Title: Flowers: Leveraging Warp Mechanics for Neural Partial Differential Equation Solvers

Abstract:

We present Flowers, a novel neural architecture designed to learn operators for solving partial differential equations (PDEs). This framework is constructed exclusively using multihead warping mechanisms. Unlike traditional models, Flowers eschew Fourier multipliers, dot-product attention, and convolutional mixing, relying instead on pointwise channel mixing and a multiscale scaffold. In this approach, each attention head computes a displacement field and applies it to warp the mixed input features.

Driven by principles of physics and computational efficiency, these displacements are calculated pointwise, avoiding any form of spatial aggregation. Nonlocal interactions are introduced solely through sparse sampling at source coordinates, with one sample allocated per head. By stacking these warping layers within multiscale residual blocks, Flowers facilitate adaptive, global interactions at a linear computational cost.

We justify this architectural choice through three theoretical perspectives: flow maps relevant to conservation laws, wave propagation in inhomogeneous media, and a kinetic-theoretic continuum limit. Empirical evaluations demonstrate that Flowers deliver superior performance across a wide range of 2D and 3D time-dependent PDE benchmarks, with notable strengths in modeling flows and waves. Specifically, a compact variant with 17 million parameters consistently surpasses baseline models based on Fourier transforms, convolutions, and attention mechanisms of comparable size. Furthermore, a larger 150 million parameter version outperforms recent transformer-based foundation models, despite those competitors utilizing significantly greater parameters, data volumes, and training resources.

Source: arXiv Generated at: 2026-06-02 00:00:00 UTC