A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theory

TL;DR

This paper introduces new neural network architectures and theoretical results for learning collections of operators between function spaces, with applications to parametric PDEs, demonstrating strong approximation capabilities and efficiency.

Contribution

It proposes two architectures for multiple operator learning, establishes universal approximation results, and provides scaling laws and efficiency frameworks for learning multiple operators.

Findings

Proposed MNO and MONet architectures effectively approximate operators.

Universal approximation results hold for continuous, integrable, and Lipschitz operators.

Empirical results confirm the architectures' expressive power and efficiency.

Abstract

While many problems in machine learning focus on learning mappings between finite-dimensional spaces, scientific applications require approximating mappings between function spaces, i.e., operators. We study the problem of learning collections of operators and provide both theoretical and empirical advances. We distinguish between two regimes: (i) multiple operator learning, where a single network represents a continuum of operators parameterized by a parametric function, and (ii) learning several distinct single operators, where each operator is learned independently. For the multiple operator case, we introduce two new architectures, $\mathrm{MNO}$ and $\mathrm{MONet}$, and establish universal approximation results in three settings: continuous, integrable, or Lipschitz operators. For the latter, we further derive explicit scaling laws that quantify how the network size must grow to achieve a target approximation accuracy. For learning several single operators, we develop a framework for balancing architectural complexity across subnetworks and show how approximation order determines computational efficiency. Empirical experiments on parametric PDE benchmarks confirm the strong expressive power and efficiency of the proposed architectures. Overall, this work establishes a unified theoretical and practical foundation for scalable neural operator learning across multiple operators.