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arXiv · 2510.04304

Wave-PDE Nets: Trainable Wave-Equation Layers as an Alternative to Attention

Harshil Vejendla
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Abstract

We introduce Wave-PDE Nets, a neural architecture whose elementary operation is a differentiable simulation of the second-order wave equation. Each layer propagates its hidden state as a continuous field through a medium with trainable spatial velocity c(x) and damping {\gamma}(x). A symplectic spectral solver based on FFTs realises this propagation in O(nlog n) time. This oscillatory, global mechanism provides a powerful alternative to attention and first-order state-space models. We prove that a single Wave-PDE layer is a universal approximator. On language and vision benchmarks, Wave-PDE Nets match or exceed Transformer performance while demonstrating superior practical efficiency, reducing wall-clock time by up to 30% and peak memory by 25%. Ablation studies confirm the critical role of symplectic integration and a spectral Laplacian for stability and performance. Visualizations of the learned physical parameters reveal that the model learns intuitive strategies for information propagation. These results position Wave-PDE Nets as a computationally efficient and robust architecture with a strong physical inductive bias.

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Harshil Vejendla. 2025-10-05. Wave-PDE Nets: Trainable Wave-Equation Layers as an Alternative to Attention. https://arxiv.org/abs/2510.04304

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