# DimINO: Dimension-Informed Neural Operator Learning

arXiv: https://arxiv.org/abs/2410.05894 · HTML: https://mlsanity.com/paper/2410.05894/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 25 references that are in the map.
- First posted on arXiv 8 Oct 2024.

## Builds on (most cited first, 13 of 13)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Layer Normalization](https://mlsanity.com/paper/1607.06450/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)

## Abstract (arXiv)

In computational physics, a longstanding challenge lies in finding numerical solutions to partial differential equations (PDEs). Recently, research attention has increasingly focused on Neural Operator methods, which are notable for their ability to approximate operators-mappings between functions. Although neural operators benefit from a universal approximation theorem, achieving reliable error bounds often necessitates large model architectures, such as deep stacks of Fourier layers. This raises a natural question: Can we design lightweight models without sacrificing generalization? To address this, we introduce DimINO (Dimension-Informed Neural Operators), a framework inspired by dimensional analysis. DimINO incorporates two key components, DimNorm and a redimensionalization operation, which can be seamlessly integrated into existing neural operator architectures. These components enhance the model's ability to generalize across datasets with varying physical parameters. Theoretically, we establish a universal approximation theorem for DimINO and prove that it satisfies a critical property we term Similar Transformation Invariance (STI). Empirically, DimINO achieves up to 76.3% performance gain on PDE datasets while exhibiting clear evidence of the STI property.