Thermodynamic Computing

Generative thermodynamic computers turn thermal noise into structured data through Langevin dynamics. We train these systems with a local update at each integration step. The reverse-path Onsager-Machlup objective yields a coupling gradient that is a symmetric sum of local residual-state correlations. We apply this gradient immediately rather than accumulating it over a full trajectory. In digital simulations using MNIST prototypes, online and trajectory-batch training reach similar validation losses on fixed noising paths. Models trained online release less heat on average in all five independently seeded pairs, with both models' parameters held fixed during sampling. Auxiliary classifier and nearest-prototype measures change modestly, while pairwise diversity decreases. The response to noise depends strongly on where the errors enter: independent zero-mean errors in the formed updates produce little heat change over a finite range of noise amplitudes, whereas residual offset and temporal correlation have much larger effects. Storing trained couplings requires substantially less precision than resolving deterministic updates during training. Together, these results establish a local online training method and show how update timing, noise structure, and precision affect generative thermodynamic computing.

Thermodynamic computers are stochastic physical devices designed to perform calculations at the thermal energy scale. Their operation is constrained by the equations of stochastic thermodynamics, among which are a set of bounds, known as speed limits, that relate a thermodynamic computer's run time to its computational progress and the heat it dissipates. Using the Wasserstein speed limit we assess the thermodynamic efficiency of a simulation model of a thermodynamic computer trained to perform a standard machine-learning classification task. On this task the thermodynamic computer is as capable as a simple multilayer perceptron. We show that different inference protocols allow the computer to operate within 40% of the thermodynamic limit of efficiency without loss of accuracy, or to perform inference increasingly rapidly at fixed accuracy and thermodynamic efficiency. These results indicate that a thermodynamic computer designed for a particular task retains considerable freedom in its thermodynamic operation.

We present a set of tools for mapping general stochastic programs to thermodynamic hardware designed for energy-efficient stochastic sampling. Given a target stochastic program expressed as a Directed Factor Graph (DFG) of stochastic channels, or equivalently as a Parametrized Stochastic Circuit (PSC), we first introduce a method to approximately compile each factor in the DFG to an Energy-Based Model (EBM) that is native to the hardware. We then analyze how the error of the compiled DFG accumulates from the per-factor errors, and introduce two training refinements, context matching and trajectory-level REINFORCE post-training, which can reduce the residual error left by training each factor in isolation. The \texttt{thermalizers} framework takes a stochastic program expressed in the \texttt{torx} library and replaces its factors with thermodynamic kernels implemented and sampled using the \texttt{thrml} library. We demonstrate it on several example applications, including a market simulator that learns the joint day-to-day dynamics of a panel of financial time series from recorded market history alone, a probabilistic model from mathematical ecology, Gibbs sampling of an EBM the hardware cannot natively express, and a sequential Bayesian design loop over a Gaussian stochastic circuit.

Thermodynamic computing is an emerging hardware paradigm, in which stochastic physical dynamics serve as the direct computational primitive. The recent explosion of generative AI has only sharpened the search for alternative approaches to compute, and, as we show in this work, thermodynamic computing turns out to be well suited to this space. An important class of methods realises a function as the stationary expectation of an ergodic stochastic process: the answer is encoded in the time-averaged statistics of an equilibrating trajectory. To date, this equilibration-style class has been formulated exclusively through Langevin dynamics, restricting its implementations to analogue substrates and the engineering challenges those bring. In this work, we propose a substrate-independent formalisation of the equilibration-style formulation, in which the only object of design is the dynamical generator L* of an arbitrary ergodic process. The formalisation makes three hardware-level properties of the formulation explicit: the precision of a result is a knob set by how long the dynamics are run, sample averages decompose across independent trajectories, and dependent stages of a computation operate concurrently rather than serially, a property we call sequential parallelism. We instantiate the formalisation by fabricating a prototype digital thermodynamic computing chip, named CN101, that implements the formulation through discrete accumulator dynamics on standard CMOS using stochastic computing principles. We characterise CN101's success across conventional generative AI workloads in the form of VAEs and flow matching, applied to both image generation and scientific problems. Together, the formalisation and its digital instantiation show that the equilibration-style formulation is substrate-independent, and that its computational properties can be exploited on standard digital hardware.

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.

Nonlinear thermodynamic computers based on Langevin dynamics exploit thermal fluctuations as a physical substrate for computation. Recent work has shown that quartic-confined fluctuating degrees of freedom can act as thermodynamic neurons capable of nonlinear function approximation at finite observation times. Here we extend this paradigm from mean-only readout to moment-resolved readout. Instead of representing each driven reservoir solely by its first moment, we construct a response vector from the elementwise raw polynomial moments $\mathbb{E}[\bm{x}]$, $\mathbb{E}[\bm{x}^{\odot 2}]$, and $\mathbb{E}[\bm{x}^{\odot 4}]$. These observables combine displacement and central-shape contributions and are naturally aligned with the linear, quadratic, and quartic terms of the local driven dynamics. We further introduce a heterogeneous multi-reservoir architecture in which three reservoirs with distinct initialization and training histories form a joint $2304$-dimensional response representation. Under the fixed MNIST $60000/10000$ reproduction protocol, feature-level fusion achieves the best observed accuracy of $9695/10000=96.95\%$, compared with $9682/10000=96.82\%$ for the strongest single-reservoir model and $9684/10000=96.84\%$ for equal-weight logit averaging. An exact paired McNemar test does not establish a statistically significant improvement over the strongest single reservoir, but the ablation and wrong-set overlap results provide suggestive evidence of complementary classification errors. These results motivate higher-order polynomial-moment readout and reservoir heterogeneity as candidate design principles for finite-time Langevin computing.

