Flowers: A Warp Drive for Neural PDE Solvers

• URL: http://arxiv.org/abs/2603.04430v1
• Date: Tue, 17 Feb 2026 15:06:28 GMT
• Title: Flowers: A Warp Drive for Neural PDE Solvers
• Abstract summary: We introduce Flowers, a neural architecture for learning PDE solution operators built entirely from multihead warps.<n>Flowers achieve excellent performance on a broad suite of 2D and 3D time-dependent PDE benchmarks, particularly flows and waves.
• Score: 47.10397857711356
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce Flowers, a neural architecture for learning PDE solution operators built entirely from multihead warps. Aside from pointwise channel mixing and a multiscale scaffold, Flowers use no Fourier multipliers, no dot-product attention, and no convolutional mixing. Each head predicts a displacement field and warps the mixed input features. Motivated by physics and computational efficiency, displacements are predicted pointwise, without any spatial aggregation, and nonlocality enters \emph{only} through sparse sampling at source coordinates, \emph{one} per head. Stacking warps in multiscale residual blocks yields Flowers, which implement adaptive, global interactions at linear cost. We theoretically motivate this design through three complementary lenses: flow maps for conservation laws, waves in inhomogeneous media, and a kinetic-theoretic continuum limit. Flowers achieve excellent performance on a broad suite of 2D and 3D time-dependent PDE benchmarks, particularly flows and waves. A compact 17M-parameter model consistently outperforms Fourier, convolution, and attention-based baselines of similar size, while a 150M-parameter variant improves over recent transformer-based foundation models with much more parameters, data, and training compute.

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