NOMAD: Nonlinear Manifold Decoders for Operator Learning

06/07/2022
by   Jacob H. Seidman, et al.
109

Supervised learning in function spaces is an emerging area of machine learning research with applications to the prediction of complex physical systems such as fluid flows, solid mechanics, and climate modeling. By directly learning maps (operators) between infinite dimensional function spaces, these models are able to learn discretization invariant representations of target functions. A common approach is to represent such target functions as linear combinations of basis elements learned from data. However, there are simple scenarios where, even though the target functions form a low dimensional submanifold, a very large number of basis elements is needed for an accurate linear representation. Here we present NOMAD, a novel operator learning framework with a nonlinear decoder map capable of learning finite dimensional representations of nonlinear submanifolds in function spaces. We show this method is able to accurately learn low dimensional representations of solution manifolds to partial differential equations while outperforming linear models of larger size. Additionally, we compare to state-of-the-art operator learning methods on a complex fluid dynamics benchmark and achieve competitive performance with a significantly smaller model size and training cost.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
04/15/2023

Learning in latent spaces improves the predictive accuracy of deep neural operators

Operator regression provides a powerful means of constructing discretiza...
research
07/11/2022

Neural and gpc operator surrogates: construction and expression rate bounds

Approximation rates are analyzed for deep surrogates of maps between inf...
research
02/01/2023

Learning Functional Transduction

Research in Machine Learning has polarized into two general regression a...
research
01/04/2022

Learning Operators with Coupled Attention

Supervised operator learning is an emerging machine learning paradigm wi...
research
02/20/2023

Variational Autoencoding Neural Operators

Unsupervised learning with functional data is an emerging paradigm of ma...
research
05/07/2019

Variational training of neural network approximations of solution maps for physical models

A novel solve-training framework is proposed to train neural network in ...
research
12/08/2021

KoopmanizingFlows: Diffeomorphically Learning Stable Koopman Operators

We propose a novel framework for constructing linear time-invariant (LTI...

Please sign up or login with your details

Forgot password? Click here to reset