  Operator learning techniques have recently emerged as a powerful tool for
learning maps between infinite-dimensional Banach spaces. Trained under
appropriate constraints, they can also be effective in learning the solution
operator of partial differential equations (PDEs) in an entirely
self-supervised manner. In this work we analyze the training dynamics of deep
operator networks (DeepONets) through the lens of Neural Tangent Kernel (NTK)
theory, and reveal a bias that favors the approximation of functions with
larger magnitudes. To correct this bias we propose to adaptively re-weight the
importance of each training example, and demonstrate how this procedure can
effectively balance the magnitude of back-propagated gradients during training
via gradient descent. We also propose a novel network architecture that is more
resilient to vanishing gradient pathologies. Taken together, our developments
provide new insights into the training of DeepONets and consistently improve
their predictive accuracy by a factor of 10-50x, demonstrated in the
challenging setting of learning PDE solution operators in the absence of paired
input-output observations. All code and data accompanying this manuscript are
publicly available at
\url{https://github.com/PredictiveIntelligenceLab/ImprovedDeepONets.}
