# A Lagrangian Dual-based Theory-guided Deep Neural Network

arXiv: https://arxiv.org/abs/2008.10159 · HTML: https://mlsanity.com/paper/2008.10159/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Aug 2020.

## Built on by (most cited first, 8 of 8)

- [Machine Learning in Aerodynamic Shape Optimization](https://mlsanity.com/paper/2202.07141/) (2022)
- [Robust discovery of partial differential equations in complex situations](https://mlsanity.com/paper/2106.00008/) (2021)
- [AutoKE: An automatic knowledge embedding framework for scientific machine learning](https://mlsanity.com/paper/2205.05390/) (2022)
- [A Dual Perspective on Decision-Focused Learning: Scalable Training via Dual-Guided Surrogates](https://mlsanity.com/paper/2511.04909/) (2025)
- [Domain-guided Machine Learning for Remotely Sensed In-Season Crop Growth Estimation](https://mlsanity.com/paper/2106.13323/) (2021)
- [Filtered Partial Differential Equations: a robust surrogate constraint in physics-informed deep learning framework](https://mlsanity.com/paper/2311.03776/) (2023)
- [Worth of knowledge in deep learning](https://mlsanity.com/paper/2307.00712/) (2023)
- [NASP-T: A Fuzzy Neuro-Symbolic Transformer for Logic-Constrained Aviation Safety Report Classification](https://mlsanity.com/paper/2510.05451/) (2025)

## Abstract (arXiv)

The theory-guided neural network (TgNN) is a kind of method which improves the effectiveness and efficiency of neural network architectures by incorporating scientific knowledge or physical information. Despite its great success, the theory-guided (deep) neural network possesses certain limits when maintaining a tradeoff between training data and domain knowledge during the training process. In this paper, the Lagrangian dual-based TgNN (TgNN-LD) is proposed to improve the effectiveness of TgNN. We convert the original loss function into a constrained form with fewer items, in which partial differential equations (PDEs), engineering controls (ECs), and expert knowledge (EK) are regarded as constraints, with one Lagrangian variable per constraint. These Lagrangian variables are incorporated to achieve an equitable tradeoff between observation data and corresponding constraints, in order to improve prediction accuracy, and conserve time and computational resources adjusted by an ad-hoc procedure. To investigate the performance of the proposed method, the original TgNN model with a set of optimized weight values adjusted by ad-hoc procedures is compared on a subsurface flow problem, with their L2 error, R square (R2), and computational time being analyzed. Experimental results demonstrate the superiority of the Lagrangian dual-based TgNN.