Operator learning in the overparameterized regime

Neural Operators that directly learn mappings between function spaces have received considerable recent attention. Deep Operator Networks (DeepONets), a popular recent class of operator networks have shown promising preliminary results in approximating solution operators of parametric partial differential equations. Despite the universal approximation guarantees there is yet no optimization convergence guarantee for DeepONets based on gradient descent (GD). In this thesis, we establish such guarantees and show that overparameterization based on wide layers provably helps. In particular, we present two types of optimization convergence analysis: first, for smooth activations, we bound the spectral norm of the Hessian of DeepONets and use the bound to show geometric convergence of GD based on restricted strong convexity (RSC); and second, for ReLU activations, we show the neural tangent kernel (NTK) of DeepONets at initialization is positive definite, which can be used with the standard NTK analysis to imply geometric convergence. Further, we present empirical results on three canonical operator learning problems: Antiderivative, DiffusionReaction equation, and Burger’s equation, and show that wider DeepONets lead to lower training loss on all the problems, thereby supporting the theoretical results

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Computer Science

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2023

Author dc:creator

• Shrimali, Bhavesh

Contributors dc:contributor

• Banerjee, Arindam

• Overparameterization
• Optimization
• Deep Operator Networks

Statement dc:rights

• Copyright 2023 Bhavesh Shrimali

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/120438

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F120438