Dynamic Quadtree Tokenization and Transformer for Adaptive Mesh PDE Forecasting

Abstract:The quadratic attention cost of Vision Transformers (ViTs) forces a trade-off between spatial resolution and rollout horizon, particularly for fine-scale PDEs where shocks, reaction fronts, and material interfaces occupy small, evolving regions of the domain. Conventional neural surrogates also lack mechanisms to adapt resolution dynamically. We propose WAMRViT, a ViT that tokenizes inputs as balanced quadtrees using a wavelet-inspired refinement criterion, jointly encodes position and refinement level with 3D rotary positional embeddings, and regrids in cell space during inference for stable long-horizon rollouts. A multi-scale variant retains each leaf at its native source resolution and lets the model learn across resolution levels. Unlike fixed-budget adaptive-tokenization methods, WAMRViT imposes no predetermined token count and supports fully adaptive topology throughout autoregressive rollout. To our knowledge, it is the first machine-learning surrogate to natively tokenize multi-level Adaptive Mesh Refinement (AMR) data. On uniform-grid benchmarks, uniform-patch WAMRViT improves finest-level region-of-interest VRMSE over a finest-patch uniform ViT while using substantially fewer tokens. The multi-scale variant achieves the lowest first-step full-field RMSE and VRMSE on both benchmarks and improves rollout-averaged full-field and refined-region accuracy at long horizons. With parallelized regridding, its end-to-end rollout cost lies between finest-patch and approximately token-matched coarser-patch ViTs. On a complex AMR combustion problem whose finest features cannot be represented natively by the evaluated uniform-grid baselines, WAMRViT operates directly on adaptive cells and substantially reduces finest-level error at matched transformer capacity. Code: this https URL

Computer Vision and Pattern Recognition (cs.CV); Machine Learning (cs.LG)

arXiv:2610.04044 [cs.CV]

(or arXiv:2610.04044v1 [cs.CV] for this version)

https://doi.org/10.48550/arXiv.2610.04044 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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