# Phy-Taylor: Physics-Model-Based Deep Neural Networks

arXiv: https://arxiv.org/abs/2209.13511 · HTML: https://mlsanity.com/paper/2209.13511/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 53 references that are in the map.
- First posted on arXiv 27 Sep 2022.

## Builds on (most cited first, 30 of 32)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [TensorFlow: Large-Scale Machine Learning on Heterogeneous Distributed Systems](https://mlsanity.com/paper/1603.04467/) (2016)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Geometric deep learning on graphs and manifolds using mixture model CNNs](https://mlsanity.com/paper/1611.08402/) (2016)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Control Barrier Functions: Theory and Applications](https://mlsanity.com/paper/1903.11199/) (2019)
- [An Investigation of Why Overparameterization Exacerbates Spurious Correlations](https://mlsanity.com/paper/2005.04345/) (2020)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Towards Physics-informed Deep Learning for Turbulent Flow Prediction](https://mlsanity.com/paper/1911.08655/) (2019)
- [Disentangling Physical Dynamics from Unknown Factors for Unsupervised Video Prediction](https://mlsanity.com/paper/2003.01460/) (2020)
- [Integrating Scientific Knowledge with Machine Learning for Engineering and Environmental Systems](https://mlsanity.com/paper/2003.04919/) (2020)
- [Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction](https://mlsanity.com/paper/2007.04439/) (2020)
- [Physics-guided Neural Networks (PGNN): An Application in Lake Temperature Modeling](https://mlsanity.com/paper/1710.11431/) (2017)
- [Incorporating Symmetry into Deep Dynamics Models for Improved Generalization](https://mlsanity.com/paper/2002.03061/) (2020)
- [Simplifying Hamiltonian and Lagrangian Neural Networks via Explicit Constraints](https://mlsanity.com/paper/2010.13581/) (2020)
- [Augmenting Physical Models with Deep Networks for Complex Dynamics Forecasting](https://mlsanity.com/paper/2010.04456/) (2020)
- [Information Theoretic Model Predictive Control: Theory and Applications to Autonomous Driving](https://mlsanity.com/paper/1707.02342/) (2017)
- [Learning Compositional Koopman Operators for Model-Based Control](https://mlsanity.com/paper/1910.08264/) (2019)
- [Physics-Guided Machine Learning for Scientific Discovery: An Application in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/2001.11086/) (2020)
- [Physics Guided RNNs for Modeling Dynamical Systems: A Case Study in Simulating Lake Temperature Profiles](https://mlsanity.com/paper/1810.13075/) (2018)
- [Safety-Critical Model Predictive Control with Discrete-Time Control Barrier Function](https://mlsanity.com/paper/2007.11718/) (2020)
- [Deep learning of free boundary and Stefan problems](https://mlsanity.com/paper/2006.05311/) (2020)
- [Combining Generative and Discriminative Models for Hybrid Inference](https://mlsanity.com/paper/1906.02547/) (2019)
- [Deep Learning of Subsurface Flow via Theory-guided Neural Network](https://mlsanity.com/paper/1911.00103/) (2019)
- [Theory-guided hard constraint projection (HCP): a knowledge-based data-driven scientific machine learning method](https://mlsanity.com/paper/2012.06148/) (2020)
- [Understanding Rare Spurious Correlations in Neural Networks](https://mlsanity.com/paper/2202.05189/) (2022)
- [HybridNet: Integrating Model-based and Data-driven Learning to Predict Evolution of Dynamical Systems](https://mlsanity.com/paper/1806.07439/) (2018)
- [DR-RNN: A deep residual recurrent neural network for model reduction](https://mlsanity.com/paper/1709.00939/) (2017)
- [Isometric Transformation Invariant and Equivariant Graph Convolutional Networks](https://mlsanity.com/paper/2005.06316/) (2020)

## Abstract (arXiv)

Purely data-driven deep neural networks (DNNs) applied to physical engineering systems can infer relations that violate physics laws, thus leading to unexpected consequences. To address this challenge, we propose a physics-model-based DNN framework, called Phy-Taylor, that accelerates learning compliant representations with physical knowledge. The Phy-Taylor framework makes two key contributions; it introduces a new architectural Physics-compatible neural network (PhN), and features a novel compliance mechanism, we call em Physics-guided Neural Network Editing. The PhN aims to directly capture nonlinearities inspired by physical quantities, such as kinetic energy, potential energy, electrical power, and aerodynamic drag force. To do so, the PhN augments neural network layers with two key components: (i) monomials of Taylor series expansion of nonlinear functions capturing physical knowledge, and (ii) a suppressor for mitigating the influence of noise. The neural-network editing mechanism further modifies network links and activation functions consistently with physical knowledge. As an extension, we also propose a self-correcting Phy-Taylor framework that introduces two additional capabilities: (i) physics-model-based safety relationship learning, and (ii) automatic output correction when violations of safety occur. Through experiments, we show that (by expressing hard-to-learn nonlinearities directly and by constraining dependencies) Phy-Taylor features considerably fewer parameters, and a remarkably accelerated training process, while offering enhanced model robustness and accuracy.