# Replacing Automatic Differentiation by Sobolev Cubatures fastens Physics Informed Neural Nets and strengthens their Approximation Power

arXiv: https://arxiv.org/abs/2211.15443 · HTML: https://mlsanity.com/paper/2211.15443/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 23 Nov 2022.

## 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)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [hp-VPINNs: Variational Physics-Informed Neural Networks With Domain Decomposition](https://mlsanity.com/paper/2003.05385/) (2020)
- [Variational Physics-Informed Neural Networks For Solving Partial Differential Equations](https://mlsanity.com/paper/1912.00873/) (2019)
- [Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations](https://mlsanity.com/paper/1811.02033/) (2018)
- [Inverse-Dirichlet Weighting Enables Reliable Training of Physics Informed Neural Networks](https://mlsanity.com/paper/2107.00940/) (2021)

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

- [Continuous Spatiotemporal Transformers](https://mlsanity.com/paper/2301.13338/) (2023)
- [Ensuring Topological Data-Structure Preservation under Autoencoder Compression due to Latent Space Regularization in Gauss--Legendre nodes](https://mlsanity.com/paper/2309.08228/) (2023)
- [Learning Partial Differential Equations by Spectral Approximates of General Sobolev Spaces](https://mlsanity.com/paper/2301.04887/) (2023)
- [Zero Coordinate Shift: Whetted Automatic Differentiation for Physics-informed Operator Learning](https://mlsanity.com/paper/2311.00860/) (2023)

## Abstract (arXiv)

We present a novel class of approximations for variational losses, being applicable for the training of physics-informed neural nets (PINNs). The loss formulation reflects classic Sobolev space theory for partial differential equations and their weak formulations. The loss computation rests on an extension of Gauss-Legendre cubatures, we term Sobolev cubatures, replacing automatic differentiation (A.D.). We prove the runtime complexity of training the resulting Soblev-PINNs (SC-PINNs) to be less than required by PINNs relying on A.D. On top of one-to-two order of magnitude speed-up the SC-PINNs are demonstrated to achieve closer solution approximations for prominent forward and inverse PDE problems than established PINNs achieve.