# Improving the accuracy of physics-informed neural networks via last-layer retraining

arXiv: https://arxiv.org/abs/2603.04672 · HTML: https://mlsanity.com/paper/2603.04672/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 17 of its 28 references that are in the map.
- First posted on arXiv 4 Mar 2026.

## Builds on (most cited first, 17 of 17)

- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Weak baselines and reporting biases lead to overoptimism in machine learning for fluid-related partial differential equations](https://mlsanity.com/paper/2407.07218/) (2024)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Failure-informed adaptive sampling for PINNs](https://mlsanity.com/paper/2210.00279/) (2022)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Multifidelity Modeling for Physics-Informed Neural Networks (PINNs)](https://mlsanity.com/paper/2106.13361/) (2021)
- [Galerkin Neural Networks: A Framework for Approximating Variational Equations with Error Control](https://mlsanity.com/paper/2105.14094/) (2021)
- [Stacked networks improve physics-informed training: applications to neural networks and deep operator networks](https://mlsanity.com/paper/2311.06483/) (2023)
- [Self-adaptive weights based on balanced residual decay rate for physics-informed neural networks and deep operator networks](https://mlsanity.com/paper/2407.01613/) (2024)
- [Fast training of accurate physics-informed neural networks without gradient descent](https://mlsanity.com/paper/2405.20836/) (2024)
- [Efficient kernel surrogates for neural network-based regression](https://mlsanity.com/paper/2310.18612/) (2023)
- [Self-adaptive weighting and sampling for physics-informed neural networks](https://mlsanity.com/paper/2511.05452/) (2025)
- [Beyond Derivative Pathology of PINNs: Variable Splitting Strategy with Convergence Analysis](https://mlsanity.com/paper/2409.20383/) (2024)
- [What do physics-informed DeepONets learn? Understanding and improving training for scientific computing applications](https://mlsanity.com/paper/2411.18459/) (2024)

## Abstract (arXiv)

Physics-informed neural networks (PINNs) are a versatile tool in the burgeoning field of scientific machine learning for solving partial differential equations (PDEs). However, determining suitable training strategies for them is not obvious, with the result that they typically yield moderately accurate solutions. In this article, we propose a method for improving the accuracy of PINNs by coupling them with a post-processing step that seeks the best approximation in a function space associated with the network. We find that our method yields errors four to five orders of magnitude lower than those of the parent PINNs across architectures and dimensions. Moreover, we can reuse the basis functions for the linear space in more complex settings, such as time-dependent and nonlinear problems, allowing for transfer learning. Our approach also provides a residual-based metric that allows us to optimally choose the number of basis functions employed.