Capacity Bounds for the DeepONet Method of Solving Differential Equations

In recent times machine learning methods have made significant advances in becoming a useful tool for analyzing physical systems. A particularly active area in this theme has been "physics informed machine learning" [1] which focuses on using neural nets for numerically solving differential equations. Among all the proposals for solving differential equations using deep-learning, in this paper we aim to advance the theory of generalization error for DeepONets - which is unique among all the available ideas because of its particularly intriguing structure of having an inner-product of two neural nets. Our key contribution is to give a bound on the Rademacher complexity for a large class of DeepONets. Our bound does not explicitly scale with the number of parameters of the nets involved and is thus a step towards explaining the efficacy of overparameterized DeepONets. Additionally, a capacity bound such as ours suggests a novel regularizer on the neural net weights that can help in training DeepONets - irrespective of the differential equation being solved. [1] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics-informed machine learning. Nature Reviews Physics, 2021.