  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.
