Topological DeepONets Advance Operator Learning with Functional Measurements

Deep Operator Networks (DeepONets) are a class of neural networks designed to learn operators between infinite-dimensional function spaces. Traditionally, these networks represent input functions using discrete point samples. This new research extends the Topological DeepONet framework by replacing these point samples with continuous linear functionals. This means the network processes measurements drawn from the continuous dual of a Hausdorff locally convex space, which offers a more generalized and robust way to encode complex input functions, especially those in non-normable spaces. The proposed framework develops both fixed and adaptive functional measurement systems. These measurements are integrated with a coefficient-space Two-Step procedure and stabilized by a training-only decoder and regularization. The paper provides a detailed error decomposition, separating errors arising from measurement, output-basis, and neural approximation, and refines the Barron-rate. This theoretical foundation underpins the practical improvements observed. Evaluations across various challenging problems, including the antiderivative operator, heterogeneous Darcy flow, and Navier-Stokes vorticity operators, demonstrate the efficacy of this approach. For instance, in the fixed-time Navier-Stokes problem, the Adaptive Topological DeepONet achieved superior accuracy compared to other DeepONet-based models, with significantly lower memory requirements and training time than a comparable Fourier Neural Operator (FNO), while maintaining competitive error rates. This innovation provides compact, interpretable, and discretization-portable coordinates, particularly beneficial for complex scientific and engineering simulations.