# Accelerated Training of Physics-Informed Neural Networks (PINNs) using Meshless Discretizations

**Authors:** Ramansh Sharma, Varun Shankar  
**arXiv:** [2205.09332](https://arxiv.org/abs/2205.09332)  
**Date:** 2022-05-19  
**Summary page:** https://www.summarizepaper.com/en/paper/2205.09332v6/

## Key points

- The paper presents a new technique called discretely-trained PINNs (DT-PINNs) for accelerating the training of physics-informed neural networks (PINNs).
- DT-PINNs are trained using high-order accurate numerical discretizations computed using meshless radial basis function-finite differences (RBF-FD) and applied via sparse-matrix vector multiplication.
- This approach allows for DT-PINNs to be trained even on point cloud samples placed on irregular domain geometries.
- Using fp64 on the GPU leads to significantly faster training times than fp32 vanilla-PINNs with comparable accuracy.
- RBF-FD approximations of third order accuracy and above are more efficient while being sufficiently accurate for network depth.
- DT-PINNs achieve similar losses with 2–4x faster training times compared to vanilla-PINNs on both linear and nonlinear Poisson equations on a consumer GPU.
- Similar results can be obtained for the PINN solution to the heat equation by discretizing spatial derivatives using RBF-FD and using automatic differentiation for temporal derivatives.
- Fp64 DT-PINNs offer a superior cost–accuracy profile compared to fp32 vanilla–PINNs, which could have implications for accelerating scientific computing tasks that rely on PINNs such as fluid dynamics simulations and inverse problems in medical imaging.

## Summary

The paper presents a new technique for accelerating the training of physics-informed neural networks (PINNs) called discretely-trained PINNs (DT-PINNs). DT-PINNs are trained by replacing exact spatial derivatives with high-order accurate numerical discretizations computed using meshless radial basis function-finite differences (RBF-FD) and applied via sparse-matrix vector multiplication. This approach allows for DT-PINNs to be trained even on point cloud samples placed on irregular domain geometries. The authors show that using fp64 on the GPU leads to significantly faster training times than fp32 vanilla-PINNs with comparable accuracy. The efficiency and accuracy of DT-PINNs are demonstrated through a series of experiments. First, the authors explore the effect of network depth and show that RBF-FD approximations of third order accuracy and above are more efficient while being sufficiently accurate. Next, they compare DT-PINNs to vanilla-PINNs on both linear and nonlinear Poisson equations and show that DT-PINNs achieve similar losses with 2–4x faster training times on a consumer GPU. Finally, they demonstrate that similar results can be obtained for the PINN solution to the heat equation by discretizing spatial derivatives using RBF-FD and using automatic differentiation for temporal derivatives. Overall, their results show that fp64 DT-PINNs offer a superior cost–accuracy profile compared to fp32 vanilla–PINNs. The authors conclude by highlighting how this work could have implications for accelerating scientific computing tasks that rely on PINNs such as fluid dynamics simulations and inverse problems in medical imaging.