# Error Decomposition

Blockwise distillation architecture search picks child networks by summing each replacement block's isolated score, an additive assumption rarely tested directly. On DeiT-S and DeiT-B, as the block-latency budget drops from 60% to 55%, the additive integer linear program begins to trail a pairwise integer quadratic program that models block interactions, by 3.3–16.6 top-1 points on ImageNet-100 after fine-tuning. The error splits into a metric part from the score's nonlinearity and a functional interaction part, dominant on DeiT-S and DeiT-B respectively; a hybrid search that ranks with single-block logits and measures only a few candidates needs no dedicated pair measurements and beats the pairwise search in 13 of 14 full ImageNet-1k settings.

Blockwise distillation architecture search picks child networks by summing each replacement block's isolated score, an additive assumption rarely tested directly. On DeiT-S and DeiT-B, as the block-latency budget drops from 60% to 55%, the additive integer linear program begins to trail a pairwise integer quadratic program that models block interactions, by 3.3–16.6 top-1 points on ImageNet-100 after fine-tuning. The error splits into a metric part from the score's nonlinearity and a functional interaction part, dominant on DeiT-S and DeiT-B respectively; a hybrid search that ranks with single-block logits and measures only a few candidates needs no dedicated pair measurements and beats the pairwise search in 13 of 14 full ImageNet-1k settings.

Aggregation-diffusion equations are recast as a constrained optimal control problem whose control is the transport velocity, parameterized by a neural network and trained on a variational objective evaluated along particle trajectories, so mass conservation and density positivity hold by construction without a spatial mesh. Earlier structure-preserving schemes mostly relied on mesh-based discretization whose cost grows quickly with dimension; the framework splits total error into neural approximation, optimization, Monte Carlo sampling and time discretization, and is tested on Doi-Onsager, Patlak-Keller-Segel, porous-medium and Fokker-Planck models.

Aggregation-diffusion equations are recast as a constrained optimal control problem whose control is the transport velocity, parameterized by a neural network and trained on a variational objective evaluated along particle trajectories, so mass conservation and density positivity hold by construction without a spatial mesh. Earlier structure-preserving schemes mostly relied on mesh-based discretization whose cost grows quickly with dimension; the framework splits total error into neural approximation, optimization, Monte Carlo sampling and time discretization, and is tested on Doi-Onsager, Patlak-Keller-Segel, porous-medium and Fokker-Planck models.