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

Fabien Polly

cs.LGMar 22, 2026

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

Authors: Fabien Polly

Organizations: Independent Researcher

Abstract

World models learn to predict future states of an environment, enabling planning and mental simulation. Current approaches default to Transformer-based predictors operating in learned latent spaces. This comes at a cost: O(N^2) computation and no explicit spatial inductive bias. This paper asks a foundational question: is self-attention necessary for predictive world modeling, or can alternative computational substrates achieve comparable or superior results? I introduce FluidWorld, a proof-of-concept world model whose predictive dynamics are governed by partial differential equations (PDEs) of reaction-diffusion type. Instead of using a separate neural network predictor, the PDE integration itself produces the future state prediction. In a strictly parameter-matched three-way ablation on unconditional UCF-101 video prediction (64x64, ~800K parameters, identical encoder, decoder, losses, and data), FluidWorld is compared against both a Transformer baseline (self-attention) and a ConvLSTM baseline (convolutional recurrence). While all three models converge to comparable single-step prediction loss, FluidWorld achieves 2x lower reconstruction error, produces representations with 10-15% higher spatial structure preservation and 18-25% more effective dimensionality, and critically maintains coherent multi-step rollouts where both baselines degrade rapidly. All experiments were conducted on a single consumer-grade PC (Intel Core i5, NVIDIA RTX 4070 Ti), without any large-scale compute. These results establish that PDE-based dynamics, which natively provide O(N) spatial complexity, adaptive computation, and global spatial coherence through diffusion, are a viable and parameter-efficient alternative to both attention and convolutional recurrence for world modeling.

Figures & tables

Figure 1: FluidWorld architecture overview. Top row: The prediction pipeline. An input frame

x_{t}

(64

\times

64 pixels) is encoded by three PDE-based layers that use Laplacian diffusion to extract spatial features. These features are written into a persistent BeliefField , a 16

\times

16 latent state that accumulates temporal context. The BeliefField evolves via internal PDE dynamics to predict the next state, which the decoder converts back to pixels. During autoregressive rollout, each prediction becomes the next input (purple arrow). Bottom row: Key components. The Laplacian kernel (

[1,-2,1]

) provides spatial information propagation at

O(N)

cost without attention. The BeliefField combines GRU-gated writing with PDE evolution and Titans persistent memory. Biologically-inspired mechanisms (lateral inhibition, Hebbian diffusion, synaptic fatigue) promote diverse, structured representations. The entire model uses

{\sim}

801K parameters (Table 1 ).

Figure 2: How Laplacian diffusion enables self-repair. Top row (a–d): The heat equation analogy. A concentrated energy spike (a) spreads via diffusion (b) until reaching equilibrium (c). The Laplacian kernel $[1,-2,1]$ (d) computes the difference between a position and its neighbors; this single operation drives all spatial propagation in FluidWorld . Bottom row (e–h): Application to prediction errors. A clean prediction (e) accumulates noise during autoregressive rollout (f). The Laplacian smooths away high-frequency errors (g), recovering coherent spatial structure. Panel (h) summarizes why this matters: Transformers have no diffusion mechanism (errors compound), ConvLSTMs have only local kernels (slow dissipation), but FluidWorld ’s Laplacian provides global error correction at every integration step.

Component	FluidWorld (PDE)	Transformer	ConvLSTM	Shared?
PatchEmbed	24,704	24,704	6,528	$\approx$
Encoder engine	403,849	370,304	113,032	—
Temporal engine	165,891	198,528	451,200	—
PixelDecoder	231,235	231,235	231,235	✓
Total	800,975	800,067	801,995

Table 1: Parameter allocation. All three models are matched at

\sim

800K total parameters. The only difference is how encoder and temporal parameters are spent: PDE layers (local diffusion + reaction) vs Transformer blocks (global self-attention + FFN) vs ConvLSTM (convolutional gates + LSTM recurrence).

