Thermodynamic Diffusion Inference with Minimal Digital Conditioning

Aditi De

cs.LGApr 15, 2026

Thermodynamic Diffusion Inference with Minimal Digital Conditioning

Authors: Aditi De

Abstract

Diffusion-model inference and overdamped Langevin dynamics are formally identical. A physical substrate that encodes the score function therefore equilibrates to the correct output by thermodynamics alone, requiring no digital arithmetic during inference and potentially achieving a $10{,}000\times$ reduction in energy relative to a GPU. Two fundamental barriers have until now prevented this equivalence from being realized at production scale: non-local skip connections, which locally coupled analog substrates cannot represent, and input conditioning, in which the coupling constants carry roughly $2{,}600\times$ too little signal to anchor the system to a specific input. We resolve both obstacles. Hierarchical bilinear coupling encodes U-Net skip connections as rank- $k$ inter-module interactions derived directly from the singular structure of the encoder and decoder Gram matrices, requiring only $O(Dk)$ physical connections instead of $O(D^2)$ . A minimal digital interface -- a 4-dimensional bottleneck encoder together with a 16-unit transfer network, totalling 2,560 parameters -- overcomes the conditioning barrier. When evaluated on activations drawn from a trained denoising U-Net, the complete system attains a decoder cosine similarity of 0.9906 against an oracle upper bound of 1.0000, while preserving theoretical net energy savings of approximately $10^7\times$ over GPU inference. These results constitute the first demonstration of trained-weight, production-scale thermodynamic diffusion inference.

Figures & tables

Figure 1: System architecture. Left: Conventional GPU inference executes the full score network digitally at $\sim$ 1–10 J per image. Right: A 2,560-parameter digital conditioning interface (0.032% of U-Net cost) computes the bias vectors $\mathbf{b}_{\rm enc}$ and $\mathbf{b}_{\rm dec}$ ; hierarchical bilinear skip coupling routes information physically; the Langevin substrate equilibrates under thermal noise to produce the denoised output (cosine similarity 0.9906 versus digital baseline). Theoretical net energy gain: $\sim\!10^{7}\times$ .

Figure 2: Skip-coupling analysis (trained regime). Trained U-Net, $D=64$ , real correlated activations. Panel (a): coupling matrix $\mathbf{J}_{\rm skip}$ . Panel (b): per-dimension decoder shift $\Delta y_{j}^{*}$ (mean $\pm$ std, 64 samples). Panel (c): $\rho_{\rm skip}$ versus rank $k$ . $\rho_{\rm skip}=12.74\%$ , CV $1.4\%$ . The larger shift confirms that training strengthens encoder–decoder singular alignment.

Figure 3: Signal-deficit analysis. Left: Gram-coupling eigenspectrum spanning six decades ( $\lambda_{\rm max}=0.034$ , mean $=0.009$ ), quantifying why naive biases are $\sim\!2{,}600\times$ too small. Right: Naive bias $b=xJ^{\top}$ (red) is uncorrelated with the target activation; the correct oracle bias $b=Ax^{*}$ (green) lies on the identity line. The standard-deviation ratio here is $13\times$ ; the full deficit across all $D$ dimensions reaches $2{,}600\times$ .

Figure 4: Conditioning-interface sweep. Left: Decoder cosine similarity saturates at the oracle level for $k=4$ with both linear and MLP encoders; the binding constraint is the quality of the Gram approximation. Right: Net energy savings ( $\sim\!10^{7}\times$ ) are independent of bottleneck dimension—encoder overhead remains negligible at every scale.

Figure 5: Production test (analytical regime, $D=128$ ). Left: Four conditioning regimes; oracle and full pipeline are visually indistinguishable. The skip-only collapse isolates the conditioning signal carried by rank-16 coupling in the absence of trained-weight structure. Center: Target versus equilibrium scatter lies on the identity line. Right: Per-sample alignment; full-pipeline variance is essentially zero.

Regime	$k=2$	$k=4$	$k=8$	$k=16$	$k=32$	$k=48$	$k=64$
Analytical ( $D=128$ )	21.3	12.5	8.9	7.5	9.0	9.1	9.0
Trained ( $D=64$ )	32.5	21.2	15.0	12.7	12.8	—	—

Table 1: Rank sweep —relative decoder shift

\rho_{\rm skip}

(%). Rank 16 is the operating point.

Regime	Configuration	Dec. cos. $\uparrow$	Params	Theory gain
Analytical $D=128$ random weights	Skip coupling (rank 16)	$7.48\%^{\dagger}$	—	—
	Oracle (ceiling)	0.9966	—	$\sim\!10^{7}\times$
	Skip only ‡	0.0144	256	—
	Full pipeline	0.9954	5,396	$\mathbf{\sim\!10^{7}\times}$
Trained $D=64$ real activations	Skip coupling (rank 16)	$12.74\%^{\dagger}$	—	—
	Oracle (ceiling)	1.0000	—	$\sim\!10^{7}\times$

Table 2: Main results. Both experimental regimes. † Relative shift, not cosine. ‡ Learned encoder + skip coupling as decoder conditioning (no transfer network).

Analog Computing

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Symmetric Equilibrium Propagation for Thermodynamic Diffusion Training

Aditi De

Equilibrium Propagation Thermodynamic Computing Score-Based Generative Modeling Diffusion Models

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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 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)

\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.

Abstract

Analog Computing

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