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Neural Operator: Graph Kernel Network for Partial Differential Equations (2003.03485v1)

Published 7 Mar 2020 in cs.LG, cs.NA, math.NA, and stat.ML

Abstract: The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces. We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. The kernel integration is computed by message passing on graph networks. This approach has substantial practical consequences which we will illustrate in the context of mappings between input data to partial differential equations (PDEs) and their solutions. In this context, such learned networks can generalize among different approximation methods for the PDE (such as finite difference or finite element methods) and among approximations corresponding to different underlying levels of resolution and discretization. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.

Citations (623)

Summary

  • The paper introduces a neural operator framework that extends traditional neural networks to map infinite-dimensional spaces.
  • It employs graph kernel networks and message passing to integrate nonlinear activations and achieve mesh resolution invariance.
  • The approach demonstrates data efficiency and robust, transferable solutions for solving diverse partial differential equations.

Neural Operator: Graph Kernel Network for Partial Differential Equations

The paper introduces a novel approach for learning mappings between infinite-dimensional spaces using neural networks, with a particular focus on applications to partial differential equations (PDEs). This work extends classical neural network architectures, typically used between finite-dimensional spaces, to operate between infinite-dimensional spaces, termed as "neural operators."

Key Concepts and Approach

The central innovation presented is neural operator methodology, which utilizes graph kernel networks to approximate mappings between spaces of functions. These neural operators can generalize across different discretizations and mesh resolutions, maintaining consistent performance. The approach leverages graph networks to compute kernel integrations via message passing, linking this process to Nyström approximation for kernel functions.

Neural Network Framework

  1. Neural Architecture: The proposed network comprises nonlinear activation functions and integral operators, adapting traditional neural network ideas for mapping between infinite-dimensional function spaces. The kernel is integrated using a graph-based approach.
  2. PDE Application: The method is demonstrated within the context of PDEs, providing solutions that can generalize across different numerical approximation methods such as finite difference or finite element methods, by learning discretization-independent mappings.
  3. Graph Neural Networks (GNNs): These act on graph-structured data and are adapted here to learn non-local solution operators of PDEs, providing an innovative way to handle the integral operations required by the neural operator framework.

Experimental Outcomes

The experiments highlight several advantages:

  • Resolution Invariance: The method's error remains stable across different mesh resolutions when trained on a specific grid size.
  • Data Efficiency: A small number of training samples is sufficient to achieve competitive results, indicating the network's effective use of data.
  • Consistent Performance: The network shows robust performance compared to existing methods, such as fully convolutional networks and reduced basis methods, particularly on larger discretizations where traditional methods struggle with mesh dependencies.

Implications and Future Directions

This work has significant implications for the future of neural networks in scientific computing:

  • General Purpose Solvers: The neural operator could serve as a foundation for developing general purpose, mesh-independent solvers for PDEs and other operator learning tasks.
  • Transferability: The methodology allows for transferring solutions between different meshes, which is a crucial advantage in practical engineering applications involving complex geometries.
  • Potential Extensions: Further exploration into multi-grid approaches and time-dependent PDEs could enhance the scalability and applicability of this method.

The paper presents a promising direction for developing machine learning systems capable of efficiently learning mappings in complex domains, opening avenues for advancements in areas requiring high-dimensional and computationally intensive modeling. The neural operator provides a framework not only for solving existing numerical challenges but also as a valuable tool for future AI-driven scientific discovery.

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