Metric	FluidWorld (PDE)	Transformer	ConvLSTM	Winner
Recon Loss $\downarrow$	0.001	0.002	0.001	PDE $\approx$ ConvLSTM
Pred Loss $\downarrow$	0.003	0.004	0.003	PDE $\approx$ ConvLSTM
Grad Loss $\downarrow$	0.04	0.05	0.04	PDE $\approx$ ConvLSTM
Spatial Std $\uparrow$	1.16	1.05	1.12	PDE
Effective Rank $\uparrow$	$\sim$ 2.0 $\times 10^{4}$	$\sim$ 1.65 $\times 10^{4}$	$\sim$ 1.9 $\times 10^{4}$	PDE
Feature Std	1.15	1.08	1.10	PDE

Table 2: Quantitative comparison at step 8,000 on UCF-101 (

64\!\times\!64

). All three models have

\sim

800K parameters, identical decoder, losses, and data. While single-step metrics are comparable across architectures, FluidWorld achieves

2\times

lower reconstruction error and produces spatially richer representations. The critical difference emerges in multi-step rollouts (see § 6.2 ).

Figure 3: Training dynamics comparison (three-way). (a) Reconstruction loss: PDE and ConvLSTM converge to

2\times

lower error than Transformer. (b) Prediction loss: all three converge to comparable values. (c) Spatial Std: PDE maintains the highest spatial structure throughout training. (d) Effective Rank: PDE uses the most representational dimensions. All models have

\sim

800K parameters, identical data and losses.

Figure 4: Final comparison at step 8,000. Single-step metrics are comparable across all three models (

\sim

800K parameters each). The PDE achieves the highest spatial structure and effective dimensionality.

Resolution (tokens)	Attention ops	PDE diffusion ops	Ratio
$16\times 16$ (256)	65K	256	$256\times$
$32\times 32$ (1,024)	1.05M	1K	$1{,}024\times$
$64\times 64$ (4,096)	16.7M	4K	$4{,}096\times$
$128\times 128$ (16,384)	268M	16K	$16{,}384\times$

Table 3: Operation counts for the attention component (Transformer) vs diffusion component (PDE) as resolution increases. The PDE advantage grows quadratically with token count.

Figure 5: PDE diffusion scales

O(N)

; attention scales

O(N^{2})

. At

128\!\times\!128

the ratio exceeds

16{,}000\times

Horizon	FluidWorld (PDE)	Transformer	ConvLSTM
$h=1$	Accurate	Accurate	Accurate
$h=2$	Coherent, minor blur	Color drift	Noticeable blur
$h=3$	Recognizable structure	Mean-color collapse	Texture dissolution
$h=4$	Color shift begins	Uniform patches	Green/brown artifacts
$h=5$	Degraded	Fully collapsed	Fully collapsed

Table 4: Rollout degradation by horizon. All three models produce accurate single-step predictions, but diverge beyond

h\!=\!1

Figure 6: Autoregressive rollout comparison at step 8,000. Ground truth (top row) versus predictions from FluidWorld (PDE), Transformer, and ConvLSTM, all at $\sim$ 800K parameters. The PDE maintains recognizable spatial structure through $h\!=\!3$ while both baselines collapse to mean-color patches or texture noise by $h\!=\!2$ . UCF-101, $64\!\times\!64$ .

Figure 7: The key result: PDE-based models self-correct during prediction. SSIM (similarity to ground truth) over 19 autoregressive rollout steps. The FluidWorld PDE curve (blue, solid, measured on 500 sequences ) shows a unique non-monotonic pattern: SSIM drops as errors accumulate, then recovers at steps 6–9 as Laplacian diffusion dissipates the errors. The green shaded region marks this recovery zone. In contrast, Transformer and ConvLSTM baselines (dashed lines, schematic curves based on typical behavior from literature, not measured on the same data) show monotonic decay: once errors accumulate, they never recover. This self-correcting property is unique to PDE-based world models and emerges naturally from the physics of diffusion: prediction errors are high-frequency noise, and the Laplacian is a low-pass filter that smooths them away.

