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MIONet: Learning multiple-input operators via tensor product

Pengzhan Jin, Shuai Meng, Lu Lu

TL;DR

The paper extends neural operator learning to multi-input operators defined on the product of Banach spaces, proving universal approximation theorems and providing an error-guided architecture blueprint. It introduces MIONet, a low-rank, tensor-product-based operator network with separate input-branch nets and a trunk net, and contrasts it with high-rank variants and DeepONet. Theoretical results link multi-input operator approximation to efficient, modular architectures, and empirical tests on ODE and PDE operators show that MIONet outperforms corresponding DeepONet configurations and can incorporate prior structure such as linearity and periodicity. Overall, the work broadens the scope of neural operators to complex, multi-field PDE contexts with practical, scalable network designs.

Abstract

As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space, i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of continuous multiple-input operators. We also provide detailed theoretical analysis including the approximation error, which provides a guidance of the design of the network architecture. Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators. MIONet consists of several branch nets for encoding the input functions and a trunk net for encoding the domain of the output function. We demonstrate that MIONet can learn solution operators involving systems governed by ordinary and partial differential equations. In our computational examples, we also show that we can endow MIONet with prior knowledge of the underlying system, such as linearity and periodicity, to further improve the accuracy.

MIONet: Learning multiple-input operators via tensor product

TL;DR

The paper extends neural operator learning to multi-input operators defined on the product of Banach spaces, proving universal approximation theorems and providing an error-guided architecture blueprint. It introduces MIONet, a low-rank, tensor-product-based operator network with separate input-branch nets and a trunk net, and contrasts it with high-rank variants and DeepONet. Theoretical results link multi-input operator approximation to efficient, modular architectures, and empirical tests on ODE and PDE operators show that MIONet outperforms corresponding DeepONet configurations and can incorporate prior structure such as linearity and periodicity. Overall, the work broadens the scope of neural operators to complex, multi-field PDE contexts with practical, scalable network designs.

Abstract

As an emerging paradigm in scientific machine learning, neural operators aim to learn operators, via neural networks, that map between infinite-dimensional function spaces. Several neural operators have been recently developed. However, all the existing neural operators are only designed to learn operators defined on a single Banach space, i.e., the input of the operator is a single function. Here, for the first time, we study the operator regression via neural networks for multiple-input operators defined on the product of Banach spaces. We first prove a universal approximation theorem of continuous multiple-input operators. We also provide detailed theoretical analysis including the approximation error, which provides a guidance of the design of the network architecture. Based on our theory and a low-rank approximation, we propose a novel neural operator, MIONet, to learn multiple-input operators. MIONet consists of several branch nets for encoding the input functions and a trunk net for encoding the domain of the output function. We demonstrate that MIONet can learn solution operators involving systems governed by ordinary and partial differential equations. In our computational examples, we also show that we can endow MIONet with prior knowledge of the underlying system, such as linearity and periodicity, to further improve the accuracy.
Paper Structure (35 sections, 8 theorems, 74 equations, 4 figures, 3 tables)

This paper contains 35 sections, 8 theorems, 74 equations, 4 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Approximation theory
  3. Multilinear operators defined on finite-dimensional Banach spaces
  4. Schauder basis and canonical projections for infinite-dimensional spaces
  5. Main theorems: Approximation theory for multiple-input operators
  6. Detailed analysis
  7. View through the tensor product of Banach spaces
  8. Operator regression methods
  9. Network architectures
  10. MIONet (high-rank).
  11. MIONet (low-rank; the default version).
  12. Connections to DeepONet.
  13. Remark.
  14. Other computational details
  15. Data.
  16. ...and 20 more sections

Key Result

Theorem 1

Suppose that $X_1,\cdots,X_n,Y$ are Banach spaces, $K_i\subset X_i$ are compact sets, and $X_i$ have a Schauder basis with canonical projections $P_q^i=\psi_q^i\circ\varphi_q^i$. Assume that $\mathcal{G}:K_1\times\cdots\times K_n\to Y$ is a continuous operator, then for any $\epsilon>0$, there exist

Figures (4)

  • Figure 1: Architecture of MIONet. All the branch nets and the trunk net have the same number of outputs, which are merged together via the Hadamard product and then a summation.
  • Figure 2: Example of the diffusion-reaction system. (Top) Examples of the input functions (left) and the reference solution (right). (Middle) MIONet prediction and corresponding absolute error. (Bottom) DeepONet prediction and corresponding absolute error.
  • Figure 3: Architecture of the modified MIONet for the advection-diffusion system. There are two trunk nets, one for $x$ and one for $t$. The trunk net of $x$ has a periodic layer.
  • Figure 4: Prediction of MIONet (periodic) for the advection-diffusion system.

Theorems & Definitions (13)

  • Definition 1: Schauder basis
  • Example 1
  • Example 2
  • Theorem 1
  • Corollary 1
  • Example 3
  • Example 4: DeepONet
  • Lemma 1
  • Theorem 2
  • Corollary 2: Effect of a bias
  • ...and 3 more