# INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation

arXiv: https://arxiv.org/abs/2212.14365 · HTML: https://mlsanity.com/paper/2212.14365/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 63 references that are in the map.
- First posted on arXiv 29 Dec 2022.

## Builds on (most cited first, 30 of 31)

- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Spectral Networks and Locally Connected Networks on Graphs](https://mlsanity.com/paper/1312.6203/) (2013)
- [E(n) Equivariant Graph Neural Networks](https://mlsanity.com/paper/2102.09844/) (2021)
- [Tensor field networks: Rotation- and translation-equivariant neural networks for 3D point clouds](https://mlsanity.com/paper/1802.08219/) (2018)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [SE(3)-Transformers: 3D Roto-Translation Equivariant Attention Networks](https://mlsanity.com/paper/2006.10503/) (2020)
- [Directional Message Passing for Molecular Graphs](https://mlsanity.com/paper/2003.03123/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [General E(2)-Equivariant Steerable CNNs](https://mlsanity.com/paper/1911.08251/) (2019)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Message Passing Neural PDE Solvers](https://mlsanity.com/paper/2202.03376/) (2022)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Cormorant: Covariant Molecular Neural Networks](https://mlsanity.com/paper/1906.04015/) (2019)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Learning Deep Implicit Fourier Neural Operators (IFNOs) with Applications to Heterogeneous Material Modeling](https://mlsanity.com/paper/2203.08205/) (2022)
- [Group Equivariant Stand-Alone Self-Attention For Vision](https://mlsanity.com/paper/2010.00977/) (2020)
- [Relevance of Rotationally Equivariant Convolutions for Predicting Molecular Properties](https://mlsanity.com/paper/2008.08461/) (2020)
- [Interfacing Finite Elements with Deep Neural Operators for Fast Multiscale Modeling of Mechanics Problems](https://mlsanity.com/paper/2203.00003/) (2022)
- [A Wigner-Eckart Theorem for Group Equivariant Convolution Kernels](https://mlsanity.com/paper/2010.10952/) (2020)
- [Nonlocal Kernel Network (NKN): a Stable and Resolution-Independent Deep Neural Network](https://mlsanity.com/paper/2201.02217/) (2022)
- [Simulating progressive intramural damage leading to aortic dissection using an operator-regression neural network](https://mlsanity.com/paper/2108.11985/) (2021)
- [A Physics-Guided Neural Operator Learning Approach to Model Biological Tissues from Digital Image Correlation Measurements](https://mlsanity.com/paper/2204.00205/) (2022)
- [IAE-Net: Integral Autoencoders for Discretization-Invariant Learning](https://mlsanity.com/paper/2203.05142/) (2022)

## Built on by (most cited first, 28 of 28)

- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [Gradient-Free Generation for Hard-Constrained Systems](https://mlsanity.com/paper/2412.01786/) (2024)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery](https://mlsanity.com/paper/2408.07307/) (2024)
- [Accelerating PDE Data Generation via Differential Operator Action in Solution Space](https://mlsanity.com/paper/2402.05957/) (2024)
- [Using Uncertainty Quantification to Characterize and Improve Out-of-Domain Learning for PDEs](https://mlsanity.com/paper/2403.10642/) (2024)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [Accelerating Data Generation for Neural Operators via Krylov Subspace Recycling](https://mlsanity.com/paper/2401.09516/) (2024)
- [GNN-based physics solver for time-independent PDEs](https://mlsanity.com/paper/2303.15681/) (2023)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements](https://mlsanity.com/paper/2403.18597/) (2024)
- [Spectral Convolutional Conditional Neural Processes](https://mlsanity.com/paper/2404.13182/) (2024)
- [Disentangled Representation Learning for Parametric Partial Differential Equations](https://mlsanity.com/paper/2410.02136/) (2024)
- [Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions](https://mlsanity.com/paper/2505.01060/) (2025)
- [Large language models, physics-based modeling, experimental measurements: the trinity of data-scarce learning of polymer properties](https://mlsanity.com/paper/2407.02770/) (2024)
- [Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery](https://mlsanity.com/paper/2505.23106/) (2025)
- [Physics-embedded Fourier Neural Network for Partial Differential Equations](https://mlsanity.com/paper/2407.11158/) (2024)
- [Adaptive Correction for Ensuring Conservation Laws in Neural Operators](https://mlsanity.com/paper/2505.24579/) (2025)
- [Can Data-Driven Dynamics Reveal Hidden Physics? There Is A Need for Interpretable Neural Operators](https://mlsanity.com/paper/2510.02683/) (2025)
- [Accelerating Eigenvalue Dataset Generation via Chebyshev Subspace Filter](https://mlsanity.com/paper/2510.23215/) (2025)
- [Physically consistent model learning for reaction-diffusion systems](https://mlsanity.com/paper/2512.14240/) (2025)
- [Embedded Nonlocal Operator Regression (ENOR): Quantifying model error in learning nonlocal operators](https://mlsanity.com/paper/2410.20331/) (2024)
- [A Message Passing Neural Network Surrogate Model for Bond-Associated Peridynamic Material Correspondence Formulation](https://mlsanity.com/paper/2411.08911/) (2024)
- [Accelerating IC Thermal Simulation Data Generation via Block Krylov and Operator Action](https://mlsanity.com/paper/2510.23221/) (2025)
- [Prediction of Steady-State Flow through Porous Media Using Machine Learning Models](https://mlsanity.com/paper/2603.06762/) (2026)
- [Multi-Head Residual-Gated DeepONet for Coherent Nonlinear Wave Dynamics](https://mlsanity.com/paper/2604.11972/) (2026)
- [Physics-Integrated Operator Learning via Gaussian Splatting Representations](https://mlsanity.com/paper/2608.24049/) (2026)

## Abstract (arXiv)

Neural operators, which emerge as implicit solution operators of hidden governing equations, have recently become popular tools for learning responses of complex real-world physical systems. Nevertheless, the majority of neural operator applications has thus far been data-driven, which neglects the intrinsic preservation of fundamental physical laws in data. In this paper, we introduce a novel integral neural operator architecture, to learn physical models with fundamental conservation laws automatically guaranteed. In particular, by replacing the frame-dependent position information with its invariant counterpart in the kernel space, the proposed neural operator is by design translation- and rotation-invariant, and consequently abides by the conservation laws of linear and angular momentums. As applications, we demonstrate the expressivity and efficacy of our model in learning complex material behaviors from both synthetic and experimental datasets, and show that, by automatically satisfying these essential physical laws, our learned neural operator is not only generalizable in handling translated and rotated datasets, but also achieves state-of-the-art accuracy and efficiency as compared to baseline neural operator models.