Integral equations (IEs) are equations that model spatiotemporal systems with non-local interactions. They have found important applications throughout theoretical and applied sciences, including in physics, chemistry, biology, and engineering. While efficient algorithms exist for solving given IEs, no method exists that can learn an IE and its associated dynamics from data alone. In this paper, we introduce Neural Integral Equations (NIE), a method that learns an unknown integral operator from data through an IE solver. We also introduce Attentional Neural Integral Equations (ANIE), where the integral is replaced by self-attention, which improves scalability and model capacity. We demonstrate that (A)NIE outperforms other methods in both speed and accuracy on several benchmark tasks in ODE, PDE, and IE systems of synthetic and real-world data.

翻译：积分方程（IEs）是描述具有非局域相互作用的时空系统的方程，在理论科学与应用科学（包括物理学、化学、生物学和工程学）中均有重要应用。尽管存在高效算法用于求解给定积分方程，但目前尚无方法能仅从数据中学习积分方程及其相关动力学。本文提出神经积分方程（NIE）——一种通过积分方程求解器从数据中学习未知积分算子的方法。我们还提出了注意力神经积分方程（ANIE），其中积分运算被自注意力机制替代，从而提升了可扩展性与模型容量。实验表明，在常微分方程、偏微分方程及积分方程系统的合成数据与真实数据基准任务中，（A）NIE在速度与精度上均优于其他方法。