NOMAD: Nonlinear Manifold Decoders for Operator Learning

TL;DR

NOMAD introduces a nonlinear decoder framework for operator learning in function spaces, enabling efficient low-dimensional representations of complex solution manifolds, outperforming linear models and competing with state-of-the-art methods.

Contribution

The paper presents NOMAD, a novel nonlinear decoder approach that effectively captures nonlinear submanifolds in function spaces for operator learning.

Findings

Accurately learns low-dimensional solution manifolds for PDEs.

Outperforms larger linear models in accuracy and efficiency.

Achieves competitive results on fluid dynamics benchmarks with smaller models.

Abstract

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.