Denoising Thermodynamic Models (DTMs)

• DTMs are frameworks that use nonequilibrium thermodynamics to transform noise via reversible stochastic processes.
• They integrate score-based modeling, energy-based diffusion, and physical dynamics to achieve efficient, high-dimensional generative sampling.
• DTMs improve energy efficiency by employing thermodynamically optimal, hardware-based implementations that minimize entropy production.

Denoising Thermodynamic Models (DTMs) comprise a class of generative and inference frameworks that instantiate denoising transformations via the principles and physical dynamics of nonequilibrium thermodynamics. DTMs encompass algorithms, theoretical models, and hardware realizations in which structure emerges from noise through thermodynamically-motivated processes—typically governed by stochastic differential equations (e.g., Langevin dynamics), energy-based diffusion, or Markov chains involving conservative or learnable interactions. DTMs unify approaches that perform denoising via optimized physical or model couplings, emphasizing thermodynamic reversibility, minimal entropy production, and efficient sampling across high-dimensional spaces.

1. Foundational Principles and Theoretical Basis

DTMs are rooted in nonequilibrium statistical mechanics, particularly the concepts of time-reversible stochastic processes and thermodynamic relaxation. The core paradigm is a forward trajectory that transforms structured data into noise (e.g., via diffusion or thermal fluctuation) and a reverse trajectory that reconstructs data by progressively removing noise—a process analogous to inverting entropy increase. The denoising pathway is mathematically linked to score functions, energy gradients, or physical couplings, ensuring that the process is conservative and often thermodynamically optimal.

Within the DTM framework, model instantiations typically include:

• Score-based models: Denoising is realized via score matching, where the gradient of the log-probability density (the score) provides the restoring force under a time-dependent energy $U_t^\theta$ (Máté et al., 4 Jun 2024).
• Energy-based diffusion models (DDMs): Interpolation between a simple reference distribution (e.g., ideal gas) and the target distribution is achieved via a stochastic diffusion process parameterized by a neural or physical energy function.
• Physical dynamical systems: Realized through Langevin equations, where the structure evolves from noise according to trained couplings and biases, often in analog hardware (Whitelam, 18 Jun 2025).

2. Mathematical Formulation and Denoising Process

The canonical DTM process involves two Markov chains:

• Forward process: Gradual stochastic transformation of structured data into noise. For example, in Gaussian diffusion:

$$q(x_t|x_{t-1}) = \mathcal{N}\left(x_t; \sqrt{1-\beta_t} x_{t-1}, \beta_t I\right)$$

for a noise schedule $\beta_t$ (Ho et al., 2020).

• Reverse process (denoising): Learned or physically enacted trajectory that reconstructs data from noise. The generative likelihood maximization or score matching objective drives the parameterization, often as:

$$s(x, t) = \nabla_x \log \rho_t(x) = -\beta \nabla_x U_t^\theta(x)$$

and the noise prediction formulation:

$$s_\theta(x, t) = -\frac{\epsilon_\theta(x, t)}{\sigma_t}$$

where $\epsilon_\theta$ predicts the noise added at time $t$ (Máté et al., 4 Jun 2024, Ho et al., 2020).

Training proceeds by maximizing the likelihood of generating the reverse-time trajectory or minimizing KL divergence between the model and the reference process, often resulting in thermodynamic optimality conditions: $$- \ln \tilde{P}_J[\text{reversed trajectory}] = \sum_{i=1}^N \frac{ \left( -\Delta x_i + \mu \partial_i V(\mathbf{x}') \Delta t \right)^2 }{4 \mu k_B T \Delta t }$$ with analytic gradients for physical couplings (Whitelam, 18 Jun 2025).

