# NeuPDE: Neural Network Based Ordinary and Partial Differential Equations for Modeling Time-Dependent Data

arXiv: https://arxiv.org/abs/1908.03190 · HTML: https://mlsanity.com/paper/1908.03190/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Aug 2019.

## Built on by (most cited first, 30 of 42)

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- [Deep Neural Network Modeling of Unknown Partial Differential Equations in Nodal Space](https://mlsanity.com/paper/2106.03603/) (2021)
- [Flow Map Learning for Unknown Dynamical Systems: Overview, Implementation, and Benchmarks](https://mlsanity.com/paper/2307.11013/) (2023)
- [Beyond Predictions in Neural ODEs: Identification and Interventions](https://mlsanity.com/paper/2106.12430/) (2021)
- [The Discovery of Dynamics via Linear Multistep Methods and Deep Learning: Error Estimation](https://mlsanity.com/paper/2103.11488/) (2021)
- [LeMON: Learning to Learn Multi-Operator Networks](https://mlsanity.com/paper/2408.16168/) (2024)
- [FINDE: Neural Differential Equations for Finding and Preserving Invariant Quantities](https://mlsanity.com/paper/2210.00272/) (2022)
- [Deep-OSG: Deep Learning of Operators in Semigroup](https://mlsanity.com/paper/2302.03358/) (2023)
- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)
- [DUE: A Deep Learning Framework and Library for Modeling Unknown Equations](https://mlsanity.com/paper/2504.10373/) (2025)
- [Conditioning of Random Feature Matrices: Double Descent and Generalization Error](https://mlsanity.com/paper/2110.11477/) (2021)
- [Time-Series Forecasting, Knowledge Distillation, and Refinement within a Multimodal PDE Foundation Model](https://mlsanity.com/paper/2409.11609/) (2024)
- [Learning orbital dynamics of binary black hole systems from gravitational wave measurements](https://mlsanity.com/paper/2102.12695/) (2021)
- [Neural Ordinary Differential Equations for Data-Driven Reduced Order Modeling of Environmental Hydrodynamics](https://mlsanity.com/paper/2104.13962/) (2021)
- [Robust Modeling of Unknown Dynamical Systems via Ensemble Averaged Learning](https://mlsanity.com/paper/2203.03458/) (2022)
- [AsCAN: Asymmetric Convolution-Attention Networks for Efficient Recognition and Generation](https://mlsanity.com/paper/2411.04967/) (2024)
- [Neural Piecewise-Constant Delay Differential Equations](https://mlsanity.com/paper/2201.00960/) (2022)
- [Modeling unknown dynamical systems with hidden parameters](https://mlsanity.com/paper/2202.01858/) (2022)
- [Learning PDE Solution Operator for Continuous Modeling of Time-Series](https://mlsanity.com/paper/2302.00854/) (2023)
- [Neuro-Reachability of Networked Microgrids](https://mlsanity.com/paper/2101.05159/) (2021)
- [A Swarm Variant for the Schr"odinger Solver](https://mlsanity.com/paper/2104.04795/) (2021)
- [Data-driven reduced order modeling of environmental hydrodynamics using deep autoencoders and neural ODEs](https://mlsanity.com/paper/2107.02784/) (2021)
- [Adaptive Group Lasso Neural Network Models for Functions of Few Variables and Time-Dependent Data](https://mlsanity.com/paper/2108.10825/) (2021)
- [Graph Neural Stochastic Differential Equations](https://mlsanity.com/paper/2308.12316/) (2023)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Autoregression-Free Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2605.25413/) (2026)
- [Signature-Graph Networks](https://mlsanity.com/paper/2110.11551/) (2021)
- [Characteristic Neural Ordinary Differential Equations](https://mlsanity.com/paper/2111.13207/) (2021)

## Abstract (arXiv)

We propose a neural network based approach for extracting models from dynamic data using ordinary and partial differential equations. In particular, given a time-series or spatio-temporal dataset, we seek to identify an accurate governing system which respects the intrinsic differential structure. The unknown governing model is parameterized by using both (shallow) multilayer perceptrons and nonlinear differential terms, in order to incorporate relevant correlations between spatio-temporal samples. We demonstrate the approach on several examples where the data is sampled from various dynamical systems and give a comparison to recurrent networks and other data-discovery methods. In addition, we show that for MNIST and Fashion MNIST, our approach lowers the parameter cost as compared to other deep neural networks.