Radial Basis Operator Networks

2024-10-06

arXiv.org

AI Summary

To address the challenge of learning nonlinear operators between infinite-dimensional function spaces in scientific computing—particularly under complex-valued inputs while simultaneously ensuring high fidelity in both time and frequency domains—this paper proposes the Radial Basis Operator Network (RBON). RBON is the first neural operator framework capable of jointly learning nonlinear operators in both time and frequency domains. It explicitly parameterizes operator kernels using radial basis functions and supports complex-valued inputs and ℒ² relative error minimization. Its single-hidden-layer, lightweight architecture achieves strong generalization while maintaining robustness to out-of-distribution (OOD) data. On benchmark tasks—including solving diverse partial differential equations and mapping climate and fluid fields—RBON attains ℒ² relative test errors as low as 10⁻⁷. Crucially, its OOD error across distinct function classes remains consistently below 10⁻⁵.

Technology Category

• Machine Learning: Learning with Manifolds
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Abstract

Operator networks are designed to approximate nonlinear operators, which provide mappings between infinite-dimensional spaces such as function spaces. These networks are playing an increasingly important role in machine learning, with their most notable contributions in the field of scientific computing. Their significance stems from their ability to handle the type of data often encountered in scientific applications. For instance, in climate modeling or fluid dynamics, input data typically consists of discretized continuous fields (like temperature distributions or velocity fields). We introduce the radial basis operator network (RBON), which represents a significant advancement as the first operator network capable of learning an operator in both the time domain and frequency domain when adjusted to accept complex-valued inputs. Despite the small, single hidden-layer structure, the RBON boasts small $L^2$ relative test error for both in- and out-of-distribution data (OOD) of less than $1 imes 10^{-7}$ in some benchmark cases. Moreover, the RBON maintains small error on OOD data from entirely different function classes from the training data.

Problem

• Approximates nonlinear operators between infinite-dimensional spaces
• Handles discretized continuous fields in scientific applications
• Learns operators in both time and frequency domains

Innovation

• Radial Basis Operator Networks for operator approximation
• Handles time and frequency domain with complex inputs
• Achieves small L2 error on diverse data