Spatial Transport of Integration Error in Generative ODEs

A trained flow or diffusion model is usually run with only a handful of solver steps, and the integration error this leaves behind is unevenly distributed across the image. We ask where that error is injected and how it reaches the endpoint, and answer with a signed source-and-transport accounting of few-step integration error, tested to first order. A perturbation experiment on five models at 256^2 resolution shows the learned dynamics spread local disturbances widely: near the start of sampling, under 10% of the summed endpoint response remains at the source. Signed one-step truncation residuals, propagated through the model's own linearized dynamics, reconstruct much of the endpoint error's direction and regional structure (cosine 0.81-0.87), and a region's error owes more to what arrives from elsewhere than to its own injection. Structure-destroying nulls, with protocols frozen before evaluation, locate what carries the account: randomizing contribution signs halves it, and reassigning which region receives each contribution, with content, norms, and signs intact, destroys it entirely. Where the injections land is readable from the model itself. The variation of its velocity or prediction field along the trajectory, a structure that emerges during training, predicts the final per-region gap (within-image rho of 0.57-0.70 on fine trajectories, weaker from the cheap solve alone). The prediction is partial because endpoint error depends not only on injected magnitude but on its sign, timing, and transport through the learned dynamics. A training penalty on the injected variation lowers few-step error, so the structure is one a model can be trained to change.

MosaicMem: Hybrid Spatial Memory for Controllable Video World Models

Video diffusion models are moving beyond short, plausible clips toward world simulators that must remain consistent under camera motion, revisits, and intervention. Yet spatial memory remains a key bottleneck: explicit 3D structures can improve reprojection-based consistency but struggle to depict moving objects, while implicit memory often produces inaccurate camera motion even with correct poses. We propose Mosaic Memory (MosaicMem), a hybrid spatial memory that lifts patches into 3D for reliable localization and targeted retrieval, while exploiting the model's native conditioning to preserve prompt-following generation. MosaicMem composes spatially aligned patches in the queried view via a patch-and-compose interface, preserving what should persist while allowing the model to inpaint what should evolve. With PRoPE camera conditioning and two new memory alignment methods, experiments show improved pose adherence compared to implicit memory and stronger dynamic modeling than explicit baselines. MosaicMem further enables minute-level navigation, memory-based scene editing, and autoregressive rollout.