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, capacity, and results in an interpretable model. 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（IE）是描述具有非局部相互作用的时空系统的方程。它们在理论科学和应用科学（包括物理学、化学、生物学和工程学）中具有重要的应用价值。尽管存在求解给定积分方程的高效算法，但目前尚无方法能够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），这是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。我们还提出注意力神经积分方程（ANIE），其中积分被自注意力机制替代，从而提高了可扩展性和容量，并生成了一个可解释的模型。我们证明，（A）NIE在多个基准任务（包括ODE、PDE和IE系统）的合成数据与真实数据上，在速度和精度方面均优于其他方法。