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# 🌊 LiquidDiffusion
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##
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|---------|-------------|
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| **No Attention** | All spatial mixing via multi-scale depthwise convolutions (3×3, 5×5, 7×7) + global average pooling |
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| **Fully Parallelizable** | No sequential ODE solving — CfC closed-form solution eliminates the computational bottleneck of Neural ODEs |
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| **CfC × Diffusion Bridge** | The diffusion noise level `t` IS the liquid time constant — natural mathematical correspondence |
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| **Liquid Relaxation Residuals** | Time-aware skip connections: `α·input + (1-α)·output` where `α = exp(-λ·t)` adapts to noise level |
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| **Fits 16GB VRAM** | Tiny model (8M params) fits in ~4GB; designed for Colab free tier T4 |
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``
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Input: noisy image [B, 3, H, W] + timestep t ∈ [0, 1]
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Time Embedding: Sinusoidal PE → MLP → t_emb [B, dim]
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Conv Stem: 3×3 conv → SiLU → 3×3 conv
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Stage 1: [LiquidDiffusionBlock × N₁] → DownSample (stride-2 conv)
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Stage 2: [LiquidDiffusionBlock × N₂] → DownSample
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Stage 3: [LiquidDiffusionBlock × N₃]
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Stage 3: UpSample → SkipFusion → [LiquidDiffusionBlock × N₃]
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Stage 2: UpSample → SkipFusion → [LiquidDiffusionBlock × N₂]
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Stage 1: [LiquidDiffusionBlock × N₁]
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Output: GroupNorm → SiLU → 3×3 conv → velocity prediction [B, 3, H, W]
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```
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→ AdaLN(t) → FeedForward → +residual
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```
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###
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``
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g = DWConv→SiLU→Conv1x1(backbone) # "from" state
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h = DWConv→SiLU→Conv1x1(backbone) # "to" state (attractor)
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gate = σ(time_a(t_emb) · f - time_b(t_emb)) # liquid time gate
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cfc_out = gate · g + (1-gate) · h # CfC interpolation
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# Liquid relaxation residual:
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α = exp(-softplus(ρ) · |t|) # time-aware weight
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output = α · input + (1-α) · cfc_out # noise-adaptive residual
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```
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## 📊 Model Configurations
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| Config | Channels | Blocks | Params | Resolution | VRAM (fp16) |
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|--------|----------|--------|--------|-----------|-------------|
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| **tiny** | [64, 128, 256] | [2, 2, 4] | ~8M | 256×256 | ~4GB |
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| **small** | [96, 192, 384] | [2, 3, 6] | ~25M | 256×256 | ~8GB |
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| **base** | [128, 256, 512] | [2, 4, 8] | ~65M | 512×512 | ~14GB |
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| **large** | [128, 256, 512, 768] | [2, 4, 8, 4] | ~120M | 512×512 | ~24GB |
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##
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```
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x_t = (1-t) · x_data + t · noise, t ~ U[0,1]
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Loss = ||model(x_t, t) - (noise - x_data)||²
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```
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No noise schedule. No variance. Just MSE on a straight-line velocity.
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### Sampling (Euler ODE)
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```python
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z = z - model(z, t) / N # Euler step
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```
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Typically 25-50 steps.
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### Quick Start
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```python
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from liquid_diffusion import liquid_diffusion_tiny, RectifiedFlowTrainer
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model = liquid_diffusion_tiny()
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trainer = RectifiedFlowTrainer(model, lr=1e-4, device='cuda')
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# Training step
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images = get_batch() # [B, 3, 256, 256] in [-1, 1]
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metrics = trainer.train_step(images)
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print(f"Loss: {metrics['loss']:.4f}")
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#
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```
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- **CelebA-HQ** (`huggan/CelebA-HQ`) — 30K face images, 256px
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- **Flowers-102** (`huggan/flowers-102-categories`) — botanical images
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- **AFHQ** — 15K animal faces (cats, dogs, wildlife)
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- Any folder of images
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##
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```
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dx/dt = -[1/τ + f(x,I,θ)] · x + f(x,I,θ) · A
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```
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Key: system time constant `τ_sys = τ/(1 + τ·f)` is **input-dependent** — neurons adapt their response speed.
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### CfC: Closed-form Solution
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*Hasani et al., Nature Machine Intelligence 2022 — [arxiv:2106.13898](https://arxiv.org/abs/2106.13898)*
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```
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Eliminates ODE solver → **fully parallelizable**, one order of magnitude faster.
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##
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We observe that CfC's time parameter `t` and diffusion's noise level `t` serve analogous roles:
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- CfC: `t` controls interpolation between "from" (g) and "to" (h) states
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- Diffusion: `t` controls the noise level the denoiser must handle
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##
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*[arxiv:2604.18274](https://arxiv.org/abs/2604.18274)*
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```
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```
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When `t` is large (noisy): α ≈ 0 → rely on CfC output (needs strong processing).
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When `t` is small (clean): α ≈ 1 → preserve input (only minor refinement needed).
