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docs/index.md

@@ -154,6 +154,14 @@ As for the activation functions, we use $$\sin(x)$$. In general, the choice of a
 
 Let us now consider the case where we do not have access to direct measurements of the lift force $$f_L$$. In this case, we can use measurements of the velocity field, obtained for instance via [Particle Image Velocimetry](https://en.wikipedia.org/wiki/Particle_image_velocimetry) (PIV) or [Particle Tracking Velocimetry](https://en.wikipedia.org/wiki/Particle_tracking_velocimetry) (PTV), to reconstruct the velocity field, the pressure, and consequently the drag and lift forces. A representative snapshot of the data on the velocity field is depicted in the top left and top middle panels of the following figure. The neural network architectures used here consist of 10 layers with 32 neurons in each hidden layer. 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 velocity field; however, a more intriguing result stems from the network's ability to provide an accurate prediction of the entire pressure field $$p(t,x,y)$$ in the absence of any training data on the pressure itself (see the following figure). A visual comparison against the exact pressure is presented in the following figure for a representative snapshot of the pressure. It is worth noticing that the difference in magnitude between the exact and the predicted pressure is justified by the very nature of incompressible Navier-Stokes equations, since the pressure field is only identifiable up to a constant. This result of inferring a continuous quantity of interest from auxiliary measurements by leveraging the underlying physics is a great example of the enhanced  capabilities that our approach 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_data_on_velocities_results.png)
+> _VIV-I (Velocity Measurements):_ A representative snapshot of the data on the velocity field is depicted in the top left and top middle panels of this figure. The algorithm is capable of accurately reconstructing the velocity field and more importantly the pressure without having access to even a single observation on the pressure itself. 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.
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+The trained neural networks representing the velocity field and the pressure can be used to compute the drag and lift forces by employing 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 the first loss function intoduced in the current work. Upon training, the proposed framework is capable of identifying the correct values for the structural parameters $$b$$ and $$k$$ with remarkable accuracy. The learned values for the damping and stiffness parameters are $$b = 0.0844064$$ and $$k = 2.1938791$$. This corresponds to around $$0.48\%$$ and $$0.37\%$$ relative errors in the estimated values for $$b$$ and $$k$$, respectively.
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+![](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.
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