Simone Brugiapaglia, Nick Dexter, Sebastian Moraga and I have just uploaded a new paper on learning Hilbert-valued functions from limited data using deep neural networks. This problem arises in many important problems in computational science and engineering, notably the solution of parametric PDEs for UQ. In the paper, we first present a novel practical existence theorem that shows there is a DNN architecture and training procedure that is guaranteed to perform as well the current state-of-the-art methods in terms of sample complexity. We also quantify all errors in the process, including the measurement error and physical space discretization error. We then present results from initial numerical investigations on parametric PDE problems. These results are promising, and show that even simpler DNNs and training can achieve competitive and sometimes better results than current best-in-class schemes.
Stand by for more work in this direction in the near future! In the meantime, the paper can be found here: