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@@ -162,29 +162,39 @@ The trained neural networks representing the velocity field and the pressure can
 ![](http://www.dam.brown.edu/people/mraissi/assets/img/VIV_Case2_data_on_velocities_lift_drag.png)
 > _VIV-I (Velocity Measurements):_ In this figure, the resulting lift (left) and drag (right) forces are compared to the exact ones.
 
+**Inferring Lift and Drag Forces from Flow Visualizations**
+
+Let us continue with the case where we do not have access to direct observations of the lift force $$f_L$$. This time rather than using data on the velocity field, we use measurements of the concentration of a passive scalar (e.g., dye or smoke) injected into the system. In the following, we are going to employ such data to reconstruct the velocity field, the pressure, and consequently the drag and lift forces. A representative snapshot of the data on the concentration of the passive scalar is depicted in the top left panel of the following figure. The neural networks' architectures used here consist of 10 layers with 32 neurons per each hidden layer per output variable. A summary of our results is presented in the following figure. The proposed framework is capable of accurately (of the order of $$10^{-3}$$) reconstructing the concentration. However, a truly intriguing result stems from the network's ability to provide accurate predictions of the entire velocity vector field as well as the pressure, in the absence of sufficient training data on the pressure and the velocity field themselves (see the following figure). A visual comparison against the exact quantities is presented in the following figure for a representative snapshot of the velocity field and the pressure. This result of inferring multiple hidden quantities of interest from auxiliary measurements by leveraging the underlying physics is a great example of the enhanced capabilities that physics-informed deep learning has to offer, and highlights its potential in solving high-dimensional data-assimilation and inverse problems.
+
+![](http://www.dam.brown.edu/people/mraissi/assets/img/VIV_case2_concentration_results.png)
+> _VIV-II (Flow Visualizations Data):_ A representative snapshot of the data on the concentration of the passive scalar is depicted in the top left panel of this figure. The algorithm is capable of accurately reconstructing the concentration of the passive scalar and more importantly the velocity field as well as the pressure without having access to enough observations of these quantities themselves. To compute the difference between the predicted and exact pressure fields we had to subtract the spacial average pressure from both predicted and exact fields because for incompressible fluids the pressure is unique only up to a constant.
+
+Following the same procedure as in the previous example, the trained neural networks representing the velocity field and the pressure can be used to compute the drag and lift forces by employing the equations for drag and lift, respectively. The resulting drag and lift forces are compared to the exact ones in the following figure. In the following, we are going to use the computed lift force to generate the required training data on $$f_L$$ and estimate the structural parameters $$b$$ and $$k$$ by minimizing the first loss function introduced in the current work. Upon training, the proposed framework is capable of identifying the correct values for the structural parameters $$b$$ and $$k$$ with surprising accuracy. The learned values for the damping and stiffness parameters are $$b = 0.08600664$$ and $$k = 2.2395933$$. This corresponds to around $$2.39\%$$ and $$1.71\%$$ relative errors in the estimated values for $$b$$ and $$k$$, respectively.
 
+![](http://www.dam.brown.edu/people/mraissi/assets/img/VIV_case2_concentration_lift_drag.png)
+> _VIV-II (Flow Visualizations Data):_ In this figure, the resulting lift (left) and drag (right) forces are compared to the exact ones.
 
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 **Summary and Discussion**
 
-In this work, we put forth a deep learning approach for solving coupled forward-backward stochastic differential equations and their corresponding high-dimensional partial differential equations. The resulting methodology showcases a series of promising results for a diverse collection of benchmark problems. As deep learning technology is continuing to grow rapidly both in terms of methodological, algorithmic, and infrastructural developments, we believe that this is a timely contribution that can benefit practitioners across a wide range of scientific domains. Specific applications that can readily enjoy these benefits include, but are not limited to, stochastic control, theoretical economics, and mathematical finance.
+We have considered the classical coupled problem of a freely vibrating cylinder due to lift forces and demonstrated how physics informed deep learning can be used to infer quantities of interest from scattered data in space-time. In the *first VIV learning problem*, we inferred the pressure field and structural parameters, and hence the lift and drag on the vibrating cylinder using velocity and displacement data in time-space. In the *second VIV learning problem*, we inferred the velocity and pressure fields as well as the structural parameters given data on a passive scalar in space-time. The framework we propose here represents a *paradigm shift* in fluid mechanics simulation as it uses the governing equations as regularization mechanisms in the loss function of the corresponding minimization problem. It is particularly effective for multi-physics problems as the coupling between fields can be readily accomplished by sharing parameters among the  multiple neural networks -- here a neural network outputting 4 variables for the first problem and 5 variables for the second one -- and for more general coupled problems by also including coupled terms in the loss function. There are many questions that this new type of modeling raises, both theoretical and practical, e.g. efficiency, solution uniqueness, accuracy, etc. We have considered such questions here in the present context as well as in our previous work in the context of physics-informed learning machines but admittedly at the present time it is not possible to rigorously answer such questions. We hope, however, that our present work will ignite interest in physics-informed deep learning that can be used effectively for many different fields of multi-physics fluid mechanics.
 
-In terms of future work, one could straightforwardly extend the proposed framework in the current work to solve second-order backward stochastic differential equations. The key is to leverage the fundamental relationships between second-order backward stochastic differential equations and fully-nonlinear second-order partial differential equations. Moreover, our method can be used to solve [stochastic control](https://en.wikipedia.org/wiki/Stochastic_control) problems, where in general, to obtain a candidate for an optimal control, one needs to solve a coupled forward-backward stochastic differential equation, where the backward components influence the dynamics of the forward component.
+Moreover, it must be mentioned that we are avoiding the regimes where the Navier-Stokes equations become chaotic and turbulent (e.g., as the Reynolds number increases). In fact, it should not be difficult for a plain vanilla neural network to approximate the types of complicated functions that naturally appear in turbulence. However, as we compute the derivatives required in the computation of the physics informed neural networks, minimizing the loss functions might become a challenge, where the optimizer may fail to converge to the right values for the parameters of the neural networks. It might be the case that the resulting optimization problem inherits the complicated nature of the turbulent Navier-Stokes equations. Hence, inference of turbulent velocity and pressure fields should be considered in future extensions of this line of research. Moreover, in this work we have been operating under the assumption of Newtonian and incompressible fluid flow governed by the Navier-Stokes equations. However, the proposed algorithm can also be used when the underlying physics is non-Newtonian, compressible, or partially known. This, in fact, is one of the advantages of our algorithm in which other unknown parameters such as the Reynolds and Péclet number numbers can be inferred in addition to the velocity and pressure fields.
 
 * * * * *
 
 **Acknowledgements**
 
-This work received support by the DARPA EQUiPS grant N66001-15-2-4055 and the AFOSR grant FA9550-17-1-0013. All data and codes are publicly available on [GitHub](https://github.com/maziarraissi/FBSNNs).
+This work received support by the DARPA EQUiPS grant N66001-15-2-4055 and the AFOSR grant FA9550-17-1-0013. All data and codes are publicly available on [GitHub](https://github.com/maziarraissi/DeepVIV).
 
 * * * * *
 ## Citation
 
-	@article{raissi2018forwardbackward,
-	  title={Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations},
-	  author={Raissi, Maziar},
-	  journal={arXiv preprint arXiv:1804.07010},
+	@article{raissi2018deepVIV,
+	  title={Deep Learning of Vortex Induced Vibrations},
+	  author={Raissi, Maziar and Wang, Zhicheng and Triantafyllou, Michael S and Karniadakis, George Em},
+	  journal={arXiv preprint arXiv:1808.08952},
 	  year={2018}
 	}