Efficient operator learning with derivative-enhanced parameter sensitivity information and hybrid optimization

Abstract We present two complementary strategies that improve the performance of Deep Operator Networks (DeepONets) for parametric partial differential equations (PDEs). The first one enriches the loss function with parameter derivatives, inspired by Hermite interpolation, thereby enhancing generalization even when only limited training data are available. We generate both solutions and parameter sensitivities by extending the OpenFOAM finite volume method (FVM) solver with automatic differentiation. The second strategy is a hybrid training scheme that combines gradient descent (GD) with least-squares (LS) optimization. By isolating the final linear layer and solving it exactly via LS at each GD step, we accelerate convergence and reduce training error. A distinctive feature of our formulation is the treatment of parametric inputs: rather than evaluating high-dimensional fields across the entire domain, we project them onto a coarse parameter mesh before passing them to the branch network, substantially lowering input dimension and network complexity while retaining accuracy. Numerical experiments on convection–diffusion problems with heterogeneous diffusivity and velocity fields confirm that this framework consistently reduces error and improves efficiency, highlighting the promise of derivative-enhanced, hybrid-trained DeepONets for complex parametric PDEs.