Nonlinear thermodynamic computing out of equilibrium

TL;DR

This work demonstrates a nonlinear thermodynamic computer that can perform arbitrary nonlinear computations at specified observation times by leveraging a network of thermodynamic neurons with a quartic nonlinearity and Langevin dynamics. By training a digital twin of layered thermodynamic networks with a genetic algorithm, the authors show learning of nontrivial functions (e.g., cosine) and MNIST digit classification at fixed observation times, even away from thermal equilibrium. The key contributions include (i) design and analysis of equilibrium and nonequilibrium neuron activation, (ii) a scalable digital model for programmable thermodynamic neural networks, and (iii) demonstrations of nonlinear function approximation and vision-task classification powered by thermal fluctuations, with discussion of noise robustness and hardware feasibility.

Abstract

We present the design for a thermodynamic computer that can perform arbitrary nonlinear calculations in or out of equilibrium. Simple thermodynamic circuits, fluctuating degrees of freedom in contact with a thermal bath and confined by a quartic potential, display an activity that is a nonlinear function of their input. Such circuits can therefore be regarded as thermodynamic neurons, and can serve as the building blocks of networked structures that act as thermodynamic neural networks, universal function approximators whose operation is powered by thermal fluctuations. We simulate a digital model of a thermodynamic neural network, and show that its parameters can be adjusted by genetic algorithm to perform nonlinear calculations at specified observation times, regardless of whether the system has attained thermal equilibrium. This work expands the field of thermodynamic computing beyond the regime of thermal equilibrium, enabling fully nonlinear computations, analogous to those performed by classical neural networks, at specified observation times.

Figures (16)

• Figure 1: (a) A thermodynamic circuit whose interaction energy is given by Eq. (\ref{['nrg']}) can function as a thermodynamic neuron. The circle represents a scalar degree of freedom $x$. The curved line represents its intrinsic energy, the terms in ${\bm J}=(J_2,J_3,J_4)$ in Eq. (\ref{['nrg']}). The straight line represent an input signal or bias, the term in $I$ in Eq. (\ref{['nrg']}). (b) Equilibrium activation function $\langle x \rangle_0$ of the neuron, Eq. (\ref{['act']}), as a function of the neuron input $I$, for the case $\beta=1$. The vector ${\bm J}=(J_2,J_3,J_4)$ sets the values of the intrinsic couplings of the neuron. The quadratic-potential activation function is linear, while the quartic-potential activation function is nonlinear. (c) Dynamical evolution (\ref{['lang1']}) of the quadratic-potential neuron, for $\beta=100$, for 11 evenly-spaced values of $I$. (d) The same for the quartic-potential neuron. For times longer than some short threshold, the finite-time response is a nonlinear function of $I$.
• Figure 2: (a,b) Elements of a thermodynamic computer analogous to a neural network. (a) The thermodynamic neurons described in Fig. \ref{['fig1']} are connected by bilinear couplings. Top: a single circle implies a neuron $x_i$ of the type described in Fig. \ref{['fig1']}, with an input (bias) $I=b_i$ and a nonlinear potential parameterized by ${\bm J}=(J_1,J_2,J_3)$. Bottom: lines between circles imply a bilinear coupling $J_{ij} x_i x_j$. (b) With the visual shorthand described in panel (a), we consider layered networks of such neurons, with adjacent layers coupled all-to-all, having total potential energy (\ref{['pot_tot']}). (c) Training a simulation model of a thermodynamic computer to express a nonlinear function at a specified observation time. We show loss (\ref{['phi']}) as a function of evolutionary time $n$ for a layered thermodynamic computer built from quadratic neurons (gray) or quadratic-quartic neurons (green). (d) Output (\ref{['out']}) at observation time $t_{\rm f}=1$ of the linear computer (gray) and the nonlinear computer (green), as a function of the input $z$, averaged over $M=10^3$ samples. The target function is shown as a black line. (e) Mean neuron activations measured at observation time $t_{\rm f}$ as a function of the neuron inputs at the same time, for the nonlinear model. The color band denotes $\pm$ one standard deviation. (f) Output (\ref{['out']}) at time $t_{\rm f}=1$ of the trained nonlinear thermodynamic computer as a function of input $z$, computed using $M$ samples. The target function is shown as a black line (training was done using $M=10^3$ samples). (g) Output (\ref{['out']}) at various observation times $t_{\rm f}$ of the trained nonlinear thermodynamic computer, as a function of input $z$, computed using $M=10^3$ samples. The target function is shown as a black line. The computer is trained so that it reproduces the target function when observed at time $t_{\rm f}=1$.
• Figure 3: Training a simulation model of a thermodynamic computer to classify MNIST. The computer, which consists of a 3-layer network of quadratic-quartic $(1,0,1)$ neurons, is trained in reset-sampling mode, using $M=10^3$ samples taken at observation time $t_{\rm f}=1$. (a) Loss (cross-entropy) and training-set classification accuracy as a function of evolutionary time $n$. (b) For a single digit, an 8, we show the probability distribution, taken over $10^5$ samples, of the computer's per-sample class score. The mean value of each distribution, which is the value used for classification, is indicated at the top of the panel. The correct distribution is shown in green, the others in shades of blue. (c) The class probabilities of the computer, upon being shown the indicated digit, for various observation time $t_{\rm f}$. The computer is trained to classify the digit at an observation time $t_{\rm f}=1$ (vertical dotted line). (d) Test-set classification accuracy of the trained computer, as a function of the number of samples $M$ generated by the computer (each taken at observation time $t_{\rm f}=1$). (e) Test-set accuracy of the trained computer at a range of energy scales. The computer is trained at the energy scale $J=10 k_{\rm B} T$.
• Figure S1: (a) A thermodynamic circuit whose interaction energy is given by Eq. (\ref{['supp_nrg']}) can function as a thermodynamic neuron. In panels (b--e) we show the equilibrium activation function $\langle x \rangle_0$ of the neuron, Eq. (\ref{['supp_act']}), as a function of the neuron input $I$, for the case $\beta=1$. The vector ${\bm J}=(J_2,J_3,J_4)$ sets the values of the intrinsic couplings of the neuron. The top panels in (b--e) show the potential (\ref{['supp_nrg']}) at zero input, for the quadratic case (gray) and the case introduced in the lower panel (blue or green). (b) Purely quadratic (gray) and quartic (blue) cases. The purely quartic case (with $J_4>0$) is the simplest case that is thermodynamically stable and admits a nonlinear activation function. Both cases are shown for reference in the following panels. Inset: gradient of activation function on the same horizontal scale as the main panel. (c) Example in which a quadratic coupling is included with the quartic coupling. (d) Example in which a cubic coupling is included with the quartic coupling. (e) Example in which all three couplings are nonzero.
• Figure S2: (a) Equilibrium fluctuations (\ref{['supp_var']}) of the thermodynamic neuron of Fig. \ref{['supp_fig1']}(a), for the case $\beta=100$. The addition of the quadratic coupling to the quartic one (green), suppresses fluctuations relative to the pure quartic case (blue). The mean equilibrium activation functions of those two cases are similar; see Fig. \ref{['supp_fig1']}(c). The inset shows the largest fluctuations of the purely quartic neuron to be many times that of the purely quadratic neuron. (b,c) The mean (b) and variance (c) of the equilibrium activation function of the $(1,0,1)$ neuron depend on temperature.