# DUE: A Deep Learning Framework and Library for Modeling Unknown Equations

arXiv: https://arxiv.org/abs/2504.10373 · HTML: https://mlsanity.com/paper/2504.10373/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- Builds on 44 of its 70 references that are in the map.
- First posted on arXiv 14 Apr 2025.

## Builds on (most cited first, 30 of 44)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [An overview of gradient descent optimization algorithms](https://mlsanity.com/paper/1609.04747/) (2016)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Modern Koopman Theory for Dynamical Systems](https://mlsanity.com/paper/2102.12086/) (2021)
- [Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1801.06637/) (2018)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Data Driven Governing Equations Approximation Using Deep Neural Networks](https://mlsanity.com/paper/1811.05537/) (2018)
- [Multistep Neural Networks for Data-driven Discovery of Nonlinear Dynamical Systems](https://mlsanity.com/paper/1801.01236/) (2018)
- [Stiff Neural Ordinary Differential Equations](https://mlsanity.com/paper/2103.15341/) (2021)
- [Data-Driven Deep Learning of Partial Differential Equations in Modal Space](https://mlsanity.com/paper/1910.06948/) (2019)
- [DLGA-PDE: Discovery of PDEs with incomplete candidate library via combination of deep learning and genetic algorithm](https://mlsanity.com/paper/2001.07305/) (2020)
- [Multi-level Convolutional Autoencoder Networks for Parametric Prediction of Spatio-temporal Dynamics](https://mlsanity.com/paper/1912.11114/) (2019)
- [NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data](https://mlsanity.com/paper/1908.03190/) (2019)
- [Any equation is a forest: Symbolic genetic algorithm for discovering open-form partial differential equations (SGA-PDE)](https://mlsanity.com/paper/2106.11927/) (2021)
- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Learning Stochastic Dynamical System via Flow Map Operator](https://mlsanity.com/paper/2305.03874/) (2023)

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

- [Identifiable learning of dissipative dynamics](https://mlsanity.com/paper/2510.24160/) (2025)
- [Hypothesis-driven construction of mesoscopic dynamics](https://mlsanity.com/paper/2605.16211/) (2026)
- [Training-free score-based diffusion for parameter-dependent stochastic dynamical systems](https://mlsanity.com/paper/2602.02113/) (2026)
- [MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data](https://mlsanity.com/paper/2604.00333/) (2026)
- [CSympNet-ID: conformal-symplectic map learning for linearly damped Hamiltonian systems](https://mlsanity.com/paper/2607.03339/) (2026)
- [A Hybrid Framework for Healing Semigroups with Machine Learning](https://mlsanity.com/paper/2509.01763/) (2025)
- [Time-Delayed Transformers for Data-Driven Modeling of Low-Dimensional Dynamics](https://mlsanity.com/paper/2602.08478/) (2026)
- [Cost-Accuracy Trade-offs: Neural Operator vs Classical Numerical Solver](https://mlsanity.com/paper/2609.24021/) (2026)

## Abstract (arXiv)

Equations, particularly differential equations, are fundamental for understanding natural phenomena and predicting complex dynamics across various scientific and engineering disciplines. However, the governing equations for many complex systems remain unknown due to intricate underlying mechanisms. Recent advancements in machine learning and data science offer a new paradigm for modeling unknown equations from measurement or simulation data. This paradigm shift, known as data-driven discovery or modeling, stands at the forefront of AI for science, with significant progress made in recent years. In this paper, we introduce a systematic framework for data-driven modeling of unknown equations using deep learning. This versatile framework is capable of learning unknown ODEs, PDEs, DAEs, IDEs, SDEs, reduced or partially observed systems, and non-autonomous differential equations. Based on this framework, we have developed Deep Unknown Equations (DUE), an open-source software package designed to facilitate the data-driven modeling of unknown equations using modern deep learning techniques. DUE serves as an educational tool for classroom instruction, enabling students and newcomers to gain hands-on experience with differential equations, data-driven modeling, and contemporary deep learning approaches such as FNN, ResNet, generalized ResNet, operator semigroup networks (OSG-Net), and Transformers. Additionally, DUE is a versatile and accessible toolkit for researchers across various scientific and engineering fields. It is applicable not only for learning unknown equations from data but also for surrogate modeling of known, yet complex, equations that are costly to solve using traditional numerical methods. We provide detailed descriptions of DUE and demonstrate its capabilities through diverse examples, which serve as templates that can be easily adapted for other applications.