# AdjointNet: Constraining machine learning models with physics-based codes

arXiv: https://arxiv.org/abs/2109.03956 · HTML: https://mlsanity.com/paper/2109.03956/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 54 references that are in the map.
- First posted on arXiv 8 Sep 2021.

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

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [The many Shapley values for model explanation](https://mlsanity.com/paper/1908.08474/) (2019)
- [Explaining NonLinear Classification Decisions with Deep Taylor Decomposition](https://mlsanity.com/paper/1512.02479/) (2015)
- [Model-Agnostic Interpretability of Machine Learning](https://mlsanity.com/paper/1606.05386/) (2016)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [A Differentiable Programming System to Bridge Machine Learning and Scientific Computing](https://mlsanity.com/paper/1907.07587/) (2019)
- [On weight initialization in deep neural networks](https://mlsanity.com/paper/1704.08863/) (2017)
- [Generalization in Deep Networks: The Role of Distance from Initialization](https://mlsanity.com/paper/1901.01672/) (2019)
- [A type of generalization error induced by initialization in deep neural networks](https://mlsanity.com/paper/1905.07777/) (2019)
- [Physics Guided Machine Learning Methods for Hydrology](https://mlsanity.com/paper/2012.02854/) (2020)
- [Unsupervised Machine Learning Based on Non-Negative Tensor Factorization for Analyzing Reactive-Mixing](https://mlsanity.com/paper/1805.06454/) (2018)
- [Modeling nanoconfinement effects using active learning](https://mlsanity.com/paper/2005.02587/) (2020)
- [A Physics-Constrained Deep Learning Model for Simulating Multiphase Flow in 3D Heterogeneous Porous Media](https://mlsanity.com/paper/2105.09467/) (2021)

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

- [Physics-integrated generative modeling using attentive planar normalizing flow based variational autoencoder](https://mlsanity.com/paper/2404.12267/) (2024)
- [Deep Learning to Estimate Permeability using Geophysical Data](https://mlsanity.com/paper/2110.10077/) (2021)

## Abstract (arXiv)

Physics-informed Machine Learning has recently become attractive for learning physical parameters and features from simulation and observation data. However, most existing methods do not ensure that the physics, such as balance laws (e.g., mass, momentum, energy conservation), are constrained. Some recent works (e.g., physics-informed neural networks) softly enforce physics constraints by including partial differential equation (PDE)-based loss functions but need re-discretization of the PDEs using auto-differentiation. Training these neural nets on observational data showed that one could solve forward and inverse problems in one shot. They evaluate the state variables and the parameters in a PDE. This re-discretization of PDEs is not necessarily an attractive option for domain scientists that work with physics-based codes that have been developed for decades with sophisticated discretization techniques to solve complex process models and advanced equations of state. This paper proposes a physics constrained machine learning framework, AdjointNet, allowing domain scientists to embed their physics code in neural network training workflows. This embedding ensures that physics is constrained everywhere in the domain. Additionally, the mathematical properties such as consistency, stability, and convergence vital to the numerical solution of a PDE are still satisfied. We show that the proposed AdjointNet framework can be used for parameter estimation (and uncertainty quantification by extension) and experimental design using active learning. The applicability of our framework is demonstrated for four flow cases. Results show that AdjointNet-based inversion can estimate process model parameters with reasonable accuracy. These examples demonstrate the applicability of using existing software with no changes in source code to perform accurate and reliable inversion of model parameters.