METR-LA directed diffusion¶

Directed affinity lowers validation RMSE against both permuted and self-only controls in two matched seeds, with the advantage growing at 30 and 60 minutes. Self-only recurrence still has lower MAE. This supports a topology-specific long-horizon error signal, not a model that dominates its node-local control.

Factorized mesh¶

METR-LA supplies a fixed directed spatial domain and an ordered five-minute time domain:

space G                 time T
sensor u -> sensor v    ... -> t-1 -> t
             \                 /
              joint state (v,t)

The conceptual domain is G x T, following the time-vertex framework. Execution stays factorized: one lowered Graph is reused while DiffusionGRU advances hidden state through 12 snapshots. Tinymesh never constructs product adjacency or a dense node-pair tensor.

This is a time-varying traffic field on a directed sensor mesh. The DCRNN affinity is not a complete street graph, and the source does not declare the speed or distance units needed to invent physical travel times.

Inputs¶

Every input fact is available at its snapshot:

[B,12,207,6]
       |
       +-- normalized speed; missing input becomes 0 after normalization
       +-- observed mask
       +-- sin/cos daily phase
       +-- sin/cos weekly phase

Daily and weekly phase are continuous across midnight. Speed normalization, sensor fallback means, and all baseline statistics use observed training rows only. Targets, masks, target-time splits, raw-speed metrics, and persistence remain identical to the existing METR-LA forecast.

Model¶

For each period, public Tinymesh components construct three aligned channels. This is the smallest DCRNN-style recurrent test that preserves the source affinity:

local values --------------------------+
forward affinity diffusion ------------+--> DiffusionGRU --> H[t]
reverse affinity diffusion ------------+
H[t-1] --------------------------------+

H[11] --> ReLU --> Linear(32,12) --> persistence + residual

DirectedDiffusion source-normalizes the original affinity in both graph directions. DiffusionGRU keeps the local channel beside both transported channels inside its update, reset, and candidate paths. Sequential unrolling makes period t depend on earlier hidden state; unlike A3T-GCN, periods are not encoded independently from the same zero state.

At hidden width 32 the model has 11,436 parameters. Twelve recurrent steps perform 48 sparse sums after the fixed diffusion weights are prepared once. Topology storage remains O(N + E) and no [N,N] or [N,E] carrier enters the model.

Controls¶

The same initialization, parameters, features, split, batches, loss, and checkpoint policy receive:

true       original directed support + affinity
permuted   one fixed node relabeling + aligned affinity
self       self edges with unit affinity

The declared graph gate required true affinity to beat both controls on MAE and RMSE in at least two of three matched seeds. Model test stayed closed.

Model selection¶

head=residual   loss=mae        hidden=32
epochs=3        batch=512       learning_rate=0.001
seeds=0,1       checkpoint=each epoch

At seed 0, one epoch made self-only look best on both metrics. Three epochs changed the structural conclusion: every topology continued improving, while true affinity became best on RMSE. This is direct evidence that the one-epoch probe undertrained the graph model.

Hidden width 16 reached 3.8025 MAE and 7.4732 RMSE after one epoch, versus 3.7765 and 7.3958 at width 32. It removed 8,016 parameters but only 13.92 seconds end to end in those ordered runs, so width 32 earned the frozen quality budget. That timing is model-selection context, not a performance benchmark.

Validation result¶

Values are mean ± sample standard deviation across seeds 0 and 1:

True affinity lowers mean RMSE by 0.97% relative to permuted affinity and 0.90% relative to self-only. It wins all four paired overall RMSE comparisons. It does not win MAE: self-only wins both paired comparisons, and permuted affinity narrowly wins seed 1.

The RMSE separation grows with forecast distance:

At 30 and 60 minutes, true affinity beats both controls in each individual seed. At 15 minutes self-only is slightly better on mean RMSE. Huber loss repeats the tradeoff at seed 0: true affinity reaches 3.6931 MAE and 6.8997 RMSE, while self-only reaches 3.5669 and 6.9805. Changing the loss does not recover both objectives.

Decision¶

The declared all-metric graph gate fails. After true affinity lost self-only MAE in seeds 0 and 1, seed 2 could not produce the required two passing seeds and was not run. Model test remains unopened.

The narrower result is useful:

local recurrence     lower typical absolute error
directed diffusion   lower large and long-horizon error
                                  |
                                  v
factorized follow-up identifies topology but loses to the incumbent

The factorized follow-up keeps local recurrence outside graph propagation and adds a zero-initialized transported residual. True topology beats both of its matched controls, but the earlier self-only DiffusionGRU still has lower MAE and RMSE. This strengthens the topology signal without earning a model or API promotion.

Typed relations, hierarchy, and physical integration from MeshGraphNets, RIGNO, and PhyMPGN remain broader Tinymesh research, not claims supported by this result.

No public component changes. DirectedDiffusion and DiffusionGRU remain correct sparse compositions; the result constrains how a forecast should combine them.

Reproduce¶

The Metal comparison used these commands with MODEL=true, permuted, and self, once for each seed:

uv run --locked python -m experiments.run metr_la_diffusion DEV=METAL EPOCHS=3 MODEL=true HEAD=residual LOSS=mae SEED=0 BS=512 HIDDEN=32 LR=0.001 CHECKPOINT_EVERY=1
uv run --locked python -m experiments.run metr_la_diffusion DEV=METAL EPOCHS=3 MODEL=permuted HEAD=residual LOSS=mae SEED=0 BS=512 HIDDEN=32 LR=0.001 CHECKPOINT_EVERY=1
uv run --locked python -m experiments.run metr_la_diffusion DEV=METAL EPOCHS=3 MODEL=self HEAD=residual LOSS=mae SEED=0 BS=512 HIDDEN=32 LR=0.001 CHECKPOINT_EVERY=1
uv run --locked python -m experiments.run metr_la_diffusion DEV=METAL EPOCHS=3 MODEL=true HEAD=residual LOSS=mae SEED=1 BS=512 HIDDEN=32 LR=0.001 CHECKPOINT_EVERY=1
uv run --locked python -m experiments.run metr_la_diffusion DEV=METAL EPOCHS=3 MODEL=permuted HEAD=residual LOSS=mae SEED=1 BS=512 HIDDEN=32 LR=0.001 CHECKPOINT_EVERY=1
uv run --locked python -m experiments.run metr_la_diffusion DEV=METAL EPOCHS=3 MODEL=self HEAD=residual LOSS=mae SEED=1 BS=512 HIDDEN=32 LR=0.001 CHECKPOINT_EVERY=1

True and self-only ran at revision 1d81c7be. Permuted controls ran at 87dddb14, whose only execution change moves the fixed timeout into the catalog. Model, data, features, splits, optimization, references, and metrics are identical. Every model result records evaluate_test=false and test=null.

The original implementation smoke at 90f53dca processed 512 full-source windows with 11,436 parameters and 48 sparse calls.