# Variationally Mimetic Operator Networks

arXiv: https://arxiv.org/abs/2209.12871 · HTML: https://mlsanity.com/paper/2209.12871/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 52 references that are in the map.
- First posted on arXiv 26 Sep 2022.

## Builds on (most cited first, 22 of 22)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priors](https://mlsanity.com/paper/2203.03048/) (2022)
- [wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation laws](https://mlsanity.com/paper/2207.08483/) (2022)
- [Variable-Input Deep Operator Networks](https://mlsanity.com/paper/2205.11404/) (2022)
- [Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functions](https://mlsanity.com/paper/2202.08942/) (2022)
- [A Bayesian Level Set Method for Geometric Inverse Problems](https://mlsanity.com/paper/1504.00313/) (2015)

## Built on by (most cited first, 19 of 19)

- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Operator learning with PCA-Net: upper and lower complexity bounds](https://mlsanity.com/paper/2303.16317/) (2023)
- [Nonlocality and Nonlinearity Implies Universality in Operator Learning](https://mlsanity.com/paper/2304.13221/) (2023)
- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)
- [Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methods](https://mlsanity.com/paper/2311.11798/) (2023)
- [MODNO: Multi Operator Learning With Distributed Neural Operators](https://mlsanity.com/paper/2404.02892/) (2024)
- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [Operator Learning of Lipschitz Operators: An Information-Theoretic Perspective](https://mlsanity.com/paper/2406.18794/) (2024)
- [Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penalty](https://mlsanity.com/paper/2306.04895/) (2023)
- [First Order System Least Squares Neural Networks](https://mlsanity.com/paper/2409.20264/) (2024)
- [Deep Learning and Computational Physics (Lecture Notes)](https://mlsanity.com/paper/2301.00942/) (2023)
- [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025)
- [A Neural Operator Emulator for Coastal and Riverine Shallow Water Dynamics](https://mlsanity.com/paper/2502.14782/) (2025)
- [Learning WENO for entropy stable schemes to solve conservation laws](https://mlsanity.com/paper/2403.14848/) (2024)
- [Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators](https://mlsanity.com/paper/2412.04409/) (2024)
- [Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimization](https://mlsanity.com/paper/2412.04596/) (2024)
- [Multi-fidelity Hamiltonian Monte Carlo](https://mlsanity.com/paper/2405.05033/) (2024)
- [An optimal Petrov-Galerkin framework for operator networks](https://mlsanity.com/paper/2503.04024/) (2025)
- [HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems](https://mlsanity.com/paper/2606.11963/) (2026)

## Abstract (arXiv)

In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. This work describes a new architecture for operator networks that mimics the form of the numerical solution obtained from an approximate variational or weak formulation of the problem. The application of these ideas to a generic elliptic PDE leads to a variationally mimetic operator network (VarMiON). Like the conventional Deep Operator Network (DeepONet) the VarMiON is also composed of a sub-network that constructs the basis functions for the output and another that constructs the coefficients for these basis functions. However, in contrast to the DeepONet, the architecture of these sub-networks in the VarMiON is precisely determined. An analysis of the error in the VarMiON solution reveals that it contains contributions from the error in the training data, the training error, the quadrature error in sampling input and output functions, and a "covering error" that measures the distance between the test input functions and the nearest functions in the training dataset. It also depends on the stability constants for the exact solution operator and its VarMiON approximation. The application of the VarMiON to a canonical elliptic PDE and a nonlinear PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet and a recently proposed multiple-input operator network (MIONet). Further, its performance is more robust to variations in input functions, the techniques used to sample the input and output functions, the techniques used to construct the basis functions, and the number of input functions.