FluidWorld is a proof-of-concept world model built on reaction-diffusion PDEs. No attention. No KV-cache. The Laplacian operator is the computation. And it produces something no Transformer or ConvLSTM has demonstrated: autopoietic self-repair, where the model spontaneously corrects its own prediction errors through the physics of diffusion.

Trained with a latent prediction objective, the PDE substrate maintains cosine similarity of 0.827 across 19 autoregressive steps. It predicts in abstract representation space, not pixels, and encodes dynamics non-linearly.

800K parameters. One consumer GPU. Results that challenge the assumption that attention is necessary for world modeling.

Paper: Reaction-Diffusion Dynamics as a Predictive Substrate for World Models

During autoregressive rollout, prediction quality degrades. Every world model has this problem. But FluidWorld does something no attention-based model does: after degrading, it recovers.

Over 500 rollouts on Moving MNIST, SSIM drops from 0.778 to 0.287 by step 6, then climbs back to 0.508 by step 9. Two thirds of all rollouts show this recovery. The statistics leave no room for doubt:

Transformers and ConvLSTMs only decay. They never recover. The difference is structural: the Laplacian is a low-pass filter that smooths prediction errors at every integration step. Blurry intermediate predictions are not a defect. They are the self-repair mechanism.

Corrupt 50% of the BeliefField (the model's persistent latent state) during rollout. Inject noise, zero out channels, shuffle spatial positions. The model recovers in 3 to 7 steps. No explicit repair mechanism. No retraining. The Laplacian fills corrupted regions via diffusion from intact neighbors, and RMSNorm re-normalizes amplitude.

Push corruption to 90%. The model still works. No cliff effect. Graceful degradation all the way up. This kind of robustness has implications for deployed world models operating under sensor noise or partial observability.

Training with a JEPA-style latent objective (target encoder + VICReg, inspired by Y. LeCun's JEPA framework) produces abstract representations that barely degrade over time:

For comparison, Pixel and Random models sit near zero. The PDE dynamics produce stable internal predictions without any pixel-level supervision.

An MLP probe confirms the representations encode velocity non-linearly (R^2 jumps from 0.29 to 0.60), while Pixel and Random both drop under the same probe. The JEPA-style PDE is the only model whose dynamics encoding improves with a non-linear readout.

At current resolution the difference is negligible. At the resolutions real-world robotics and autonomous driving require, PDE diffusion is orders of magnitude cheaper than attention.

All three models: ~800K parameters, identical encoder, identical decoder, identical losses, identical data (UCF-101 64x64). The only difference is the predictive engine.

Single-step metrics are intentionally comparable. That was the point: isolate what happens when you chain predictions. The PDE maintains spatial structure 1 to 2 steps longer than both baselines. The ConvLSTM has spatial bias too (convolutions), yet it fails just as fast as the Transformer. Spatial bias alone is not enough. Continuous diffusion with global reach is what matters.

Input frame (3, 64, 64)
       |
 PatchEmbed (stride 4)
       |
 (128, 16, 16) latent grid
       |
 FluidLayer2D x3               Reaction-diffusion PDE
 |  Multi-scale Laplacian          dilations {1, 4, 16}
 |  Learned reaction MLP           position-wise nonlinearity
 |  Bio mechanisms                 Hebbian, lateral inhibition, fatigue
       |
 BeliefField                    Persistent temporal state
 |  write(observation)             GRU-gated integration
 |  evolve(state)                  PDE + DeltaNet + Titans
 |  read(state)                    Spatial readout
       |
 PixelDecoder
       |
 Output frame (3, 64, 64)

One equation, two roles. The same reaction-diffusion PDE governs both spatial encoding (in the encoder layers) and temporal prediction (in the BeliefField). The core update:

u(t+1) = u(t) + dt * [D * laplacian(u) + Reaction(u) + Memory_terms]

What falls out naturally: O(N) computation, adaptive early stopping, spatial coherence via continuous diffusion, and self-repair as a free byproduct of the Laplacian low-pass filter.

The BeliefField adds biologically-inspired mechanisms: lateral inhibition (retinal processing), synaptic fatigue (prevents channel collapse), and Hebbian diffusion (co-activated pathways strengthen). The system operates at the edge of chaos (Lyapunov exponent 0.0033), bounded by RMSNorm homeostasis.

FluidWorld is the third iteration of a research program replacing attention with reaction-diffusion PDEs across modalities:

• FluidLM (2024) : Language modeling with 1D Laplacian on token sequences
• FluidVLA (2025) : Vision and robotic control with 2D Laplacian (40ms inference on RTX 4070). Extended to 3D for medical imaging (CT/MRI segmentation)
• FluidWorld (2026) : World modeling. The PDE is now both encoder and temporal predictor

Same core equation. Same O(N) complexity. Three modalities.

15 numbered Jupyter notebooks, ordered for sequential execution. Training notebooks (01 to 03) need a GPU. Analysis notebooks (04 to 15) load saved checkpoints.

Clone the repository and run the numbered notebooks in experiments/ sequentially. Training notebooks (01 to 03) require a GPU (tested on RTX 4070 Ti, ~3.5h for 30 epochs). Analysis notebooks (04 to 15) load saved checkpoints and run on CPU or GPU.

Full setup instructions and dataset preparation are documented inside each notebook.

• fluidworld/core/ : model source code (PDE, BeliefField, baselines, decoder)
• experiments/ : 15 numbered notebooks + saved analysis data
• paper/ : LaTeX source, all figures, bibliography

@article{polly2026fluidworld,
    title={FluidWorld: Reaction-Diffusion Dynamics as a Predictive Substrate for World Models},
    author={Polly, Fabien},
    year={2026},
    note={Preprint}
}

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