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Variable-Input Deep Operator Networks

Michael Prasthofer, Tim De Ryck, Siddhartha Mishra

TL;DR

This work introduces Variable-Input Deep Operator Networks (VIDON), a neural operator framework that accepts variable and random sensor inputs while preserving permutation invariance. The authors prove a universal approximation theorem for VIDON and establish polynomial scaling guarantees for PDE operators, with detailed results for Darcy flow, Allen-Cahn, and Navier-Stokes equations. Empirical studies show VIDON robustly learns operators across diverse sensor configurations, often outperforming or matching traditional DeepONet and FNO baselines when sensor inputs vary. The approach enables learning operators from heterogeneous measurements and simulations, broadening applicability to real world data with irregular sensor deployments.

Abstract

Existing architectures for operator learning require that the number and locations of sensors (where the input functions are evaluated) remain the same across all training and test samples, significantly restricting the range of their applicability. We address this issue by proposing a novel operator learning framework, termed Variable-Input Deep Operator Network (VIDON), which allows for random sensors whose number and locations can vary across samples. VIDON is invariant to permutations of sensor locations and is proved to be universal in approximating a class of continuous operators. We also prove that VIDON can efficiently approximate operators arising in PDEs. Numerical experiments with a diverse set of PDEs are presented to illustrate the robust performance of VIDON in learning operators.

Variable-Input Deep Operator Networks

TL;DR

This work introduces Variable-Input Deep Operator Networks (VIDON), a neural operator framework that accepts variable and random sensor inputs while preserving permutation invariance. The authors prove a universal approximation theorem for VIDON and establish polynomial scaling guarantees for PDE operators, with detailed results for Darcy flow, Allen-Cahn, and Navier-Stokes equations. Empirical studies show VIDON robustly learns operators across diverse sensor configurations, often outperforming or matching traditional DeepONet and FNO baselines when sensor inputs vary. The approach enables learning operators from heterogeneous measurements and simulations, broadening applicability to real world data with irregular sensor deployments.

Abstract

Existing architectures for operator learning require that the number and locations of sensors (where the input functions are evaluated) remain the same across all training and test samples, significantly restricting the range of their applicability. We address this issue by proposing a novel operator learning framework, termed Variable-Input Deep Operator Network (VIDON), which allows for random sensors whose number and locations can vary across samples. VIDON is invariant to permutations of sensor locations and is proved to be universal in approximating a class of continuous operators. We also prove that VIDON can efficiently approximate operators arising in PDEs. Numerical experiments with a diverse set of PDEs are presented to illustrate the robust performance of VIDON in learning operators.
Paper Structure (38 sections, 11 theorems, 81 equations, 5 figures, 12 tables)

This paper contains 38 sections, 11 theorems, 81 equations, 5 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Variable-Input Deep Operator Networks
  3. Setting.
  4. DeepONets.
  5. Functions on sets.
  6. Input encoding.
  7. Head(s).
  8. Variable-Input Deep Operator Network.
  9. Related work.
  10. Rigorous analysis of Variable-Input Deep Operator Networks
  11. Universal approximation theorem.
  12. Computational complexity of VIDON.
  13. Darcy-type elliptic PDE.
  14. Nonlinear parabolic PDE: Allen-Cahn equation.
  15. Navier-Stokes equations.
  16. ...and 23 more sections

Key Result

Theorem 3.1

Let $\mathcal{G}:\mathcal{X}\to H^s(U)$ be an $\alpha$-Hölder continuous operator, let $\mu$ be a measure on $L^2(D)$ whose covariance operator has a bounded eigenbasis with eigenvectors $\{\lambda_j\}_{j\in\mathbb{N}}$ and let $\mathcal{E}:\mathcal{X} \to (\mathbb{R}^{d+d_v})^{\leq M}$ for $M\in\ma where the constant $C>0$ only depends on $\mathcal{G}$ and $\mu$.

Figures (5)

  • Figure 1: Learning the operator mapping the permeability coefficient $a$ (Input) to the solution $u$ (Target) for Darcy flow \ref{['eq:darcy']}, for two different realizations (samples) of the input. The input has to be evaluated at sensor points. FNO requires a Cartesian grid of sensors for each sample whereas as DeepONets allow for a random cloud of sensors. However, the number and location of sensors has to be the same for all training and test samples. In contrast, VIDON \ref{['eq:tb-deeponet']} allows for random sensor points, whose location and number can vary for each sample.
  • Figure 2: Structure of VIDON \ref{['eq:tb-deeponet']}. The coordinates and values at each sensor are encoded through MLPs and are processed through another set of MLPs to compute weights and values. The resulting convex combination of the outputs from different sensors constitutes the output of a single head. Multiple heads are concatenated and combined through another MLP to yield the branch net, which is then combined with the trunk net to obtain the VIDON output. Light blue shading represents a MLP.
  • Figure 3: Illustration of sensor locations for configuration where sensors are not on a grid. See Section \ref{['sec:4']} for the nomenclature. We consider two sample inputs for the Darcy flow test case. See columns 2 and 3 in Figure \ref{['fig:1']} for illustrations of regular and irregular grids respectively and plot the other sensor configurations here.
  • Figure 4: A sample illustrating the solution operator for the Allen-Cahn equation \ref{['eq:AC']}. The input is given by the initial conditions and the output is given by the time-history (up to time $T=0.05$) of the rotated travelling-wave solution \ref{['eq:AC_IC']} of the PDE.
  • Figure 5: A sample illustrating the operator for the Navier-Stokes equations \ref{['eq:NS']}. The input to the operator is given by the initial vorticity and the output is the vorticity at time $T=5$.

Theorems & Definitions (23)

  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • Theorem 3.4
  • Theorem 3.5
  • Lemma A.1: Continuous Sobolev embedding
  • Lemma A.2
  • proof
  • Lemma A.3
  • proof
  • ...and 13 more