# Fixed-budget online adaptive learning for physics-informed neural networks. Towards parameterized problem inference

arXiv: https://arxiv.org/abs/2212.11776 · HTML: https://mlsanity.com/paper/2212.11776/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 21 references that are in the map.
- First posted on arXiv 22 Dec 2022.

## Builds on (most cited first, 7 of 7)

- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [A unified deep artificial neural network approach to partial differential equations in complex geometries](https://mlsanity.com/paper/1711.06464/) (2017)
- [Efficient training of physics-informed neural networks via importance sampling](https://mlsanity.com/paper/2104.12325/) (2021)
- [RANG: A Residual-based Adaptive Node Generation Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2205.01051/) (2022)
- [Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process](https://mlsanity.com/paper/2201.13389/) (2022)

## Built on by (most cited first, 3 of 3)

- [HyperSBINN: A Hypernetwork-Enhanced Systems Biology-Informed Neural Network for Efficient Drug Cardiosafety Assessment](https://mlsanity.com/paper/2408.14266/) (2024)
- [Geometry-aware framework for deep energy method: an application to structural mechanics with hyperelastic materials](https://mlsanity.com/paper/2405.03427/) (2024)
- [Beyond PINNs: A Unified Gauss--Newton and Petrov--Galerkin Framework for Neural and Hybrid PDE Solvers](https://mlsanity.com/paper/2609.20641/) (2026)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINNs) have gained much attention in various fields of engineering thanks to their capability of incorporating physical laws into the models. PINNs integrate the physical constraints by minimizing the partial differential equations (PDEs) residuals on a set of collocation points. The distribution of these collocation points appears to have a huge impact on the performance of PINNs and the assessment of the sampling methods for these points is still an active topic. In this paper, we propose a Fixed-Budget Online Adaptive Learning (FBOAL) method, which decomposes the domain into sub-domains, for training collocation points based on local maxima and local minima of the PDEs residuals. The effectiveness of FBOAL is demonstrated for non-parameterized and parameterized problems. The comparison with other adaptive sampling methods is also illustrated. The numerical results demonstrate important gains in terms of the accuracy and computational cost of PINNs with FBOAL over the classical PINNs with non-adaptive collocation points. We also apply FBOAL in a complex industrial application involving coupling between mechanical and thermal fields. We show that FBOAL is able to identify the high-gradient locations and even give better predictions for some physical fields than the classical PINNs with collocation points sampled on a pre-adapted finite element mesh built thanks to numerical expert knowledge. From the present study, it is expected that the use of FBOAL will help to improve the conventional numerical solver in the construction of the mesh.