# Physics-driven machine learning models coupling PyTorch and Firedrake

arXiv: https://arxiv.org/abs/2303.06871 · HTML: https://mlsanity.com/paper/2303.06871/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 2 of its 19 references that are in the map.
- First posted on arXiv 13 Mar 2023.

## Builds on (most cited first, 2 of 2)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [TNet: A Model-Constrained Tikhonov Network Approach for Inverse Problems](https://mlsanity.com/paper/2105.12033/) (2021)

## Built on by (most cited first, 2 of 2)

- [G-Adaptivity: optimised graph-based mesh relocation for finite element methods](https://mlsanity.com/paper/2407.04516/) (2024)
- [Structure-Preserving Operator Learning](https://mlsanity.com/paper/2410.01065/) (2024)

## Abstract (arXiv)

Partial differential equations (PDEs) are central to describing and modelling complex physical systems that arise in many disciplines across science and engineering. However, in many realistic applications PDE modelling provides an incomplete description of the physics of interest. PDE-based machine learning techniques are designed to address this limitation. In this approach, the PDE is used as an inductive bias enabling the coupled model to rely on fundamental physical laws while requiring less training data. The deployment of high-performance simulations coupling PDEs and machine learning to complex problems necessitates the composition of capabilities provided by machine learning and PDE-based frameworks. We present a simple yet effective coupling between the machine learning framework PyTorch and the PDE system Firedrake that provides researchers, engineers and domain specialists with a high productive way of specifying coupled models while only requiring trivial changes to existing code.