# Physics-Driven Regularization of Deep Neural Networks for Enhanced Engineering Design and Analysis

arXiv: https://arxiv.org/abs/1810.05547 · HTML: https://mlsanity.com/paper/1810.05547/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Oct 2018.

## Built on by (most cited first, 6 of 6)

- [CAN-PINN: A Fast Physics-Informed Neural Network Based on Coupled-Automatic-Numerical Differentiation Method](https://mlsanity.com/paper/2110.15832/) (2021)
- [A Deep Collocation Method for the Bending Analysis of Kirchhoff Plate](https://mlsanity.com/paper/2102.02617/) (2021)
- [Physics-Informed Neural Networks for Dynamic Process Operations with Limited Physical Knowledge and Data](https://mlsanity.com/paper/2406.01528/) (2024)
- [Demystifying the Data Need of ML-surrogates for CFD Simulations](https://mlsanity.com/paper/2205.08355/) (2022)
- [Robustness of Physics-Informed Neural Networks to Noise in Sensor Data](https://mlsanity.com/paper/2211.12042/) (2022)
- [Description of Structural Biases and Associated Data in Sensor-Rich Environments](https://mlsanity.com/paper/2104.04885/) (2021)

## Abstract (arXiv)

In this paper, we introduce a physics-driven regularization method for training of deep neural networks (DNNs) for use in engineering design and analysis problems. In particular, we focus on prediction of a physical system, for which in addition to training data, partial or complete information on a set of governing laws is also available. These laws often appear in the form of differential equations, derived from first principles, empirically-validated laws, or domain expertise, and are usually neglected in data-driven prediction of engineering systems. We propose a training approach that utilizes the known governing laws and regularizes data-driven DNN models by penalizing divergence from those laws. The first two numerical examples are synthetic examples, where we show that in constructing a DNN model that best fits the measurements from a physical system, the use of our proposed regularization results in DNNs that are more interpretable with smaller generalization errors, compared to other common regularization methods. The last two examples concern metamodeling for a random Burgers' system and for aerodynamic analysis of passenger vehicles, where we demonstrate that the proposed regularization provides superior generalization accuracy compared to other common alternatives.