# Spectrally Adapted Physics-Informed Neural Networks for Solving Unbounded Domain Problems

arXiv: https://arxiv.org/abs/2202.02710 · HTML: https://mlsanity.com/paper/2202.02710/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 70 references that are in the map.
- First posted on arXiv 6 Feb 2022.

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

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [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)
- [B-PINNs: Bayesian Physics-Informed Neural Networks for Forward and Inverse PDE Problems with Noisy Data](https://mlsanity.com/paper/2003.06097/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Deep Lagrangian Networks: Using Physics as Model Prior for Deep Learning](https://mlsanity.com/paper/1907.04490/) (2019)
- [Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control](https://mlsanity.com/paper/1909.12077/) (2019)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Minimum Width for Universal Approximation](https://mlsanity.com/paper/2006.08859/) (2020)
- [Regularization for Deep Learning: A Taxonomy](https://mlsanity.com/paper/1710.10686/) (2017)
- [How Much Over-parameterization Is Sufficient to Learn Deep ReLU Networks?](https://mlsanity.com/paper/1911.12360/) (2019)
- [Physics-Informed Neural Networks for Power Systems](https://mlsanity.com/paper/1911.03737/) (2019)
- [Modeling System Dynamics with Physics-Informed Neural Networks Based on Lagrangian Mechanics](https://mlsanity.com/paper/2005.14617/) (2020)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [Implicit energy regularization of neural ordinary-differential-equation control](https://mlsanity.com/paper/2103.06525/) (2021)
- [Neural Ordinary Differential Equation Control of Dynamics on Graphs](https://mlsanity.com/paper/2006.09773/) (2020)
- [Recipes for when Physics Fails: Recovering Robust Learning of Physics Informed Neural Networks](https://mlsanity.com/paper/2110.13330/) (2021)
- [Adversarial Multi-task Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations](https://mlsanity.com/paper/2104.14320/) (2021)

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

- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Neural Spectral Methods: Self-supervised learning in the spectral domain](https://mlsanity.com/paper/2312.05225/) (2023)
- [Spectral Informed Neural Network: An Efficient and Low-Memory PINN](https://mlsanity.com/paper/2408.16414/) (2024)
- [EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale](https://mlsanity.com/paper/2510.24173/) (2025)
- [Numerical Methods For PDEs Over Manifolds Using Spectral Physics Informed Neural Networks](https://mlsanity.com/paper/2302.05322/) (2023)
- [A comprehensive analysis of PINNs: Variants, Applications, and Challenges](https://mlsanity.com/paper/2505.22761/) (2025)
- [A Spectral Approach for Learning Spatiotemporal Neural Differential Equations](https://mlsanity.com/paper/2309.16131/) (2023)
- [Statistical Mechanics and Artificial Neural Networks: Principles, Models, and Applications](https://mlsanity.com/paper/2405.10957/) (2024)
- [Solving nonlinear subsonic compressible flow in infinite domain via multi-stage neural networks](https://mlsanity.com/paper/2601.00342/) (2026)

## Abstract (arXiv)

Solving analytically intractable partial differential equations (PDEs) that involve at least one variable defined on an unbounded domain arises in numerous physical applications. Accurately solving unbounded domain PDEs requires efficient numerical methods that can resolve the dependence of the PDE on the unbounded variable over at least several orders of magnitude. We propose a solution to such problems by combining two classes of numerical methods: (i) adaptive spectral methods and (ii) physics-informed neural networks (PINNs). The numerical approach that we develop takes advantage of the ability of physics-informed neural networks to easily implement high-order numerical schemes to efficiently solve PDEs and extrapolate numerical solutions at any point in space and time. We then show how recently introduced adaptive techniques for spectral methods can be integrated into PINN-based PDE solvers to obtain numerical solutions of unbounded domain problems that cannot be efficiently approximated by standard PINNs. Through a number of examples, we demonstrate the advantages of the proposed spectrally adapted PINNs in solving PDEs and estimating model parameters from noisy observations in unbounded domains.