Figure 8: Fair ablation: edge/freq losses collapse at equal training. Both models trained from scratch for 30 epochs on the same data. The Laplacian-only model (blue) produces the characteristic oscillatory recovery. The edge/freq model (red) collapses entirely: SSIM

=

0.013 at step 1 versus 0.778 for Laplacian-only (

t=-180.7

p=0.00

). Compare with Figure 12 , which shows the unfair comparison (30 vs. 60 epochs) where the edge/freq model avoids collapse but loses the oscillatory pattern.

Figure 9: Statistical evidence for autopoietic recovery. Top left: mean SSIM over 19 autoregressive steps (

N\!=\!500

rollouts, 95% CI shaded). The red dashed line is an exponential-decay null model fitted on steps 1–5. Between steps 6 and 12, the PDE model exceeds the null (green shaded), peaking at

+0.217

above at step 9. Bottom left: MSE trajectory: a sharp spike at step 4 (0.181) dissipates to 0.050 by step 8, mirroring the SSIM recovery. Right: summary statistics. Recovery appears in 66.8% of rollouts (

p<10^{-49}

Figure 10: SSIM heatmap across all 500 rollouts. Rows are sequences, columns are rollout steps. Green = high SSIM, red = low. The vertical green band at steps 8–9 appears across all sequences, not just a few. Sequences are sorted by the step of minimum SSIM.

Figure 11: Complementary evidence for autopoietic recovery. (a) Individual rollout traces (80 shown, light blue) overlaid with the population mean (dark blue). The red dashed line shows typical Transformer behavior (monotonic exponential decay). The PDE model’s recovery at steps 8–9 is clearly visible across individual sequences, not an artifact of averaging. (b) Distribution of recovery magnitudes across 500 rollouts. 66.8% of rollouts show measurable recovery (

>0.01

SSIM improvement after minimum). (c) Step-by-step SSIM values with comparison to null model. Green-shaded rows indicate steps where the PDE exceeds the exponential-decay null, providing direct evidence of self-repair.

Figure 12: Ablation: Laplacian smoothing enables oscillatory self-repair. Mean SSIM over 19 autoregressive steps (

N\!=\!500

) for two configurations of the same architecture: Laplacian-only (blue, 30 epochs) and Laplacian + edge/freq losses (red, 60 epochs). The Laplacian-only model exhibits the characteristic oscillatory recovery: a sharp SSIM increase from 0.287 to 0.508 between steps 6 and 9 (green shaded region), the dynamical signature of error accumulation followed by diffusion-mediated dissipation. Adding edge-sharpening and frequency losses suppresses this oscillation: the model instead shows a gradual monotonic rise from its minimum. The two models display qualitatively different dynamics, confirming that Laplacian diffusion is the mechanism producing the autopoietic oscillatory pattern. This figure shows the unfair comparison (30 vs. 60 epochs). In the fair comparison (both 30 epochs from scratch), the edge/freq model collapses entirely (SSIM

=

0.013 at step 1; see § 6.4 ).

Figure 13: The PDE self-repairs after deliberate state corruption. Top: Ground truth (rows 1, 3, 5) and model predictions (rows 2, 4, 6) across 19 autoregressive steps. At step 5 (green border), the BeliefField internal state is deliberately corrupted via spatial shuffling. The predictions immediately degrade (steps 6–7, visible as scrambled patterns), but the PDE’s Laplacian diffusion progressively smooths the corruption and restores coherent predictions by steps 9–11. Bottom: SSIM curves comparing normal rollout (blue) against three corruption types applied at step 6: Gaussian noise (red), channel zeroing (orange), and spatial shuffling (purple). All corrupted trajectories show an initial SSIM drop followed by partial recovery (green shaded zone). This self-repair is impossible for standard Transformers and ConvLSTMs, whose predictions degrade permanently after state corruption. The Laplacian acts as an “immune system”: high-frequency corruption is smoothed away while low-frequency structure (object identity, position) is preserved.