Thermodynamic computing devices based on the Ising model show great promise for low-power AI inference and edge computing, but scalable methods for training large models for such hardware remain limited. Prior theory shows that the time-averaged behavior of high-temperature Gibbs-sampled Ising systems can implement feed-forward neural inference. We turn this theoretical correspondence into a scalable and purely backpropagation-based algorithm for training deep convolutional networks for thermodynamic inference on Ising machine hardware. Our image classification models achieve accuracies of 94.9% on CIFAR-10 and 76.0% on CIFAR-100 under binary Gibbs sampling. We then develop and experimentally validate a mathematical theory relating inference cost to accuracy and controlling autocorrelation times. Subsequently, we calculate asymptotic results showing that inference cost is bounded by a well-controlled tradeoff with performance and exhibit algorithms for computing optimal inference schedules. Finally, we discuss implications for hardware development and the future of high-temperature thermodynamic AI models.

The growing energy demand for computation is becoming increasingly unsustainable. Thermodynamic computing, which harnesses physical thermal fluctuations as a computational resource rather than suppressing them, offers orders-of-magnitude energy savings for probabilistic and combinatorial tasks. Pharmaceutical R&D, heavily reliant on computational optimization and sampling, is a natural application domain. Here we present what is, to our knowledge, the first concrete pharmaceutical application mapped to thermodynamic hardware with energy estimates grounded in prototype measurements. We reduce mRNA codon optimization, a combinatorial problem routinely solved in drug development, to sampling from an Ising model, making it directly executable on a thermodynamic sampling unit (TSU). Benchmarking three approaches (Potts sampling, Ising sampling, and a genetic algorithm baseline) on the SARS-CoV-2 spike protein, we find that all achieve comparable optimization quality (scores ~234-240), but energy estimates based on validated hardware models indicate that a TSU could solve this problem using approximately 10e6 times less energy than a conventional GPU. All code is released under an open-source license.

We introduce thermodynamic networks, a general framework for autonomous, physics-based computation using non-equilibrium steady states. These networks are modeled as a collection of finite-size reservoirs that exchange conserved quantities--such as electric charge or molecular number--while relaxing to a non-equilibrium steady state, which encodes the solution of a computational problem. We identify Negative Differential Conductance (NDC) as the critical physical property governing the computational expressivity of the thermodynamic network. While networks lacking NDC are restricted to computing monotonic functions, the presence of NDC enables universal function approximation. For the training of the network, we use protocols that take advantage of the natural tendency of the system to equilibrate. We illustrate the versatility of our approach via two different platforms: quantum dot networks and enzymatic reaction networks. Both systems can be engineered to have NDC, enabling high performance in standard benchmarks, including sine function approximation and MNIST digit classification. Overall, our work establishes a rigorous link between non-equilibrium steady states and computational expressivity.

The reverse process in score-based diffusion models is formally equivalent to overdamped Langevin dynamics in a time-dependent energy landscape. In our prior work we showed that a bilinearly-coupled analog substrate can physically realize this dynamics at a projected three-to-four orders of magnitude energy advantage over digital inference by replacing dense skip connections with low-rank inter-module couplings. Whether the \emph{training} loop can be closed on the same substrate -- without routing gradients through an external digital accelerator -- has remained open. We resolve this affirmatively: Equilibrium Propagation applied directly to the bilinear energy yields an unbiased estimator of the denoising score-matching gradient in the zero-nudge limit. For finite nudging we derive a sharp bias bound controlled solely by substrate stiffness, local curvature, and the norm of the loss-gradient signal, with a bilinear-specific corollary showing that one dominant bias term vanishes identically for coupling-parameter updates. Symmetric nudging further upgrades the leading bias from $\mathcal{O}(β)$ to $\mathcal{O}(β^2)$ at negligible extra cost. Under realistic finite-relaxation budgets this upgrade is essential, as one-sided EqProp produces anti-correlated gradients while symmetric EqProp yields well-aligned updates. Bias-variance analysis determines the optimal operating point, and end-to-end physical-unit accounting projects a $10^3$-$10^4\times$ energy advantage per training step over a matched GPU baseline. Symmetric bilinear EqProp is the first local, readout-only training rule that preserves the low-rank coupling enabling scalable thermodynamic diffusion models.

Landauer's principle bounds the heat generated by logical operations, but in practice the thermodynamic cost of computation is dominated by the control systems that implement logic. CMOS gates dissipate energy far above the Landauer bound, while laboratory demonstrations of near-Landauer erasure rely on external measurement or feedback systems whose energy costs exceed that of the logic operation by many orders of magnitude. Here we use simulations to show that a genetic algorithm can program a thermodynamic computer to implement logic operations in which the total heat emitted by the control system is of a similar order of magnitude to that of the information-bearing degrees of freedom. Moreover, the computer can be programmed so that heat is drawn away from the information-bearing degrees of freedom and dissipated within the control unit, suggesting the possibility of computing architectures in which heat management is an integral part of the program design.