DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators

DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators Lu Lu 1, Pengzhan Jin 2, and George Em Karniadakis 1 1 Division of Applied Mathematics, Brown University, Providence, RI 02912, USA 2 LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Abstract While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator [5]. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensorsx i,i 1,...,m (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size. 1 Introduction The universal approximation theorem states that neural networks can be used to approximate any continuous function to arbitrary accuracy if no constraint is placed on the width and depth of the hidden layers [7, 11]. However, another approximation result, which is yet more surprising and has not been appreciated so far, states that a neural network with a single hidden layer can approximate accurately any nonlinear continuous functional (a mapping from a space of functions into the real numbers) [3, 18, 25] or (nonlinear) operator (a mapping from a space of functions into another space of functions) [5, 4]. Before reviewing the approximation theorem for operators, we introduce some notation, which will be used through this paper. Let G be an operator taking an input functionu, and then G( u) is the corresponding output function.