# Guaranteed Approximation Bounds for Mixed-Precision Neural Operators

arXiv: https://arxiv.org/abs/2307.15034 · HTML: https://mlsanity.com/paper/2307.15034/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 43 references that are in the map.
- First posted on arXiv 27 Jul 2023.

## Builds on (most cited first, 25 of 25)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [ShapeNet: An Information-Rich 3D Model Repository](https://mlsanity.com/paper/1512.03012/) (2015)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Mixed Precision Training](https://mlsanity.com/paper/1710.03740/) (2017)
- [FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2202.11214/) (2022)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [A Study of BFLOAT16 for Deep Learning Training](https://mlsanity.com/paper/1905.12322/) (2019)
- [FP8 Formats for Deep Learning](https://mlsanity.com/paper/2209.05433/) (2022)
- [Solving ill-posed inverse problems using iterative deep neural networks](https://mlsanity.com/paper/1704.04058/) (2017)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [TensorLy: Tensor Learning in Python](https://mlsanity.com/paper/1610.09555/) (2016)
- [Highly Scalable Deep Learning Training System with Mixed-Precision: Training ImageNet in Four Minutes](https://mlsanity.com/paper/1807.11205/) (2018)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [A Survey of Methods for Low-Power Deep Learning and Computer Vision](https://mlsanity.com/paper/2003.11066/) (2020)
- [High-Accuracy Low-Precision Training](https://mlsanity.com/paper/1803.03383/) (2018)
- [Forecasting subcritical cylinder wakes with Fourier Neural Operators](https://mlsanity.com/paper/2301.08290/) (2023)
- [LNS-Madam: Low-Precision Training in Logarithmic Number System using Multiplicative Weight Update](https://mlsanity.com/paper/2106.13914/) (2021)
- [Mixed-Precision Neural Networks: A Survey](https://mlsanity.com/paper/2208.06064/) (2022)

## Built on by (most cited first, 9 of 9)

- [A Library for Learning Neural Operators](https://mlsanity.com/paper/2412.10354/) (2024)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Turbulence in Focus: Benchmarking Scaling Behavior of 3D Volumetric Super-Resolution with BLASTNet 2.0 Data](https://mlsanity.com/paper/2309.13457/) (2023)
- [Designing Large Foundation Models for Efficient Training and Inference: A Survey](https://mlsanity.com/paper/2409.01990/) (2024)
- [TensorGRaD: Tensor Gradient Robust Decomposition for Memory-Efficient Neural Operator Training](https://mlsanity.com/paper/2501.02379/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Tensor-Compressed and Fully-Quantized Training of Neural PDE Solvers](https://mlsanity.com/paper/2512.09202/) (2025)
- [Operator Boosting Produces Pareto-Efficient PDE Surrogates](https://mlsanity.com/paper/2606.17460/) (2026)
- [An Operator Learning Framework for Spatiotemporal Super-resolution of Scientific Simulations](https://mlsanity.com/paper/2311.02328/) (2023)

## Abstract (arXiv)

Neural operators, such as Fourier Neural Operators (FNO), form a principled approach for learning solution operators for PDEs and other mappings between function spaces. However, many real-world problems require high-resolution training data, and the training time and limited GPU memory pose big barriers. One solution is to train neural operators in mixed precision to reduce the memory requirement and increase training speed. However, existing mixed-precision training techniques are designed for standard neural networks, and we find that their direct application to FNO leads to numerical overflow and poor memory efficiency. Further, at first glance, it may appear that mixed precision in FNO will lead to drastic accuracy degradation since reducing the precision of the Fourier transform yields poor results in classical numerical solvers. We show that this is not the case; in fact, we prove that reducing the precision in FNO still guarantees a good approximation bound, when done in a targeted manner. Specifically, we build on the intuition that neural operator learning inherently induces an approximation error, arising from discretizing the infinite-dimensional ground-truth input function, implying that training in full precision is not needed. We formalize this intuition by rigorously characterizing the approximation and precision errors of FNO and bounding these errors for general input functions. We prove that the precision error is asymptotically comparable to the approximation error. Based on this, we design a simple method to optimize the memory-intensive half-precision tensor contractions by greedily finding the optimal contraction order. Through extensive experiments on different state-of-the-art neural operators, datasets, and GPUs, we demonstrate that our approach reduces GPU memory usage by up to 50% and improves throughput by 58% with little or no reduction in accuracy.