# The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?

arXiv: https://arxiv.org/abs/2103.09655 · HTML: https://mlsanity.com/paper/2103.09655/

## Key facts

- Cited by 61 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Mar 2021.

## Built on by (most cited first, 30 of 61)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Enhancing Computational Fluid Dynamics with Machine Learning](https://mlsanity.com/paper/2110.02085/) (2021)
- [PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs](https://mlsanity.com/paper/2306.08827/) (2023)
- [Learning in Sinusoidal Spaces with Physics-Informed Neural Networks](https://mlsanity.com/paper/2109.09338/) (2021)
- [Deep Learning in Deterministic Computational Mechanics](https://mlsanity.com/paper/2309.15421/) (2023)
- [Kronecker-Factored Approximate Curvature for Physics-Informed Neural Networks](https://mlsanity.com/paper/2405.15603/) (2024)
- [Asymptotic self-similar blow-up profile for three-dimensional axisymmetric Euler equations using neural networks](https://mlsanity.com/paper/2201.06780/) (2022)
- [Improved Training of Physics-Informed Neural Networks with Model Ensembles](https://mlsanity.com/paper/2204.05108/) (2022)
- [Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems](https://mlsanity.com/paper/2205.06704/) (2022)
- [Solving Forward and Inverse Problems of Contact Mechanics using Physics-Informed Neural Networks](https://mlsanity.com/paper/2308.12716/) (2023)
- [Auto-PINN: Understanding and Optimizing Physics-Informed Neural Architecture](https://mlsanity.com/paper/2205.13748/) (2022)
- [Data-Driven Physics-Informed Neural Networks: A Digital Twin Perspective](https://mlsanity.com/paper/2401.08667/) (2024)
- [Evolutionary Optimization of Physics-Informed Neural Networks: Evo-PINN Frontiers and Opportunities](https://mlsanity.com/paper/2501.06572/) (2025)
- [Physical Activation Functions (PAFs): An Approach for More Efficient Induction of Physics into Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2205.14630/) (2022)
- [Bayesian Physics Informed Neural Networks for Data Assimilation and Spatio-Temporal Modelling of Wildfires](https://mlsanity.com/paper/2212.00970/) (2022)
- [Physics-Informed Neural Networks for High-Frequency and Multi-Scale Problems using Transfer Learning](https://mlsanity.com/paper/2401.02810/) (2024)
- [Physics-informed Neural Networks with Unknown Measurement Noise](https://mlsanity.com/paper/2211.15498/) (2022)
- [Component Fourier Neural Operator for Singularly Perturbed Differential Equations](https://mlsanity.com/paper/2409.04779/) (2024)
- [Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process](https://mlsanity.com/paper/2201.13389/) (2022)
- [Do ReLU Networks Have An Edge When Approximating Compactly-Supported Functions?](https://mlsanity.com/paper/2204.11231/) (2022)
- [Gradient-Enhanced Physics-Informed Neural Networks for Power Systems Operational Support](https://mlsanity.com/paper/2206.10579/) (2022)
- [PINN surrogate of Li-ion battery models for parameter inference. Part I: Implementation and multi-fidelity hierarchies for the single-particle model](https://mlsanity.com/paper/2312.17329/) (2023)
- [Physics-Informed Neural Networks for Dynamic Process Operations with Limited Physical Knowledge and Data](https://mlsanity.com/paper/2406.01528/) (2024)
- [Physics-guided Data Augmentation for Learning the Solution Operator of Linear Differential Equations](https://mlsanity.com/paper/2212.04100/) (2022)
- [Error convergence and engineering-guided hyperparameter search of PINNs: towards optimized I-FENN performance](https://mlsanity.com/paper/2303.03918/) (2023)
- [Grad-Shafranov equilibria via data-free physics informed neural networks](https://mlsanity.com/paper/2311.13491/) (2023)
- [A Deep Learning-Based Particle-in-Cell Method for Plasma Simulations](https://mlsanity.com/paper/2107.02232/) (2021)
- [LSA-PINN: Linear Boundary Connectivity Loss for Solving PDEs on Complex Geometry](https://mlsanity.com/paper/2302.01518/) (2023)
- [A Critical Review of Physics-Informed Machine Learning Applications in Subsurface Energy Systems](https://mlsanity.com/paper/2308.04457/) (2023)
- [Integration of physics-informed operator learning and finite element method for parametric learning of partial differential equations](https://mlsanity.com/paper/2401.02363/) (2024)

## Abstract (arXiv)

Physics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems arising from PDEs, a task for which several traditional methods exist. In this work, we focus first on evaluating the potential of PINNs as linear solvers in the case of the Poisson equation, an omnipresent equation in scientific computing. We characterize PINN linear solvers in terms of accuracy and performance under different network configurations (depth, activation functions, input data set distribution). We highlight the critical role of transfer learning. Our results show that low-frequency components of the solution converge quickly as an effect of the F-principle. In contrast, an accurate solution of the high frequencies requires an exceedingly long time. To address this limitation, we propose integrating PINNs into traditional linear solvers. We show that this integration leads to the development of new solvers whose performance is on par with other high-performance solvers, such as PETSc conjugate gradient linear solvers, in terms of performance and accuracy. Overall, while the accuracy and computational performance are still a limiting factor for the direct use of PINN linear solvers, hybrid strategies combining old traditional linear solver approaches with new emerging deep-learning techniques are among the most promising methods for developing a new class of linear solvers.