Hybrid Least Squares/Gradient Descent Methods for DeepONets

TL;DR

This paper introduces a hybrid approach combining least squares and gradient descent to efficiently train DeepONets, addressing computational challenges by decomposing large linear systems into manageable subproblems, applicable to various loss functions.

Contribution

The paper presents a novel hybrid LS/gradient descent method for DeepONet training that decomposes large linear systems into smaller subproblems, improving efficiency and scalability.

Findings

Efficient training of DeepONets achieved through the hybrid method.

Method applicable to physics-informed and regularized loss functions.

Decomposition reduces computational complexity significantly.

Abstract

We propose an efficient hybrid least squares/gradient descent method to accelerate DeepONet training. Since the output of DeepONet can be viewed as linear with respect to the last layer parameters of the branch network, these parameters can be optimized using a least squares (LS) solve, and the remaining hidden layer parameters are updated by means of gradient descent form. However, building the LS system for all possible combinations of branch and trunk inputs yields a prohibitively large linear problem that is infeasible to solve directly. To address this issue, our method decomposes the large LS system into two smaller, more manageable subproblems $\unicode{x2014}$ one for the branch network and one for the trunk network $\unicode{x2014}$ and solves them separately. This method is generalized to a broader type of $L^2$ loss with a regularization term for the last layer parameters, including the case of unsupervised learning with physics-informed loss.