2 research outputs found
Representation Equivalent Neural Operators: a Framework for Alias-free Operator Learning
Recently, operator learning, or learning mappings between
infinite-dimensional function spaces, has garnered significant attention,
notably in relation to learning partial differential equations from data.
Conceptually clear when outlined on paper, neural operators necessitate
discretization in the transition to computer implementations. This step can
compromise their integrity, often causing them to deviate from the underlying
operators. This research offers a fresh take on neural operators with a
framework Representation equivalent Neural Operators (ReNO) designed to address
these issues. At its core is the concept of operator aliasing, which measures
inconsistency between neural operators and their discrete representations. We
explore this for widely-used operator learning techniques. Our findings detail
how aliasing introduces errors when handling different discretizations and
grids and loss of crucial continuous structures. More generally, this framework
not only sheds light on existing challenges but, given its constructive and
broad nature, also potentially offers tools for developing new neural
operators.Comment: 28 page
Convolutional Neural Operators for robust and accurate learning of PDEs
Although very successfully used in conventional machine learning, convolution
based neural network architectures -- believed to be inconsistent in function
space -- have been largely ignored in the context of learning solution
operators of PDEs. Here, we present novel adaptations for convolutional neural
networks to demonstrate that they are indeed able to process functions as
inputs and outputs. The resulting architecture, termed as convolutional neural
operators (CNOs), is designed specifically to preserve its underlying
continuous nature, even when implemented in a discretized form on a computer.
We prove a universality theorem to show that CNOs can approximate operators
arising in PDEs to desired accuracy. CNOs are tested on a novel suite of
benchmarks, encompassing a diverse set of PDEs with possibly multi-scale
solutions and are observed to significantly outperform baselines, paving the
way for an alternative framework for robust and accurate operator learning