# Strategies for training point distributions in physics-informed neural networks

arXiv: https://arxiv.org/abs/2508.13216 · HTML: https://mlsanity.com/paper/2508.13216/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 30 references that are in the map.
- First posted on arXiv 17 Aug 2025.

## Builds on (most cited first, 8 of 8)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Activation Functions in Deep Learning: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2109.14545/) (2021)
- [An Expert's Guide to Training Physics-informed Neural Networks](https://mlsanity.com/paper/2308.08468/) (2023)
- [From Activation to Initialization: Scaling Insights for Optimizing Neural Fields](https://mlsanity.com/paper/2403.19205/) (2024)
- [A Novel Adaptive Causal Sampling Method for Physics-Informed Neural Networks](https://mlsanity.com/paper/2210.12914/) (2022)
- [Computational characteristics of feedforward neural networks for solving a stiff differential equation](https://mlsanity.com/paper/2012.01867/) (2020)

## Abstract (arXiv)

Physics-informed neural networks approach the approximation of differential equations by directly incorporating their structure and given conditions in a loss function. This enables conditions like, e.g., invariants to be easily added during the modelling phase. In addition, the approach can be considered as mesh free and can be utilised to compute solutions on arbitrary grids after the training phase. Therefore, physics-informed neural networks are emerging as a promising alternative to solving differential equations with methods from numerical mathematics. However, their performance highly depends on a large variety of factors. In this paper, we systematically investigate and evaluate a core component of the approach, namely the training point distribution. We test two ordinary and two partial differential equations with five strategies for training data generation and shallow network architectures, with one and two hidden layers. In addition to common distributions, we introduce sine-based training points, which are motivated by the construction of Chebyshev nodes. The results are challenged by using certain parameter combinations like, e.g., random and fixed-seed weight initialisation for reproducibility. The results show the impact of the training point distributions on the solution accuracy and we find evidence that they are connected to the characteristics of the differential equation.