# HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork

arXiv: https://arxiv.org/abs/2312.15949 · HTML: https://mlsanity.com/paper/2312.15949/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 31 of its 47 references that are in the map.
- First posted on arXiv 26 Dec 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [Continual learning with hypernetworks](https://mlsanity.com/paper/1906.00695/) (2019)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [Prediction of Aerodynamic Flow Fields Using Convolutional Neural Networks](https://mlsanity.com/paper/1905.13166/) (2019)
- [Modulated Periodic Activations for Generalizable Local Functional Representations](https://mlsanity.com/paper/2104.03960/) (2021)
- [Approximating Continuous Functions by ReLU Nets of Minimal Width](https://mlsanity.com/paper/1710.11278/) (2017)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Machine Learning at the Network Edge: A Survey](https://mlsanity.com/paper/1908.00080/) (2019)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Hypernetwork functional image representation](https://mlsanity.com/paper/1902.10404/) (2019)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Solving PDE-constrained Control Problems Using Operator Learning](https://mlsanity.com/paper/2111.04941/) (2021)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [Neural Implicit Flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data](https://mlsanity.com/paper/2204.03216/) (2022)
- [SVD Perspectives for Augmenting DeepONet Flexibility and Interpretability](https://mlsanity.com/paper/2204.12670/) (2022)
- [Implicit Weight Uncertainty in Neural Networks](https://mlsanity.com/paper/1711.01297/) (2017)

## Built on by (most cited first, 26 of 26)

- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [tLaSDI: Thermodynamics-informed latent space dynamics identification](https://mlsanity.com/paper/2403.05848/) (2024)
- [Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learning](https://mlsanity.com/paper/2405.09285/) (2024)
- [HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions](https://mlsanity.com/paper/2509.05117/) (2025)
- [Learning time-dependent PDE via graph neural networks and deep operator network for robust accuracy on irregular grids](https://mlsanity.com/paper/2402.08187/) (2024)
- [An Advanced Physics-Informed Neural Operator for Comprehensive Design Optimization of Highly-Nonlinear Systems: An Aerospace Composites Processing Case Study](https://mlsanity.com/paper/2406.14715/) (2024)
- [Separable Operator Networks](https://mlsanity.com/paper/2407.11253/) (2024)
- [Neural Operators Can Play Dynamic Stackelberg Games](https://mlsanity.com/paper/2411.09644/) (2024)
- [PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems](https://mlsanity.com/paper/2505.01736/) (2025)
- [FD-Bench: A Modular and Fair Benchmark for Data-driven Fluid Simulation](https://mlsanity.com/paper/2505.20349/) (2025)
- [Error analysis for finite element operator learning methods for solving parametric second-order elliptic PDEs](https://mlsanity.com/paper/2404.17868/) (2024)
- [Disentangled Representation Learning for Parametric Partial Differential Equations](https://mlsanity.com/paper/2410.02136/) (2024)
- [Neural Operators Learn the Local Physics of Magnetohydrodynamics](https://mlsanity.com/paper/2404.16015/) (2024)
- [PinnDE: Physics-Informed Neural Networks for Solving Differential Equations](https://mlsanity.com/paper/2408.10011/) (2024)
- [Estimation of System Parameters Including Repeated Cross-Sectional Data through Emulator-Informed Deep Generative Model](https://mlsanity.com/paper/2412.19517/) (2024)
- [Spectral-inspired Operator Learning with Limited Data and Unknown Physics](https://mlsanity.com/paper/2505.21573/) (2025)
- [Estimating the Distribution of Parameters in Differential Equations with Repeated Cross-Sectional Data](https://mlsanity.com/paper/2404.14873/) (2024)
- [FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition](https://mlsanity.com/paper/2409.09207/) (2024)
- [Learning from Imperfect Data: Robust Inference of Dynamic Systems using Simulation-based Generative Model](https://mlsanity.com/paper/2507.10884/) (2025)
- [Low-rank adaptive physics-informed HyperDeepONets for solving differential equations](https://mlsanity.com/paper/2507.18346/) (2025)
- [Sparse FEONet: A Low-Cost, Memory-Efficient Operator Network via Finite-Element Local Sparsity for Parametric PDEs](https://mlsanity.com/paper/2601.00672/) (2026)
- [Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitions](https://mlsanity.com/paper/2605.19823/) (2026)
- [Hybrid Least Squares/Gradient Descent Methods for MIONets](https://mlsanity.com/paper/2607.06976/) (2026)
- [Deep Learning-based Surrogate Modelling of the LOD Method for Multiscale Problems](https://mlsanity.com/paper/2607.12570/) (2026)
- [Feature Interaction Modeling for Neural Operators](https://mlsanity.com/paper/2607.28762/) (2026)
- [Hypernetwork-Parameterized Spatially Adaptive Neural Operators for PDE Learning](https://mlsanity.com/paper/2609.20309/) (2026)

## Abstract (arXiv)

Fast and accurate predictions for complex physical dynamics are a significant challenge across various applications. Real-time prediction on resource-constrained hardware is even more crucial in real-world problems. The deep operator network (DeepONet) has recently been proposed as a framework for learning nonlinear mappings between function spaces. However, the DeepONet requires many parameters and has a high computational cost when learning operators, particularly those with complex (discontinuous or non-smooth) target functions. This study proposes HyperDeepONet, which uses the expressive power of the hypernetwork to enable the learning of a complex operator with a smaller set of parameters. The DeepONet and its variant models can be thought of as a method of injecting the input function information into the target function. From this perspective, these models can be viewed as a particular case of HyperDeepONet. We analyze the complexity of DeepONet and conclude that HyperDeepONet needs relatively lower complexity to obtain the desired accuracy for operator learning. HyperDeepONet successfully learned various operators with fewer computational resources compared to other benchmarks.