# Neural Parameter Regression for Explicit Representations of PDE Solution Operators

arXiv: https://arxiv.org/abs/2403.12764 · HTML: https://mlsanity.com/paper/2403.12764/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 32 references that are in the map.
- First posted on arXiv 19 Mar 2024.

## Builds on (most cited first, 23 of 23)

- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [HyperNetworks](https://mlsanity.com/paper/1609.09106/) (2016)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equations](https://mlsanity.com/paper/2107.07871/) (2021)
- [Identity Matters in Deep Learning](https://mlsanity.com/paper/1611.04231/) (2016)
- [A Brief Review of Hypernetworks in Deep Learning](https://mlsanity.com/paper/2306.06955/) (2023)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Long-time integration of parametric evolution equations with physics-informed DeepONets](https://mlsanity.com/paper/2106.05384/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Generic bounds on the approximation error for physics-informed (and) operator learning](https://mlsanity.com/paper/2205.11393/) (2022)
- [Reliable extrapolation of deep neural operators informed by physics or sparse observations](https://mlsanity.com/paper/2212.06347/) (2022)
- [Hypernetwork-based Meta-Learning for Low-Rank Physics-Informed Neural Networks](https://mlsanity.com/paper/2310.09528/) (2023)
- [HyperPINN: Learning parameterized differential equations with physics-informed hypernetworks](https://mlsanity.com/paper/2111.01008/) (2021)
- [PERP: Rethinking the Prune-Retrain Paradigm in the Era of LLMs](https://mlsanity.com/paper/2312.15230/) (2023)
- [How I Learned to Stop Worrying and Love Retraining](https://mlsanity.com/paper/2111.00843/) (2021)
- [Generating synthetic data for neural operators](https://mlsanity.com/paper/2401.02398/) (2024)
- [Improving physics-informed DeepONets with hard constraints](https://mlsanity.com/paper/2309.07899/) (2023)
- [Adaptive physics-informed neural operator for coarse-grained non-equilibrium flows](https://mlsanity.com/paper/2210.15799/) (2022)

## Built on by (most cited first, 1 of 1)

- [Neural Discovery in Mathematics: Do Machines Dream of Colored Planes?](https://mlsanity.com/paper/2501.18527/) (2025)

## Abstract (arXiv)

We introduce Neural Parameter Regression (NPR), a novel framework specifically developed for learning solution operators in Partial Differential Equations (PDEs). Tailored for operator learning, this approach surpasses traditional DeepONets (Lu et al., 2021) by employing Physics-Informed Neural Network (PINN, Raissi et al., 2019) techniques to regress Neural Network (NN) parameters. By parametrizing each solution based on specific initial conditions, it effectively approximates a mapping between function spaces. Our method enhances parameter efficiency by incorporating low-rank matrices, thereby boosting computational efficiency and scalability. The framework shows remarkable adaptability to new initial and boundary conditions, allowing for rapid fine-tuning and inference, even in cases of out-of-distribution examples.