colors of the curves have no meaning. (c) $\{v_{1};v_{2}\}$ space to describe the network dynamics for the case where $R_{1}/R_{3}<R_{2}/R_{4}$. In the described map, there is one fixed point described by the stable binary state (0,1) to which the network reaches from any initial conditions. (d) $\{v_{1};v_{2}\}$ space to describe the network dynamics for the case where $R_{1}/R_{3}>R_{2}/R_{4}$. In the described map, there is one fixed point described by the stable binary state (1,0) to which the network reaches from any initial conditions.
• Figure 3: Illustration of two numerical simulations showcasing the learning capabilities of a metamaterial composed of a bistable flow network. The elastic nodes are modeled as hyperelastic balloons, utilizing the Ogden model Ogden. These balloons incorporate a bistable regime (between 0.8Pa to 1.1Pa), transitioning from binary state '0' to '1' at a volume of 2.55cc and reverting from state '1' to '0' at 22cc. Learning parameters included learning rate $\eta=0.1$ (see Eq. (\ref{['eq: PGD']})) and $\beta=10^{-5}$. steady state convergence pressure was set at 0.9Pa. A constant flux was introduced into the system during a specific period of time, allowing it to reach equilibrium. The initial viscous resistances were set to unity. Panel (a) displays the results for the first network, indicating the obtained connection topology, colored by viscous resistance and thickness. The middle section depicts the system results in equilibrium, highlighting the inlet and target balloons. The lower part shows volume changes over time (in blue) and the entering flow (in red), obtained by numerical integration of (\ref{['eq: Equation of motion']}) using ode45 in Matlab. Those nodes that do not snap to the binary state '1' remain nearly at their initial volume, while others snap through in a sequence defined by the resistance configuration. Some snapped nodes exhibit so similar dynamic responses that their graphs appear virtually indistinguishable. Panel (b) exhibits the results for the second network, which mastered five tasks, visually displaying the digit representing the inserted flux's entry balloon in the input layer.
• Figure 4: Demonstration of bistable PNNs memory. The input layer of this network is composed of two nodes. The next layer is composed of a 5x5 node lattice, with two nodes as the final layer. We examine two identical topology structures (a), highlighted with green and orange frames. Flow signals are directed to nodes 1 and 2 in each structure's input layer. Once both networks have reached equilibrium (b), both are given the same input signal through node number 1, and are allowed to stabilize again (c). There is one column of nodes in the binary state '1' within the lattice of the first network, and all other nodes are in the binary state '0'. In contrast, the equilibrium state of the second network features two columns of nodes in the binary state '1,' which indicates that the network's response can vary significantly depending on history and, therefore, the initial state.
• Figure 5: (Caption next page.)

Theorems & Definitions (7)

• Proposition 1
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Proposition 7