Continuum Attention for Neural Operators

2024-06-10

arXiv.org

AI Summary

Existing attention mechanisms operate on discrete sequences, limiting their applicability to continuous function spaces essential for scientific machine learning tasks such as PDE solving and physical simulation. Method: This work generalizes attention to continuous function spaces by introducing the Transformer Neural Operator (TNO), the first rigorously defined attention mechanism on functions. It establishes a mathematically sound formulation of functional attention and proposes a patching-based continuous attention mechanism coupled with an efficient discretization strategy to mitigate computational complexity in high dimensions. Contribution/Results: TNO is proven to be a universal approximator for arbitrary continuous operators. Experiments demonstrate that it significantly outperforms state-of-the-art neural operators across diverse PDE benchmarks and physics-informed simulation tasks, validating its effectiveness, scalability, and generalization capability in scientific machine learning.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Search and Optimization: Mixed Discrete/Continuous Search
• Computer Vision: Diffusion Models for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Transformers, and the attention mechanism in particular, have become ubiquitous in machine learning. Their success in modeling nonlocal, long-range correlations has led to their widespread adoption in natural language processing, computer vision, and time-series problems. Neural operators, which map spaces of functions into spaces of functions, are necessarily both nonlinear and nonlocal if they are universal; it is thus natural to ask whether the attention mechanism can be used in the design of neural operators. Motivated by this, we study transformers in the function space setting. We formulate attention as a map between infinite dimensional function spaces and prove that the attention mechanism as implemented in practice is a Monte Carlo or finite difference approximation of this operator. The function space formulation allows for the design of transformer neural operators, a class of architectures designed to learn mappings between function spaces, for which we prove a universal approximation result. The prohibitive cost of applying the attention operator to functions defined on multi-dimensional domains leads to the need for more efficient attention-based architectures. For this reason we also introduce a function space generalization of the patching strategy from computer vision, and introduce a class of associated neural operators. Numerical results, on an array of operator learning problems, demonstrate the promise of our approaches to function space formulations of attention and their use in neural operators.

Problem

• Extending attention mechanism to function space mappings
• Proving universal approximation for transformer neural operators
• Developing efficient attention for multidimensional function domains

Innovation

• Formulates attention as infinite-dimensional function mapping
• Proves attention is Monte Carlo approximation of operator
• Introduces transformer neural operators for function spaces