# Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

@article{Raissi2017PhysicsID, title={Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations}, author={Maziar Raissi and Paris Perdikaris and George Em Karniadakis}, journal={ArXiv}, year={2017}, volume={abs/1711.10566} }

We introduce physics informed neural networks -- neural networks that are trained to solve supervised learning tasks while respecting any given law of physics described by general nonlinear partial differential equations. [] Key Method Depending on whether the available data is scattered in space-time or arranged in fixed temporal snapshots, we introduce two main classes of algorithms, namely continuous time and discrete time models. The effectiveness of our approach is demonstrated using a wide range of…

## 453 Citations

### Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations

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This work puts forth a deep learning approach for discovering nonlinear partial differential equations from scattered and potentially noisy observations in space and time by approximate the unknown solution as well as the nonlinear dynamics by two deep neural networks.

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A deep, feed-forward, and fully-connected neural network is used to approximate the partial differential equation, where the initial and boundary conditions are either hard or soft assigned, and the resulting physics-informed surrogate model learns to satisfy the differential operator and the initialand boundary conditions.

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Two new Physics-Informed Neural Networks (PINNs) are proposed for solving time-dependent SPDEs, namely the NN-DO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expressions for the temporal derivatives of the Do/BO modes.

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This paper proposes a novel neural network framework, finite difference neural networks (FD-Net), to learn partial differential equations from data, and to iteratively estimate the future dynamical behavior using only a few trainable parameters.

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