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## ��� References
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1. Hasani et al., "Liquid Time-constant Networks", AAAI 2021 — [arxiv:2006.04439](https://arxiv.org/abs/2006.04439)
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2. Hasani et al., "Closed-form Continuous-time Neural Networks", Nature MI 2022 — [arxiv:2106.13898](https://arxiv.org/abs/2106.13898)
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3. Lechner et al., "Neural Circuit Policies", Nature MI 2020
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4. LiquidTAD: Parallel liquid relaxation — [arxiv:2604.18274](https://arxiv.org/abs/2604.18274)
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5. USM: U-Shape Mamba for diffusion — [arxiv:2504.13499](https://arxiv.org/abs/2504.13499)
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6. DiffuSSM: Diffusion without attention — [arxiv:2311.18257](https://arxiv.org/abs/2311.18257)
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7. Liu et al., "Flow Straight and Fast: Rectified Flow", ICLR 2023 — [arxiv:2209.03003](https://arxiv.org/abs/2209.03003)
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8. Lee et al., "Improving the Training of Rectified Flows" — [arxiv:2405.20320](https://arxiv.org/abs/2405.20320)
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## License
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# 🌊 LiquidDiffusion
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**A novel attention-free image generation model based on Liquid Neural Networks**
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## What is this?
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LiquidDiffusion is a **first-of-its-kind** image generation model that replaces attention with **Parallel CfC (Closed-form Continuous-depth) blocks** from Liquid Neural Network research. No existing paper combines LNNs with image generation — this fills that gap.
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### Key Properties
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- ✅ **Zero attention layers** — fully convolutional + liquid time-gating
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- ✅ **Fully parallelizable** — no ODE solvers, no sequential scanning, no recurrence
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- ✅ **Pretrained VAE** — uses `stabilityai/sd-vae-ft-mse` for efficient latent-space training
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- ✅ **Fits 16GB VRAM** — tiny config runs 256px at batch=8 on T4 GPU
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- ✅ **Simple training** — Rectified Flow (MSE velocity prediction, no noise schedule)
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- ✅ **6 verified datasets** ready to use
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## Quick Start
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Open the Colab notebook, pick your dataset from the dropdown, run all cells:
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**`LiquidDiffusion_Training.ipynb`**
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### Verified Datasets (all tested ✓)
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| Dataset | Size | Content |
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|---------|------|---------|
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| `nielsr/CelebA-faces` | 202K | Celebrity faces |
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| `huggan/flowers-102-categories` | 8K | Flowers |
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| `reach-vb/pokemon-blip-captions` | 833 | Pokemon art |
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| `huggan/anime-faces` | 21K | Anime faces |
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| `huggan/AFHQv2` | 16K | Cat/dog/wild animals |
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| `Norod78/cartoon-blip-captions` | 2K | Cartoon characters |
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## Architecture
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```
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Input (noisy latent 4ch) → Conv Stem
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→ Encoder [LiquidDiffusionBlock × N, with downsampling]
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→ Bottleneck [LiquidDiffusionBlock × 2]
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→ Decoder [LiquidDiffusionBlock × N, with upsampling + skip fusion]
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→ Conv Head → Velocity prediction
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```
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### VAE Integration
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- **Encoder**: `stabilityai/sd-vae-ft-mse` (83M params, frozen)
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- **Latent space**: 4 channels, 8× spatial downscale
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- **256px image → 32×32×4 latent** (64× fewer pixels to process!)
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- **Pre-caching**: Encode dataset once, then train without VAE on GPU (saves ~160MB VRAM)
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### ParallelCfCBlock (Novel Contribution)
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Based on CfC Eq.10: `x(t) = σ(-f·t) ⊙ g + (1 - σ(-f·t)) ⊙ h`
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```python
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# Three CfC heads from shared backbone
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gate = sigmoid(time_a(t_emb) * f(features) - time_b(t_emb))
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cfc_out = gate * g(features) + (1 - gate) * h(features)
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# Liquid relaxation residual
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α = exp(-softplus(ρ) * |t_emb_mean|)
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output = α * input + (1 - α) * cfc_out
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```
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**Key insight**: Diffusion timestep `t` IS the liquid time constant. CfC gate naturally adapts to noise level.
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## Model Configs
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| Config | Channels | Blocks | Params | 256px VRAM | Best For |
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|--------|----------|--------|--------|------------|----------|
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| tiny | [64, 128, 256] | [2, 2, 4] | ~23M | ~6 GB | Quick experiments, T4 |
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| small | [96, 192, 384] | [2, 3, 6] | ~69M | ~10 GB | Quality 256px, T4/A10G |
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## Training Objective: Rectified Flow
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```python
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x_t = (1 - t) * x0 + t * noise # linear interpolation
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v_target = noise - x0 # constant velocity
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loss = MSE(model(x_t, t), v_target) # simple MSE — no noise schedule!
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```
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## References
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| Paper | Contribution |
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|-------|-------------|
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| [CfC Networks (Nature MI 2022)](https://arxiv.org/abs/2106.13898) | CfC Eq.10, parallelizable closed-form |
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| [LTC Networks (AAAI 2021)](https://arxiv.org/abs/2006.04439) | Liquid time-constant ODE, stability |
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| [LiquidTAD (2024)](https://arxiv.org/abs/2604.18274) | Parallel liquid relaxation |
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| [USM (CVPR 2025)](https://arxiv.org/abs/2504.13499) | U-Net + SSM for diffusion |
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| [DiffuSSM (2023)](https://arxiv.org/abs/2311.18257) | SSM beats attention in diffusion |
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| [Rectified Flow (ICLR 2023)](https://arxiv.org/abs/2209.03003) | Simple velocity training |
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## Files
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```
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├── liquid_diffusion/
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│ ├── __init__.py
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│ ├── model.py # Full model architecture
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│ └── trainer.py # Rectified Flow trainer + dataset utils
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├── LiquidDiffusion_Training.ipynb # Complete Colab notebook (VAE + 6 datasets)
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├── test_model.py
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└── README.md
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```
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## License
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