Dynamic Quadtree Tokenization and Transformer for Adaptive Mesh PDE Forecasting

Yilin Zhuang, Noah Zambrano, Karthik Duraisamy

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…