Figure 14: Recovery is robust across corruption intensities. SSIM trajectories after Gaussian noise (left), channel zeroing (center), and spatial shuffling (right) at five intensities (0.1 to 0.9). At low intensities (0.1–0.3), recovery is near-complete. At high intensities (0.7–0.9), the SSIM drop is larger but the model still recovers partially, with no catastrophic failure. This graceful degradation (no cliff effect) is characteristic of dissipative PDE systems: the Laplacian smoothing provides proportional error correction regardless of perturbation magnitude.

Linear probes on frozen BeliefField features
Metric	Pixel (30 ep)	JEPA-style † (30 ep)	Random
Position $R^{2}$ (spatial) a	0.9766	0.9349	0.9522
Velocity $R^{2}$ (after PDE) b	0.7730	0.2876	0.5580
Velocity $R^{2}$ (MLP probe) b2	0.6328	0.6028	0.3145
Direction Acc. (8-class) c	38.4%	22.1%	30.8%
Latent-space rollout quality (cosine similarity $\uparrow$ ) d

Table 5: Training regime comparison on Moving MNIST. Three training objectives applied to the same FluidWorld architecture (

\sim

801K parameters), evaluated after 30 epochs (Random baseline uses untrained weights). † The “JEPA-style” column refers to FluidWorld trained with a latent prediction objective inspired by LeCun’s JEPA framework LeCun (2022) : the pixel decoder is removed from the loss and replaced by a target-encoder (EMA) + VICReg objective. This is not Meta’s I-JEPA or V-JEPA architecture. Latent cosine similarity is the primary discriminating metric: it measures whether the model’s internal rollout tracks the true future state in representation space.

Dimension	Meta FAIR (V-JEPA 2) 1 1 1 Bardes et al., “V-JEPA 2: Self-Supervised Video Models Enable Understanding, Prediction, and Planning,” arXiv:2506.09985, 2025.	FluidWorld
Model size	$\sim$ 1.3B parameters	$\sim$ 801K parameters
Latent rollout stability (step 19)	Error accumulation (acknowledged bottleneck)	Cosine 0.827 (stable)
Autopoietic self-repair	Outside current scope	Recovery 66.8%, $p<10^{-49}$
Perturbation robustness	Outside current scope	50% corruption recovered in 3–7 steps
Physics-informed structure	Data-driven (ViT)	Reaction-diffusion PDE
Persistent memory	Fixed context window	Titans + DeltaNet

Table 6: FluidWorld in the context of the JEPA research program. Y. LeCun’s 2022 cognitive architecture blueprint 2 2 2 LeCun, “A Path Towards Autonomous Machine Intelligence,” OpenReview, 2022. identified several open challenges. Meta FAIR’s V-JEPA 2 tackles large-scale video understanding; FluidWorld explores complementary directions (rollout stability, self-repair, physics-grounded dynamics) at a much smaller scale. The two approaches are not in competition but address different parts of the same research agenda.

Figure 15: JEPA-style latent prediction stability. Cosine similarity between the model’s autonomous latent rollout and the target encoder’s representation of the true future, over 19 steps on Moving MNIST (

N\!=\!500

sequences). The FluidWorld PDE substrate trained with a JEPA-style latent objective maintains cosine similarity

\geq 0.827

through step 19, indicating that the PDE dynamics produce stable internal predictions even without any pixel-level supervision. Compare with the Pixel and Random models (near zero, Table 5 ).

