# HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks

arXiv: https://arxiv.org/abs/2111.01008 · HTML: https://mlsanity.com/paper/2111.01008/

## Key facts

- Cited by 39 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 12 references that are in the map.
- First posted on arXiv 28 Oct 2021.

## Builds on (most cited first, 10 of 10)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Physics-informed neural networks for solving Reynolds-averaged Navierunicodex2013Stokes equations](https://mlsanity.com/paper/2107.10711/) (2021)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [Active Training of Physics-Informed Neural Networks to Aggregate and Interpolate Parametric Solutions to the Navier-Stokes Equations](https://mlsanity.com/paper/2005.05092/) (2020)
- [A comparative study of physics-informed neural network models for learning unknown dynamics and constitutive relations](https://mlsanity.com/paper/1904.04058/) (2019)
- [Neural networks catching up with finite differences in solving partial differential equations in higher dimensions](https://mlsanity.com/paper/1712.05067/) (2017)

## Built on by (most cited first, 30 of 39)

- [PDEBENCH: An Extensive Benchmark for Scientific Machine Learning](https://mlsanity.com/paper/2210.07182/) (2022)
- [A Brief Review of Hypernetworks in Deep Learning](https://mlsanity.com/paper/2306.06955/) (2023)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [HyperDeepONet: learning operator with complex target function space using the limited resources via hypernetwork](https://mlsanity.com/paper/2312.15949/) (2023)
- [Deep Learning and Symbolic Regression for Discovering Parametric Equations](https://mlsanity.com/paper/2207.00529/) (2022)
- [Meta-PDE: Learning to Solve PDEs Quickly Without a Mesh](https://mlsanity.com/paper/2211.01604/) (2022)
- [Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systems](https://mlsanity.com/paper/2301.10391/) (2023)
- [CoLoRA: Continuous low-rank adaptation for reduced implicit neural modeling of parameterized partial differential equations](https://mlsanity.com/paper/2402.14646/) (2024)
- [Physics-aware Machine Learning Revolutionizes Scientific Paradigm for Machine Learning and Process-based Hydrology](https://mlsanity.com/paper/2310.05227/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [Learning a Neural Solver for Parametric PDE to Enhance Physics-Informed Methods](https://mlsanity.com/paper/2410.06820/) (2024)
- [HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions](https://mlsanity.com/paper/2509.05117/) (2025)
- [Learning Hamiltonian neural Koopman operator and simultaneously sustaining and discovering conservation law](https://mlsanity.com/paper/2406.02154/) (2024)
- [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)
- [Meta-learning of Physics-informed Neural Networks for Efficiently Solving Newly Given PDEs](https://mlsanity.com/paper/2310.13270/) (2023)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Advancing Generalizability by the Baldwin Effect](https://mlsanity.com/paper/2312.03243/) (2023)
- [Physics Informed Symbolic Networks](https://mlsanity.com/paper/2207.06240/) (2022)
- [Real-time Health Monitoring of Heat Exchangers using Hypernetworks and PINNs](https://mlsanity.com/paper/2212.10032/) (2022)
- [Are Two Hidden Layers Still Enough for the Physics-Informed Neural Networks?](https://mlsanity.com/paper/2412.19235/) (2024)
- [Multiple Case Physics-Informed Neural Network for Biomedical Tube Flows](https://mlsanity.com/paper/2309.15294/) (2023)
- [Towards Automated Knowledge Integration From Human-Interpretable Representations](https://mlsanity.com/paper/2402.16105/) (2024)
- [Neural Parameter Regression for Explicit Representations of PDE Solution Operators](https://mlsanity.com/paper/2403.12764/) (2024)
- [HyperSBINN: A Hypernetwork-Enhanced Systems Biology-Informed Neural Network for Efficient Drug Cardiosafety Assessment](https://mlsanity.com/paper/2408.14266/) (2024)
- [Physics-Informed Deep B-Spline Networks](https://mlsanity.com/paper/2503.16777/) (2025)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [Nonlinear model reduction for transport-dominated problems](https://mlsanity.com/paper/2602.01397/) (2026)
- [Physics-Informed Chebyshev Polynomial Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2602.01737/) (2026)
- [D'OH: Decoder-Only Random Hypernetworks for Implicit Neural Representations](https://mlsanity.com/paper/2403.19163/) (2024)
- [Scalable physical source-to-field inference with hypernetworks](https://mlsanity.com/paper/2405.05981/) (2024)
- [Rational-WENO: A lightweight, physically-consistent three-point weighted essentially non-oscillatory scheme](https://mlsanity.com/paper/2409.09217/) (2024)

## Abstract (arXiv)

Many types of physics-informed neural network models have been proposed in recent years as approaches for learning solutions to differential equations. When a particular task requires solving a differential equation at multiple parameterizations, this requires either re-training the model, or expanding its representation capacity to include the parameterization -- both solution that increase its computational cost. We propose the HyperPINN, which uses hypernetworks to learn to generate neural networks that can solve a differential equation from a given parameterization. We demonstrate with experiments on both a PDE and an ODE that this type of model can lead to neural network solutions to differential equations that maintain a small size, even when learning a family of solutions over a parameter space.