  The deep operator network (DeepONet) is a popular neural operator
architecture that has shown promise in solving partial differential equations
(PDEs) by using deep neural networks to map between infinite-dimensional
function spaces. In the absence of labeled datasets, we utilize the PDE
residual loss to learn the physical system, an approach known as
physics-informed DeepONet. This method faces significant computational
challenges, primarily due to the curse of dimensionality, as the computational
cost increases exponentially with finer discretization. In this paper, we
introduce the Separable DeepONet framework to address these challenges and
improve scalability for high-dimensional PDEs. Our approach involves a
factorization technique where sub-networks handle individual one-dimensional
coordinates, thereby reducing the number of forward passes and the size of the
Jacobian matrix. By using forward-mode automatic differentiation, we further
optimize the computational cost related to the Jacobian matrix. As a result,
our modifications lead to a linear scaling of computational cost with
discretization density, making Separable DeepONet suitable for high-dimensional
PDEs. We validate the effectiveness of the separable architecture through three
benchmark PDE models: the viscous Burgers equation, Biot's consolidation
theory, and a parametrized heat equation. In all cases, our proposed framework
achieves comparable or improved accuracy while significantly reducing
computational time compared to conventional DeepONet. These results demonstrate
the potential of Separable DeepONet in efficiently solving complex,
high-dimensional PDEs, advancing the field of physics-informed machine
learning.
