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Equilibrium Matching: Generative Modeling with Implicit Energy-Based Models

NeurIPS 2026 · Probabilistic and scientific modeling › Variational and generative sampling · approach: Flow matching and flow sampling · 0 citations (OpenAlex)

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Proposes Equilibrium Matching (EqM), a time-invariant gradient-field generative model trained with a single objective that is sampled by gradient descent on an implicit energy landscape, reporting FID 1.90 on ImageNet 256x256.

Problem

Diffusion and flow models learn non-equilibrium, time-conditioned dynamics that impose noise schedules and fixed integration horizons; prior attempts to remove time conditioning degrade quality and EBMs suffer from instability and poor sample quality.

Approach

Replace the time-conditioned velocity with a single time-invariant equilibrium gradient field that vanishes at data and points away from data (target (eps-x)c(gamma) with c(1)=0), so generation becomes gradient descent on an implicit energy with adaptive step sizes, optimizers, and compute.

Key results

• EqM achieves FID 1.90 on ImageNet 256x256 class-conditional generation, outperforming diffusion and flow counterparts.
• Adaptive compute lowers total sampling steps to about 40% of the fixed 250-step budget (FID 33.79 vs 32.85 on EqM-B/2).
• EqM-E dot-product variant is used for OOD detection on CIFAR-10 and achieves the best overall AUROC among compared baselines (PixelCNN++, GLOW, IGEBM).

Surprising findings

• Flow Matching needs a specific step size of 1/N to work, while EqM stays high-quality across a range of step sizes.
• On CIFAR-10 EqM outperforms the noise-unconditional flow baseline but does not beat standard noise-conditional Flow Matching.

Limitations

• CIFAR-10 results do not surpass standard Flow Matching, which the authors attribute partly to tuning of the baseline.
• Theoretical statements rely on high-dimensional approximations and perfect-training assumptions.
• Adaptive compute and explicit energy variants show small degradation (L2-norm variant harder to optimize).

Experimental setup

Datasets: ImageNet CIFAR-10

Benchmarks: ImageNet 256x256 CIFAR-10 CIFAR-10 OOD detection

Metrics: FID sFID Inception Score AUROC NFE

Baselines: StyleGAN VDM++ DiT SiT Flow Matching PixelCNN++ GLOW IGEBM Energy Matching

Builds on: Flow Matching SiT DiT Nesterov Accelerated Gradient Energy Matching

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