Metriplector: From Field Theory to Neural Architecture

Neural fields parameterize data as functions from coordinates to values, providing a unified framework for representation learning across modalities. Existing approaches are dominated by per-sample meta-learning, which scales poorly due to memory-intensive inner-loop optimization. The natural alternative -- feed-forward encoding -- typically introduces modality-specific assumptions, sacrificing the generality that makes learning with neural fields attractive. We argue that locality and hierarchy are useful priors for learning field representations that can be injected without compromising modality-agnosticism. We propose LH-NeF, a framework to learn general-purpose tokenized representations of continuous signals. A locality-preserving hierarchical encoder maps raw coordinate-value field observations to structured tokens, from which the field is reconstructed during training. By replacing meta-learning's inner loop with a single forward pass, LH-NeF uses 42$\times$ less memory and supports 133$\times$ larger batches than the strongest modality-agnostic baseline. Across images, 3D shapes, and climate fields, our learned representations match or exceed performance of modality-agnostic, modality-specific, and specialized generative neural field baselines on both reconstruction and downstream tasks.

In this paper we develop quantitative approximation results for shallow neural networks constructed using a dictionary based on metaplectic operators. First, we extend the concept of Barron spaces by considering a symplectically motivated extension of the Fourier transform, known as the metaplectic transform. Then, after establishing embedding between metaplectic Barron spaces and Sobolev spaces we consider a neural metaplectic dictionary and we prove Monte-Carlo approximation bounds for metaplectic Barron functions using finite linear combinations of atoms of the dictionary. Finally, we validate the introduction of the neural metaplectic dictionary by devising a deep neural network architecture that uses as building blocks the atoms of the dictionary. We test it to approximate solutions of time-dependent Schrödinger equations, demonstrating better performance compared to classical phyisics informed neural networks architectures.

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 only through sparse sampling at source coordinates, 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.