A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing
By Owen Lockwood, Jérémy Béjanin, Joost Bus, Christopher Chamberland, Patrick Huembeli, Frank Schäfer, Guillaume Verdon
"Proposes thermodynamic computing hardware using stochastic analog circuits for energy-efficient probabilistic ML, enabling differentiable training via equilibrium sampling."
Abstract
To address the escalating energy and latency demands of machine-learning workloads, we introduce a blueprint for an energy-efficient and fast thermodynamic computing stack that leverages stochastic analog processes in physical hardware. In this work, we focus on energy-based thermodynamic computing where the stochastic process is well described by Langevin dynamics with tunable energy potentials. The implementation of such potentials in physical hardware enables us to generate and sample from basic parameterized energy-based models. We demonstrate how to construct and train popular classes of machine learning models based on these hardware-native energy-based models, using the framework of probabilistic graphical models. We analyze the runtime and energy consumption of different models in this thermodynamic paradigm based on theoretical considerations and numerical studies. As a preliminary experimental realization of such hardware, we present our stochastic analog superconducting circuits driven by thermal noise. Together, these results outline a path toward energy-efficient thermodynamic hardware for probabilistic machine learning.
Technical Analysis & Implementation
Overview§
This paper presents a blueprint for thermodynamic computing hardware that leverages stochastic analog processes, specifically Langevin dynamics with tunable energy potentials, to perform energy-efficient probabilistic machine learning. The key idea is to directly map energy-based models (EBMs) onto physical hardware, where the stochastic dynamics naturally perform sampling, eliminating the need for costly digital simulations.
Core Methodology§
The core mathematical framework is Langevin dynamics:
$$ d\mathbf{x}(t) = -\nabla U(\mathbf{x}(t))\,dt + \sqrt{2T}\,d\mathbf{W}(t) $$
where $\mathbf{x}$ are the state variables, $U$ is the energy potential, $T$ is the temperature (noise level), and $\mathbf{W}$ is a Wiener process. The system evolves to sample from the Gibbs distribution $p(\mathbf{x}) \propto e^{-U(\mathbf{x})/T}$.
To train machine learning models, the authors treat the hardware as a differentiable sampler. Given a parameterized energy function $U_\theta(\mathbf{x})$, the gradient of the log-likelihood for a target distribution $p^*(\mathbf{x})$ is:
$$ \nabla_\theta \mathbb{E}_{p^} [\log p_\theta(\mathbf{x})] = \mathbb{E}_{p_\theta} [\nabla_\theta U_\theta(\mathbf{x})] - \mathbb{E}_{p^} [\nabla_\theta U_\theta(\mathbf{x})] $$
This requires sampling from both the model and data distributions, which the hardware provides.
Implementation Details§
The proposed hardware uses superconducting circuits with Josephson junctions, inductors, and capacitors to realize tunable energy potentials. Thermal noise at cryogenic temperatures provides the stochastic driving force. The energy function $U_\theta$ is encoded via programmable circuit elements, allowing rapid reconfiguration for different models.
Code Snippet (Simulation)§
Below is a simplified PyTorch simulation of the Langevin sampler for a harmonic oscillator potential:
import torch
def langevin_sampler(theta, x_init, n_steps, dt, T):
x = x_init.clone()
for _ in range(n_steps):
# Energy U(x) = 0.5 * theta * x^2
grad_U = theta * x
noise = torch.randn_like(x) * torch.sqrt(2 * T * dt)
x = x - grad_U * dt + noise
return x
theta = torch.tensor(1.0, requires_grad=True)
x0 = torch.randn(100)
x_samples = langevin_sampler(theta, x0, 1000, 0.01, 1.0)
# Compute gradient of log-likelihood w.r.t. theta (simplified)
loss = 0.5 * torch.mean(x_samples**2) # proxy for E_p[U]
loss.backward()
print(theta.grad)Results and Analysis§
The authors analyze runtime and energy consumption theoretically, showing orders-of-magnitude improvements over digital CMOS for probabilistic inference tasks. Preliminary experiments with superconducting circuits demonstrate basic sampling functionality.
Conclusion§
This work lays the foundation for thermodynamic computing, combining physics and machine learning to create ultra-energy-efficient hardware for probabilistic models. The differentiable training procedure is key to integrating these physical systems into modern ML pipelines.