# Data Driven Governing Equations Approximation Using Deep Neural Networks

arXiv: https://arxiv.org/abs/1811.05537 · HTML: https://mlsanity.com/paper/1811.05537/

## Key facts

- Cited by 96 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Nov 2018.

## Built on by (most cited first, 30 of 96)

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- [Online Weak-form Sparse Identification of Partial Differential Equations](https://mlsanity.com/paper/2203.03979/) (2022)
- [On Approximating the Dynamic Response of Synchronous Generators via Operator Learning: A Step Towards Building Deep Operator-based Power Grid Simulators](https://mlsanity.com/paper/2301.12538/) (2023)
- [PARC: Physics-Aware Recurrent Convolutional Neural Networks to Assimilate Meso-scale Reactive Mechanics of Energetic Materials](https://mlsanity.com/paper/2204.07234/) (2022)
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- [Physics-informed neural networks for solving parametric magnetostatic problems](https://mlsanity.com/paper/2202.04041/) (2022)
- [Model Order Reduction based on Runge-Kutta Neural Network](https://mlsanity.com/paper/2103.13805/) (2021)
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- [Deep Learning of Chaotic Systems from Partially-Observed Data](https://mlsanity.com/paper/2205.08384/) (2022)

## Abstract (arXiv)

We present a numerical framework for approximating unknown governing equations using observation data and deep neural networks (DNN). In particular, we propose to use residual network (ResNet) as the basic building block for equation approximation. We demonstrate that the ResNet block can be considered as a one-step method that is exact in temporal integration. We then present two multi-step methods, recurrent ResNet (RT-ResNet) method and recursive ReNet (RS-ResNet) method. The RT-ResNet is a multi-step method on uniform time steps, whereas the RS-ResNet is an adaptive multi-step method using variable time steps. All three methods presented here are based on integral form of the underlying dynamical system. As a result, they do not require time derivative data for equation recovery and can cope with relatively coarsely distributed trajectory data. Several numerical examples are presented to demonstrate the performance of the methods.