3. Model Architectures and Hardware Implementations

DTMs are realized in digital and analog domains:

• Neural network-based architectures: Neural Thermodynamic Integration (Neural TI) uses architectures inspired by molecular force fields (e.g., SchNet), with explicit conditioning on diffusion time and vector-space or manifold energy parameterization. These models sample directly at any intermediate state across hundreds of degrees of freedom, bypassing traditional bottlenecks in thermodynamic integration (Máté et al., 4 Jun 2024).
• Thermodynamic analog computing: Structured data generation is encoded directly in the couplings of the energy landscape of a physical system, governed by Langevin dynamics. Sampling and denoising proceed autonomously via the natural relaxation dynamics, without explicit neural networks or injected noise. This framework is realized in analog hardware such as electrical, mechanical, or superconducting oscillator networks (Whitelam, 18 Jun 2025).
• Probabilistic hardware architectures: All-transistor probabilistic computers implement DTM-like models at the hardware level as chains of energy-based models (EBMs or Boltzmann machines). Each layer is a modular, local graphical model equipped with all-transistor random number generation and Gibbs sampling logic. The system achieves energy-efficient, massively parallel sampling with diffusion-like denoising (Jelinčič et al., 28 Oct 2025).

4. Practical Applications and Performance

DTMs confer distinct advantages in computational physics, machine learning, and hardware efficiency:

• Thermodynamic Integration: Neural TI enables estimation of free-energy differences and excess chemical potentials across phase transitions by sampling all intermediate ensembles—without explicit simulation of each interpolating Hamiltonian. Demonstrated on Lennard-Jones fluids with accurate reproduction of structural and thermodynamic observables (Máté et al., 4 Jun 2024).
• Autonomous Generative Sampling: DTMs implemented via thermodynamic computing produce structured data from noise solely through physical relaxation, minimizing heat emission and entropy production. This paradigm links generative computation directly to thermodynamic cost (Whitelam, 18 Jun 2025).
• Hardware Energy Efficiency: All-transistor DTM computers achieve AI inference at up to $10^4\times$ lower energy per sample compared to GPUs, matching benchmark performance (FID score parity) for image generation tasks (e.g., Fashion-MNIST) (Jelinčič et al., 28 Oct 2025).

Denoising Approach	Sampling Operator	Hardware Realization
Neural TI (energy-based diffusion)	Learned score matching, energy grad.	GPU, digital NN
Thermodynamic computing	Langevin dynamics	Analog: electrical/mechanical
Probabilistic hardware DTMs	Gibbs sampling in Boltzmann chains	All-transistor CMOS

5. Distinction from Conventional Diffusion Models

While diffusion probabilistic models (DPMs) (Ho et al., 2020) inject artificial noise and learn an explicit neural network denoiser, DTMs may encode the entire denoising pathway structurally within the energy landscape or hardware couplings. DTMs avoid stepwise control, injected noise, and can achieve autonomous, thermodynamically efficient sampling when implemented physically. In DTM hardware, multiple simple EBM layers are chained for fast mixing, circumventing the mixing–expressivity tradeoff inherent in monolithic energy models (Jelinčič et al., 28 Oct 2025).

6. Thermodynamic Optimality and Entropy Production

A core tenet of DTM training is the minimization of entropy production and heat emission during generation, aligning the reverse path with time-reversed stochastic thermodynamics. For thermodynamic computing, maximizing the probability of generating an observed reverse trajectory is directly related to minimizing the total dissipated heat, formalized by fluctuation theorems: $$\ln \frac{P_0[\omega]}{\tilde{P}_J[\tilde{\omega}]} = -\frac{1}{2}(\beta Q_0 + \beta Q_J)$$ This principle has practical ramifications—energy-efficient generation and scientifically interpretable links between sampling dynamics and physical cost (Whitelam, 18 Jun 2025).

7. Future Directions and Impact

DTMs provide a template for hybrid thermodynamic–deterministic machine learning systems, with potential to combine neural networks for data embedding with probabilistic, energy-based hardware sampling. The modularity and scalability of DTM hardware architectures suggest expansion toward richer graphical models (e.g., Potts, mixture models), efficient data embeddings, and the realization of probabilistic computing devices with robust manufacturability (CMOS compatible). The paradigm directly connects generative modeling to physical principles, computational tractability, and sustainable AI, with ongoing research in both algorithmic and hardware co-design (Jelinčič et al., 28 Oct 2025).

A plausible implication is that as thermodynamic computing and DTM hardware mature, large-scale, autonomous generative AI and optimization could be performed with orders-of-magnitude less energy and heat, governed by the principles of fluctuation theorems and hardware-level score matching—a convergence of machine learning with non-equilibrium physical computation.