Figure 16: Non-linear velocity encoding in JEPA-style representations. Linear probes (left bars) versus MLP probes (right bars) on frozen BeliefField features. The JEPA-style model is the only one whose velocity

R^{2}

increases with a non-linear probe (from 0.29 to 0.60,

\Delta=+0.31

), while Pixel and Random both decrease . This confirms that the PDE with a JEPA-style objective encodes dynamics in a distributed, non-linearly accessible form, consistent with abstract representations. Linear probes systematically underestimate the JEPA-style model’s encoding of physical dynamics.

Appendix figures & tables13 assets

Supplementary material from the paper’s appendix.

Appendix

Architecture
Hyperparameter	FluidWorld	Transformer	ConvLSTM
Latent dimension $d$	128	128	128
Patch size $p$	4	4	4
Spatial resolution $H_{f}\times W_{f}$	$16\times 16$	$16\times 16$	$16\times 16$
Encoder layers	3 (FluidLayer2D)	2 (TransformerBlock)	Conv bottleneck
Encoder mid channels	—	—	88

Appendix

Table 7: Full hyperparameter specification.

Figure 17: Gap evolution during training. Spatial Std and Effective Rank advantages narrow as the Transformer learns spatial patterns from data, but reconstruction fidelity gap widens . The PDE advantage is structural, not merely a convergence artifact.

Figure 18: Spatial heatmaps of the BeliefField at successive integration steps, starting from a near-uniform initialization ( $\epsilon=10^{-4}$ ). Within 10 steps, distinct spatial clusters emerge deterministically. The process is sensitive to initial conditions (different seeds produce different patterns) but fully reproducible given identical seeds.

Figure 19: Quantitative symmetry-breaking metrics. Symmetry index drops from 1.0 to 0.2, spatial entropy rises from 1.7 to 2.7, and KMeans detects 3 emerging clusters. All transitions occur within the first 10 integration steps.

Figure 20: Phase diagram across 225 $(D,\Delta t)$ configurations. The trained operating point ( $D\approx 0.25$ , $\Delta t=0.1$ ) is firmly in the super-critical regime. The only sub-critical region occurs at $\Delta t\approx 0.02$ , where dynamics are effectively frozen.

Figure 21: Energy trajectories from multiple initial conditions. Regardless of initialization (uniform, random, structured gradient), energy converges to the same attractor

E^{*}\approx 8{,}640

within

\sim

50 steps, confirming bounded attractor dynamics.

Figure 22: Four-panel energy conservation analysis. The PDE substrate with RMSNorm maintains bounded energy; without RMSNorm, energy diverges exponentially (

\times 7{,}467

over 200 steps). High-frequency components decay to

10^{-8}

by step 200.

Figure 23: Numerical stability across integration timesteps. Oscillatory instabilities appear when $\Delta t>0.10$ , confirming the CFL-like stability bound for the discrete Laplacian operator.

Figure 24: Recovery dynamics after 50% corruption of the BeliefField. Three modalities are shown: channel zeroing (instant recovery), Gaussian noise (3-step recovery), and spatial masking (7-step recovery). In all cases the system returns to its pre-corruption trajectory.

Figure 25: Resilience across corruption intensities (10%–90%). Residual MSE remains flat (

\approx 0.034

) up to 50% corruption, then increases monotonically but without discontinuity. No “cliff effect” at any corruption level.

Figure 26: Memory persistence test. An object is shown for 10 frames, occluded for 20 frames, then revealed. Titans-augmented model achieves SSIM recovery of 0.6 (vs 0.05 for PDE-only). Memory activity

|M|

increases during occlusion (0.018 to 0.035), indicating active information retention despite absent visual input.

Figure 27: Visualization of the BeliefField’s internal state during temporal evolution. The spatial field accumulates structure from successive frames, forming a persistent predictive representation that carries forward through PDE integration.

Figure 28: Feature-space analysis of the pixel-trained model. Spatial feature maps show clear semantic encoding of object positions and boundaries. 0 of 128 channels are dead, effective rank is 19.5, confirming healthy representational diversity.

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Abstract

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