Flowers: A Warp Drive for Neural PDE Solvers

A research team including Till Muser and Alexandra Spitzer has introduced Flowers, a radically new neural architecture designed as a 'warp drive' for solving partial differential equations (PDEs). Unlike current dominant approaches that rely on Fourier transforms, dot-product attention, or convolutional mixing, Flowers is built entirely from multihead warps—each head predicts a displacement field and warps input features. This design is motivated by physics and computational efficiency, implementing adaptive global interactions at linear cost. The architecture introduces nonlocality only through sparse sampling at source coordinates, one per head, and stacks warps in multiscale residual blocks.

Flowers demonstrates exceptional performance on a broad suite of 2D and 3D time-dependent PDE benchmarks, particularly excelling at modeling flows and waves. The compact 17M-parameter model consistently outperforms Fourier, convolution, and attention-based baselines of similar size. More impressively, a 150M-parameter variant improves over recent transformer-based foundation models that use significantly more parameters, data, and training compute. The researchers provide theoretical motivation through three lenses: flow maps for conservation laws, waves in inhomogeneous media, and a kinetic-theoretic continuum limit. This breakthrough suggests a new paradigm for neural PDE solvers that could dramatically accelerate scientific computing and physics simulations.