Simplified Consistency Models (sCM)

• Simplified Consistency Models (sCM) are unified frameworks that merge continuous-time generative modeling with LHC dark matter models, ensuring rapid sampling and theoretical rigor.
• In generative modeling, sCMs employ ODE formulations, training-free transformations, and hybrid distillation techniques to achieve state-of-the-art performance in few-step sampling.
• For particle physics, sCMs provide minimal yet gauge invariant dark matter models that satisfy perturbative unitarity and renormalizability, validated by collider benchmarks.

Simplified Consistency Models (sCM) encompass two distinct but similarly motivated concepts in contemporary research: (i) simplified continuous-time consistency models in generative modeling, focused on diffusion and flow-based models for rapid sampling; and (ii) simplified consistency models in particle physics, particularly in the construction of simplified Dark Matter (DM) models at the LHC. This article presents both lineages with technical rigor, highlighting their mathematical formulations, key theoretical principles, and empirical results.

1. Continuous-Time Simplified Consistency Models in Generative Modeling

Mathematical Formulation

The sCM paradigm in generative modeling is defined by an ODE-centric formulation that unifies diffusion, flow-matching, and TrigFlow models as probability-flow ODEs: $$\frac{dx_t}{dt} = v_\theta(x_t, t), \quad x_{t=1} \sim \mathcal{N}(0, I)$$ Flow-matching utilizes $v_\theta(x_t, t) \approx z - x_0,$ trained using

$$\mathbb{E}_{x_0, z, t}\left[w(t)\|\boldsymbol{v}_\theta(x_t, t) - (z-x_0)\|^2\right].$$

TrigFlow introduces a spherical noise schedule $\alpha_t = \cos t$, $\sigma_t = \sin t$ with $t\in[0, \tfrac{\pi}{2}]$, and the model predicts velocities via

$$F_\theta\left(\frac{x_t}{\sigma_d}, t\right) \approx \frac{\cos t}{\sigma_d}z - \frac{\sin t}{\sigma_d}x_0,$$

leading to the ODE

$$\frac{dx_t}{dt} = \sigma_d F_\theta\left(\frac{x_t}{\sigma_d}, t\right).$$

sCMs optimize a continuous-time consistency loss: $$\mathcal{L}_{\rm sCM} = \mathbb{E}_{x_t, t}\Big[ w_\phi(t) \| f_\theta(x_t, t) - f_{\theta^{-}}(x_t, t) - \tfrac{d}{dt}f_{\theta^{-}}(x_t, t) \|_2^2 \Big],$$ where $f_{\theta^{-}}$ is a frozen (stop-gradient) copy.

Unified Parameterization and Stability

The TrigFlow parameterization is central for reconciling prior approaches, allowing any noise schedule to be mapped via $t=\arctan(\sigma_t/\alpha_t)$ and its inverse. sCMs identify three sources of instability in continuous-time training: (a) time transformation blow-up, (b) high-frequency time embeddings, and (c) normalization collapse in adaptive group norm layers. Remedies include using the identity time transform, limiting Fourier embedding scales, and replacing AdaGN with pixel-wise normalized adaptive double normalization (ADN) (Lu et al., 14 Oct 2024).

2. Training-Free Distillation and Hybridization

Training-Free Transformation

sCMs enable a lossless, algebraically exact transformation from a pre-trained flow-matching model to a TrigFlow parameterization:

• Time mapping: $$t_{\mathrm{FM}} = \frac{\sin(t_{\mathrm{Trig}})}{\cos(t_{\mathrm{Trig}})+\sin(t_{\mathrm{Trig}})}$$, $$t_{\mathrm{Trig}} = \arctan\bigl(\frac{t_{\mathrm{FM}}}{1-t_{\mathrm{FM}}}\bigr)$$.
• SNR matching: $$x_{t_{\mathrm{FM}}} = x_{t_{\mathrm{Trig}}} \sqrt{t_{\mathrm{FM}}^2 + (1-t_{\mathrm{FM}})^2}$$.
• Velocity mapping:

$$\widehat{F_\theta}(x_{t_{\mathrm{Trig}}}, t_{\mathrm{Trig}}) = \frac{1}{\sqrt{t_{\mathrm{FM}}^2 + (1-t_{\mathrm{FM}})^2}} \left[ (1-2 t_{\mathrm{FM}})x_{t_{\mathrm{FM}}} + \bigl(1-2t_{\mathrm{FM}}+2t_{\mathrm{FM}}^2\bigr)v_\theta(x_{t_{\mathrm{FM}}}, t_{\mathrm{FM}}) \right],$$

where all steps are efficiently implemented in the student's forward pathway without retraining (Chen et al., 12 Mar 2025).

