# Optimal time sampling in physics-informed neural networks

arXiv: https://arxiv.org/abs/2404.18780 · HTML: https://mlsanity.com/paper/2404.18780/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 19 references that are in the map.
- First posted on arXiv 29 Apr 2024.

## Builds on (most cited first, 9 of 9)

- [Respecting causality is all you need for training physics-informed neural networks](https://mlsanity.com/paper/2203.07404/) (2022)
- [Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks](https://mlsanity.com/paper/2007.04542/) (2020)
- [On the Almost Sure Convergence of Stochastic Gradient Descent in Non-Convex Problems](https://mlsanity.com/paper/2006.11144/) (2020)
- [Tackling the Curse of Dimensionality with Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.12306/) (2023)
- [Convergence rates for the stochastic gradient descent method for non-convex objective functions](https://mlsanity.com/paper/1904.01517/) (2019)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [Adaptive Self-supervision Algorithms for Physics-informed Neural Networks](https://mlsanity.com/paper/2207.04084/) (2022)
- [The convergence of the Stochastic Gradient Descent (SGD) : a self-contained proof](https://mlsanity.com/paper/2103.14350/) (2021)
- [Convergence of a L2 regularized Policy Gradient Algorithm for the Multi Armed Bandit](https://mlsanity.com/paper/2402.06388/) (2024)

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

- [Active Learning with Selective Time-Step Acquisition for PDEs](https://mlsanity.com/paper/2511.18107/) (2025)
- [Regime-Aware Time Weighting for Physics-Informed Neural Networks](https://mlsanity.com/paper/2407.21642/) (2024)
- [Auto-Adaptive PINNs with Applications to Phase Transitions](https://mlsanity.com/paper/2510.23999/) (2025)

## Abstract (arXiv)

Physics-informed neural networks (PINN) is a extremely powerful paradigm used to solve equations encountered in scientific computing applications. An important part of the procedure is the minimization of the equation residual which includes, when the equation is time-dependent, a time sampling. It was argued in the literature that the sampling need not be uniform but should overweight initial time instants, but no rigorous explanation was provided for this choice. In the present work we take some prototypical examples and, under standard hypothesis concerning the neural network convergence, we show that the optimal time sampling follows a (truncated) exponential distribution. In particular we explain when is best to use uniform time sampling and when one should not. The findings are illustrated with numerical examples on linear equation, Burgers' equation and the Lorenz system.