mlsanityData to 7 Oct 2026

Learning Only On Boundaries: a Physics-Informed Neural operator for Solving Parametric Partial Differential Equations in Complex Geometries

arXiv 2308.12939 ↗First posted 24 Aug 2023cs.LG cs.NA math.NAcited by 12

Not on the map. Why?

Abstract (arXiv)

Recently deep learning surrogates and neural operators have shown promise in solving partial differential equations (PDEs). However, they often require a large amount of training data and are limited to bounded domains. In this work, we present a novel physics-informed neural operator method to solve parametrized boundary value problems without labeled data. By reformulating the PDEs into boundary integral equations (BIEs), we can train the operator network solely on the boundary of the domain. This approach reduces the number of required sample points from O(Nd) to O(Nd-1), where d is the domain's dimension, leading to a significant acceleration of the training process. Additionally, our method can handle unbounded problems, which are unattainable for existing physics-informed neural networks (PINNs) and neural operators. Our numerical experiments show the effectiveness of parametrized complex geometries and unbounded problems.

Key facts. Cited by 12 papers in the map (data through 7 Oct 2026). Builds on 17 of its 40 references that are in the map. First posted on arXiv 24 Aug 2023.

Builds on

17 of 40 references.

References · 17

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  2. 2020Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domainscited by 2,170
  3. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  4. 2020Learning to Simulate Complex Physics with Graph Networkscited by 867
  5. 2015Automatic differentiation in machine learning: a surveycited by 846
  6. 2021Characterizing possible failure modes in physics-informed neural networkscited by 632
  7. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  8. 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 261
  9. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  10. 2021Multiwavelet-based Operator Learning for Differential Equationscited by 229
  11. 2019PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEscited by 116
  12. 2020The Random Feature Model for Input-Output Maps between Banach Spacescited by 90
5 more
  1. 2022Learning Operators with Coupled Attentioncited by 88
  2. 2021Improved architectures and training algorithms for deep operator networkscited by 85
  3. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 81
  4. 2019Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networkscited by 76
  5. 2020NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation frameworkcited by 58

Built on by

Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 1; 2024: 3; 2025: 2; 2026: 6.

12 papers.

  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  2. 2024An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Studycited by 6
  3. 2024Separable Operator Networkscited by 4
  4. 2025HEATNETs: Explainable Random Feature Neural Networks for High-Dimensional Parabolic PDEscited by 1
  5. 2024Fredholm Integral Equations Neural Operator (FIE-NO) for Data-Driven Boundary Value Problemscited by 0
  6. 2025PIBNet: a Physics-Inspired Boundary Network for Multiple Scattering Simulationscited by 0
  7. 2026A Boundary Integral-based Neural Operator for Mesh Deformationcited by 0
  8. 2026UniPINN: A Unified PINN Framework for Multi-task Learning of Diverse Navier-Stokes Equationscited by 0
  9. 2026Virtual boundary integral neural network for three-dimensional exterior acoustic problemscited by 0
  10. 2026Learning the Helmholtz equation operator with DeepONet for non-parametric 2D geometriescited by 0
  11. 2026Quadrature-Aware Complex-Linear Neural Operator for Boundary-to-Field Prediction in Resonant Acousticscited by 0
  12. 2026ScaGNN: a Graph Neural Network for Multiple Scattering Simulationscited by 0