Hybrid Distillation: sCM + LADD

Fully sCM-based distillation exactly enforces student-teacher path alignment, optimizing diversity and faithfulness. However, for ultra-few-step sampling, text-to-image fidelity can suffer from “trajectory truncation” artifacts. The SANA-Sprint framework introduces a hybrid objective combining sCM with a latent-space adversarial loss (LADD), defined for generator and discriminator as: $$\mathcal{L}_{\rm adv}^G(\theta) = -\,\mathbb{E}_{x_0,s,t}\sum_k D_{\psi,k}(F_{\theta^{\rm pre},k}(\hat{x}_s^{f_\theta}, s, y))$$

$$\mathcal{L}_{\rm adv}^D(\psi) = \mathbb{E}_{x_0,s}\sum_k \mathrm{ReLU}\left(1 - D_{\psi,k}(F_{\theta^{\rm pre},k}(x_s, s, y))\right) + \mathbb{E}_{x_0,s,t}\sum_k \mathrm{ReLU}\left(1 + D_{\psi,k}(F_{\theta^{\rm pre},k}(\hat{x}_s^{f_{\theta^-}}, s, y))\right)$$

with the final loss $$\mathcal{L} = \mathcal{L}_{\rm sCM} + \lambda \mathcal{L}_{\rm adv}^G,~\lambda=0.5$$. This hybridization improves one-step and few-step sample quality (Chen et al., 12 Mar 2025).

3. Empirical Performance and Step-Adaptive Behavior

The SANA-Sprint model demonstrates that unified, step-adaptive sCMs outperform other few-step methods in both speed and quality, establishing new Pareto frontiers for FID and latency. Key empirical results on 1024×1024 T2I tasks (A100), summarized in the table below, show dominance over FLUX-schnell at all step counts:

Steps	Model	FID	GenEval	Latency (s)	Images/s (batch=10)
4	FLUX-schnell (12B)	7.94	0.71	2.10	1.58
	SANA-Sprint (0.6B)	6.48	0.76	0.32	7.22
2	FLUX-schnell	7.75	0.71	1.15	-
	SANA-Sprint (0.6B)	6.54	0.76	0.25	-
1	FLUX-schnell	7.26	0.69	0.68	-
	SANA-Sprint (0.6B)	7.04	0.72	0.21	-
	SANA-Sprint (1.6B)	7.59	0.74	0.21	-

Ablation analysis reveals:

• sCM-only (4 steps): FID=8.93; LADD-only: FID=12.2; hybrid: FID=8.11.
• Loss weighting (1:0.5) and “max-time” regularization (50% noise steps) improve stability and few-step FID by ~1.1.
• Dense timestep embedding and QK-normalization are necessary for stable distillation at high resolutions and large scale (Chen et al., 12 Mar 2025).

On standard benchmarks, two-step sCMs approach the performance of their 63-step EDM² teachers, closing the FID gap to within 10% but with roughly 3% of the sampling compute (Lu et al., 14 Oct 2024).

4. Architectural and Algorithmic Stabilization in Large-Scale sCMs

Architectural stabilization leverages pixelwise normalization, adaptive double-normalization, low-frequency positional embeddings, tangent normalization, and ramped time-weighting. The sCM training loop replaces AdaGN with ADN and normalizes tangent terms to ensure stable gradients for billion-parameter models. Sampling employs one or two Euler/Trigonometric steps with step-invariant weights, yielding state-of-the-art fast generative performance (CIFAR-10 FID: 2.06 in 2 steps) (Lu et al., 14 Oct 2024).

5. Theoretical Guarantees and Discussion

sCMs guarantee continuous-time self-consistency by solving an ODE with closed-form transformations, enabling distillation from pre-trained flow-matching or TrigFlow networks without retraining. Limitations include vulnerability to trajectory truncation in extreme few-step settings and the necessity for careful control of time-embedding and normalization in the teacher for robust transfer. Extension of the sCM hybridization framework beyond flow-matching schedules demands additional proof and can introduce instability (Chen et al., 12 Mar 2025, Lu et al., 14 Oct 2024).

6. Simplified Consistency Models in LHC Dark Matter Searches

The sCM concept in particle physics denotes simplified, minimal, yet theoretically consistent models for DM at the LHC. These models are constructed to satisfy gauge invariance, perturbative unitarity, and renormalizability:

• Mediators: Vector (e.g., $V_\mu$), (pseudo-)scalar ($S$), and colored scalar (squark-like, $\tilde q$) mediators are parameterized with explicit kinetic, mass, and coupling terms. Full Lagrangians enforce $SU(3)_c \times SU(2)_L \times U(1)_Y$ invariance and anomaly cancellation, e.g., vector mediator $V_\mu$ as a $U(1)'$ gauge boson.
• Consistency: Models must respect SM gauge invariance, enforce tree-level unitarity (e.g., $m_\psi \lesssim \sqrt{\pi} M_V/g_\psi^A$ for axial couplings), and restrict all couplings to the perturbative regime.
• Benchmarking: LHC-relevant benchmarks involve s-channel “$Z'$”-like mediators (e.g., $g_\chi=1,\; g_q=0.25$), Higgs-portal scalars, and t-channel colored mediators. Key exclusions derive from monojet, dijet, and dilepton resonance searches (Morgante, 2017).

7. Outlook and Open Questions

For generative modeling, further reduction of the adversarial component or its replacement by regression terms, as well as the universality of the training-free flow-to-TrigFlow transformation across exotic schedules, remain open challenges. In LHC DM models, the intersection of gauge-invariant completion, anomaly cancellation, and collider constraints tightly defines viable sCM parameter space. The evolution of sCMs in both fields reflects ongoing efforts to balance theoretical minimalism, computational scalability, and empirical fidelity (Lu et al., 14 Oct 2024, Chen et al., 12 Mar 2025, Morgante, 2017).