  Operator learning has become a powerful tool in machine learning for modeling
complex physical systems. Although Deep Operator Networks (DeepONet) show
promise, they require extensive data acquisition. Physics-informed DeepONets
(PI-DeepONet) mitigate data scarcity but suffer from inefficient training
processes. We introduce Separable Operator Networks (SepONet), a novel
framework that significantly enhances the efficiency of physics-informed
operator learning. SepONet uses independent trunk networks to learn basis
functions separately for different coordinate axes, enabling faster and more
memory-efficient training via forward-mode automatic differentiation. We
provide theoretical guarantees for SepONet using the universal approximation
theorem and validate its performance through comprehensive benchmarking against
PI-DeepONet. Our results demonstrate that for the 1D time-dependent advection
equation, when targeting a mean relative $\ell_{2}$ error of less than 6% on
100 unseen variable coefficients, SepONet provides up to $112 \times$ training
speed-up and $82 \times$ GPU memory usage reduction compared to PI-DeepONet.
Similar computational advantages are observed across various partial
differential equations, with SepONet's efficiency gains scaling favorably as
problem complexity increases. This work paves the way for extreme-scale
learning of continuous mappings between infinite-dimensional function spaces.
