Towards Size-Independent Generalization Bounds for Deep Operator Nets

37 Pages Posted: 18 Mar 2024

See all articles by Pulkit Gopalani

Pulkit Gopalani

affiliation not provided to SSRN

Sayar Karmakar

University of Florida

Dibyakanti Kumar

Indian Institute of Technology Guwahati

Anirbit Mukherjee

Wharton (UPenn), Department of Statistics

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 PDE 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. We note that our theoretical results apply to any PDE being targeted to be solved by DeepONets.

Keywords: Rademacher complexity, DeepONets, Physics-Inspired ML, Operator Learning

Suggested Citation

Gopalani, Pulkit and Karmakar, Sayar and Kumar, Dibyakanti and Mukherjee, Anirbit, Towards Size-Independent Generalization Bounds for Deep Operator Nets. Available at SSRN: https://ssrn.com/abstract=4763746 or http://dx.doi.org/10.2139/ssrn.4763746

Pulkit Gopalani

affiliation not provided to SSRN ( email )

No Address Available

Sayar Karmakar

University of Florida ( email )

PO Box 117165, 201 Stuzin Hall
Gainesville, FL 32610-0496
United States

Dibyakanti Kumar

Indian Institute of Technology Guwahati ( email )

Anirbit Mukherjee (Contact Author)

Wharton (UPenn), Department of Statistics ( email )

Wharton School
Philadelphia, PA 19104
United States

HOME PAGE: http://https://sites.google.com/view/anirbit/home

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