mlsanityData to 7 Oct 2026

Variationally Mimetic Operator Networks

arXiv 2209.12871 ↗First posted 26 Sep 2022math.NA cs.LG cs.NAcited by 19

Not on the map. Why?

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.

Key facts. Cited by 19 papers in the map (data through 7 Oct 2026). Builds on 22 of its 52 references that are in the map. First posted on arXiv 26 Sep 2022.

Builds on

22 of 52 references.

References · 22

  1. 2014Adam: A Method for Stochastic Optimizationcited by 36,960
  2. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  5. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  6. 2022Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's nextcited by 603
  7. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  9. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  10. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  11. 2020B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Datacited by 329
  12. 2022MIONet: Learning multiple-input operators via tensor productcited by 180
10 more
  1. 2018Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problemscited by 144
  2. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEscited by 115
  3. 2021Parallel Physics-Informed Neural Networks via Domain Decompositioncited by 112
  4. 2020Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEscited by 101
  5. 2021A physics-informed variational DeepONet for predicting the crack path in brittle materialscited by 98
  6. 2022Scalable Uncertainty Quantification for Deep Operator Networks using Randomized Priorscited by 39
  7. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  8. 2022Variable-Input Deep Operator Networkscited by 24
  9. 2022Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functionscited by 11
  10. 2015A Bayesian Level Set Method for Geometric Inverse Problemscited by 4

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 5; 2024: 10; 2025: 3; 2026: 1.

19 papers.

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  1. 2024Operator Learning: Algorithms and Analysiscited by 84
  2. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  3. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  4. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  5. 2023Neural Dynamical Operator: Continuous Spatial-Temporal Model with Gradient-Based and Derivative-Free Optimization Methodscited by 14
  6. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  7. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  8. 2024Operator Learning of Lipschitz Operators: An Information-Theoretic Perspectivecited by 9
  9. 2023Solution of physics-based inverse problems using conditional generative adversarial networks with full gradient penaltycited by 7
  10. 2023Deep Learning and Computational Physics (Lecture Notes)cited by 4
  11. 2024First Order System Least Squares Neural Networkscited by 4
  12. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
7 more
  1. 2025A Neural Operator Emulator for Coastal and Riverine Shallow Water Dynamicscited by 2
  2. 2024Learning WENO for entropy stable schemes to solve conservation lawscited by 1
  3. 2024Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operatorscited by 1
  4. 2024Learning Nonlinear Finite Element Solution Operators using Multilayer Perceptrons and Energy Minimizationcited by 1
  5. 2024Multi-fidelity Hamiltonian Monte Carlocited by 0
  6. 2025An optimal Petrov-Galerkin framework for operator networkscited by 0
  7. 2026HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systemscited by 0