# h-analysis and data-parallel physics-informed neural networks

arXiv: https://arxiv.org/abs/2302.08835 · HTML: https://mlsanity.com/paper/2302.08835/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 39 references that are in the map.
- First posted on arXiv 17 Feb 2023.

## Builds on (most cited first, 15 of 15)

- [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)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Horovod: fast and easy distributed deep learning in TensorFlow](https://mlsanity.com/paper/1802.05799/) (2018)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems](https://mlsanity.com/paper/1809.08327/) (2018)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating PDEs](https://mlsanity.com/paper/2006.16144/) (2020)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Estimates on the generalization error of Physics Informed Neural Networks (PINNs) for approximating a class of inverse problems for PDEs](https://mlsanity.com/paper/2007.01138/) (2020)
- [NVIDIA SimNet^{TM}: an AI-accelerated multi-physics simulation framework](https://mlsanity.com/paper/2012.07938/) (2020)
- [When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?](https://mlsanity.com/paper/2109.09444/) (2021)
- [Scalable algorithms for physics-informed neural and graph networks](https://mlsanity.com/paper/2205.08332/) (2022)
- [Hyper-parameter tuning of physics-informed neural networks: Application to Helmholtz problems](https://mlsanity.com/paper/2205.06704/) (2022)
- [Physics-informed neural networks for operator equations with stochastic data](https://mlsanity.com/paper/2211.10344/) (2022)

## Built on by (most cited first, 4 of 4)

- [Stabilizing and Solving Unique Continuation Problems by Parameterizing Data and Learning Finite Element Solution Operators](https://mlsanity.com/paper/2412.04409/) (2024)
- [Hybrid machine learning based scale bridging framework for permeability prediction of fibrous structures](https://mlsanity.com/paper/2502.05044/) (2025)
- [Forecasting India's Demographic Transition Under Fertility Policy Scenarios Using hybrid LSTM-PINN Model](https://mlsanity.com/paper/2512.00760/) (2025)
- [Physics-Informed Residuals for Adaptive Mesh Refinement in Finite-Difference PDE Solvers](https://mlsanity.com/paper/2606.02475/) (2026)

## Abstract (arXiv)

We explore the data-parallel acceleration of physics-informed machine learning (PIML) schemes, with a focus on physics-informed neural networks (PINNs) for multiple graphics processing units (GPUs) architectures. In order to develop scale-robust and high-throughput PIML models for sophisticated applications which may require a large number of training points (e.g., involving complex and high-dimensional domains, non-linear operators or multi-physics), we detail a novel protocol based on h-analysis and data-parallel acceleration through the Horovod training framework. The protocol is backed by new convergence bounds for the generalization error and the train-test gap. We show that the acceleration is straightforward to implement, does not compromise training, and proves to be highly efficient and controllable, paving the way towards generic scale-robust PIML. Extensive numerical experiments with increasing complexity illustrate its robustness and consistency, offering a wide range of possibilities for real-world simulations.