On full-field cost, MSE-only training loses 110.829 to 95.698 million share-ticks against same-ticker, same-time-of-day historical resampling. Those are Phee and Balch's figures. The MSE-only model also trails that baseline on transport energy score, 2.152 versus 2.018, though it wins on the binwise joint-field score, 0.414 versus 0.496. An energy score takes the mean distance from draws to the realization and subtracts half the mean distance between draws. Add a transport penalty during training and the first two results turn around, to 60.074 and 1.591. That reversal matters to anyone modeling liquidity provision. The case for trading on it remains open.

OrderFlowDiT is the authors' conditional Diffusion Transformer. It divides the 10,001-tick price axis into patches, each treated as a token, and generates unnormalized bid and ask submission fields in shares. The authors train two versions of the network: Base-MSE uses the usual denoising squared-error loss; Base-UOT adds an unbalanced optimal-transport penalty. That penalty charges for moving volume across ticks and for differences in total volume. Evaluation uses a full-field cost combining transport and volume terms, alongside proper energy scores.

The data are LOBSTER full-depth messages for 30 Dow-style large caps, from 10:30 to 16:00 ET. With 58.7M parameters, the Base model trains on 10 days, June 8 to 22, 2026, or 96,300 pairs. June 23 supplies the evaluation set: 1,950 conditions, 65 per stock.

What the forecast actually counts

The target is new limit-order volume submitted over the next 300 seconds. It records shares at every one-cent tick within 5,000 ticks of the mid, on both sides jointly: a 2 × 10,001 field for each forecast. An order counts even if it is canceled two seconds later. Conditioning inputs are the current book, the previous five minutes of submissions, the spread, time of day and the ticker. The output is a distribution of fields, from which the authors draw four per condition.

Bar volume records fills. A future book nets submissions against cancellations and executions. Neither reveals all the new liquidity that arrived. In the paper's example, a ±300-tick window around the mid contains 53% of submitted volume and only 1.2% of resting depth. The distinction also stopped us from reproducing this test: the target requires message-level submissions by price and side, including orders later canceled. Our one-minute OHLCV bars contain none of that information.

Does tick distance earn its keep?

Squared error judges every tick separately. It gives a forecast spike one tick from the realized spike the same penalty as one 500 ticks away. Base-UOT instead adds an unbalanced optimal-transport term on the decoded field in shares. Moving volume costs one unit per tick, while the fields may have different totals. Its weight is 0.1, reduced at high noise levels.

Base-UOT wins on all seven reported metrics. Full-field cost drops 45.8%; transport energy score drops 26.1%. Both improve for every one of the 30 stocks.

Total volume accounts for a substantial part of the story. Relative volume error, NTVE, falls from 1.058 to 0.447. At a mean NTVE of 1.058, the MSE model's side totals miss by more than the realized total on average. Only overprediction can produce that result, and any mass-sensitive cost will punish it regardless of where the volume lands. The headline measures also resemble the training objective. Full-field cost charges 50 ticks per share of mass mismatch, and transport energy score uses tick-displacement W1. The authors point to the joint-field energy score, a strictly proper binwise measure with different geometry, and note that full-field cost differs from the entropic, KL-relaxed training loss. On the joint-field score, the gain is 0.414 to 0.387, about 6.5%, beside 26.1% on the transport score. Log-volume CRPS, a proper score for the predicted distribution of log total volume, improves from 0.460 to 0.283. The latter only ties the empirical baseline.

Both skew-normal fits beat Base-UOT on relative volume error, posting NTVE of 0.328 against 0.447. Their full-field costs are much worse: 450.515 for the single fit and 109.510 for the mixture.

GRPO draws closer together

GRPO, or group-relative policy optimization, rewards draws that approach the realization while remaining apart from each other. Starting from Base-UOT, it improves transport energy score by 6.1% and volume CRPS by 14.0%. Joint-field score worsens 5.7%, and full-field cost worsens 3.2%. Although 24 of 30 per-stock means improve, pooled full-field cost rises because a few stocks deteriorate heavily.

The energy-score components show why the measures split. Under transport geometry, mean distance to the realization falls from 2.897 to 2.510. Mean distance between draws falls from 2.611 to 2.033. The target-distance drop of 0.387 exceeds half the dispersion drop, 0.289, so the score improves. Under joint-field geometry, distance to the realization moves from 0.722 to 0.686, a drop of 0.036. Distance between draws moves from 0.670 to 0.553; half that drop is 0.059. Lost dispersion outweighs the smaller improvement toward the target. The draws converge on each other faster than on the truth. As the authors write, "A lower average distance to the target therefore need not yield a better score for the predictive distribution." GRPO also changes the sampler, from 50 deterministic steps to 20 stochastic ones. The paper does not separate that change from the reward's effect.

June 23 carries the result

Every result above comes from June 23, the day also used for checkpoint selection. The paper calls its comparisons "descriptive, using the checkpoint-selection day and one training seed per learned configuration." June 24 is reserved as a test day and not used. The authors disclose the overlap, and their abstract calls these validation conditions. Still, 45.8% is a single-seed validation result without an interval. Why leave the headline untested on the reserved June 24? The paper directs readers to its training section for the checkpoint-selection rule; we did not find it there. If selection used one of the reported metrics, that metric carries selection bias.

Calibration is weak even for Base-UOT. With four draws, a calibrated forecast should place realized side volume in the interior ranks 60% of the time. Base-UOT reaches 0.395 and GRPO 0.385. Those ensembles are too narrow or off-center on total volume, the quantity a scheduler would most want bounds on.

More capacity and data bring a smaller full-field gain. Large-UOT has 174.3M parameters and 29 training days; it improves full-field cost by 2.1% over Base-UOT. It also trains for 100,000 steps at a lower learning rate, leaving scale and training budget entangled. Its transport energy score improves 17.5% over Base-UOT. Large-MSE retains lower NTVE and log-field MSE than Large-UOT.

Execution remains untested.

The authors state the wider limit plainly: "The central limitation is that generating submission flow does not complete a financial world model." Their fields omit cancellations, executions and event ordering. Volume submitted and canceled within the window counts in full. The paper reports no execution-cost or fill-rate test.

For generating price-indexed volume fields, transport loss looks like a sound default. Base-MSE's full-field loss to same-time-of-day resampling, 110.829 vs 95.698, makes that case on its own. A rerun of the same table on June 24, together with a passive-execution test conditioned on these forecasts, would change my view of its practical value.