Towards Size-Independent Generalization Bounds for Deep Operator Nets

TL;DR

This paper develops size-independent generalization bounds for DeepONets, a neural network architecture used for solving PDEs, by analyzing their Rademacher complexity and loss functions, with theoretical and experimental validation.

Contribution

It introduces a novel size-independent Rademacher complexity bound for DeepONets and demonstrates how to choose loss functions for size-independent generalization error bounds.

Findings

Size-independent Rademacher complexity bounds for DeepONets.

Generalization error bounds that do not depend on network size.

Experimental correlation between capacity measure and generalization error.

Abstract

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" which focuses on using neural nets for numerically solving differential equations. In this work, we aim to advance the theory of measuring out-of-sample error while training DeepONets - which is among the most versatile ways to solve P.D.E systems in one-shot. Firstly, for a class of DeepONets, we prove a bound on their Rademacher complexity which does not explicitly scale with the width of the nets involved. Secondly, we use this to show how the Huber loss can be chosen so that for these DeepONet classes generalization error bounds can be obtained that have no explicit dependence on the size of the nets. The effective capacity measure for DeepONets that we thus derive is also shown to correlate with the behavior of generalization error in experiments.