deep learning for fluid mechanics using PINNs

A presentation on deep learning

1. Abstract: To simulate fluid dynamics problems we use mostly experimental methods and numerical simulating software. Numerical simulation methods are sensitive to mesh size, therefore , the cost involved is also very high. Physics-Informed Neural Networks (PINNs) is used to solve any supervised machine learning problems. PINNs is used to solve any PDEs/ODEs which may be of either a linear or nonlinear form. There are two categories of problems that we encounter in practice: Data-driven solution of PDEs Data-driven discovery of PDEs Based on the availability of the data, there are mainly two types of models, which are continuous-time models and discrete-time models.

2. Introduction: R ecent advancements in machine learning algorithms and data sciences have produced significant results for various scientific fields such as image recognition. Computational fluid dynamics (CFD) approaches like finite element methods (FEM), finite volume methods (FVM), and finite difference methods (FDM) are used to solve any fluid flow simulation problems . The above discretization schemes usually computes the values of a PDE at a set of grid points which are distributed over spatial and temporal domain. PINNs together with automatic differentiation (autodiff) do not require mesh generation to solve fluid dynamics problems. PINNs incorporate the governing equation loss, initial condition loss, and boundary condition loss into the Loss function. Different optimization schemes like ADAM or SGD is used to optimize the loss function. T raining of PINNs is time-consuming and prone to errors because we have to tune the weights and biases. It is possible to obtain a locally optimal solution if we use Gradient Descent to optimize the loss function.

3. Problem Set Up: We model the fluid dynamics by the three-dimensional NS equations with the Newtonian assumption:   , Continuity equation ,   where x = [x,y, z] ∈ Ω and t ∈ [0,T] denote space and time coordinates. The scalar h(x) is the initial constraints. u(x,y, z,t), v(x,y, z,t), and w(x,y, z,t) represent the x,y, and z component of the velocity field. p(x,y, z,t) represents the pressure.  

4. The solution of Navier-Stokes equations could be approximated by neural network û(x,t;θ). We consider the feed-forward fully connected neural network of depth M, which takes the input x,y, z,t and denote the output of the m−th layer as û [m] . The neural network can be defined as: where σ is the activation function including sigmoid, relu , and tanh. W [m] and b [m] represent the weights and biases at m−th layer.

5. The residual is defined as : We use automatic differentiation to find the partial derivatives. Different types of loss term are calculated as follows:

6. Then the total loss term is calculated as follows: where ω = {ω f ,ω b ,ω i ,ω d } are the hyperparameters which is tuned during training of neural network. L PDE penalizes the residual for the loss in the governing equations. L data penalizes the loss in the data. and L IC and L BC implies the loss in the initial condition and boundary condition respectively. The parameters N f ,N b ,N i ,N data are the total number of points. The best parameter collection θ ∗ is identified by minimizing multiple loss functions as given above.

7. PINNs for the incompressible Navier-Stokes equations algorithm is summarized as follows: Fig. 1 A schematic diagram of PINNs for solving N-S equation

8. Example: I have used following conditions to solve N-S equation using PINNs: Viscosity of fluid: µ = 0.02kg · (m · sec) -1; Density if fluid: ρ = 1kg / m 3 ; At Inlet u(0,y) = u(inlet) = y(H − y)*( 4/H2) and v(0, y) = 0 ; No slip Condition @ wall and cylinder surface ; Pressure of outlet equals to zero i.e. p(1,y) = 0 Continuity Equation (for incompressible flow): ∇ · V = 0 Steady N-S equation : ρ. (V · ∇)V = −∇p + µ∇ 2 V Or it can be expressed as Cauchy momentum equation ρ(V · ∇)V = ∇ · σ ; where σ = −p*I+ µ(∇V+ (∇V) T ) Fig. 2 A Schematic Diagram of Flow over a Cylinder

9. After solving the Navier- Stokes equation, for the given case, using PINNs we have found the variation of u(x,y) , v(x,y) and p(x,y) as follows: Model 1: Fig. 3 Variation of velocity and pressure with x direction for model 1.

10. After solving the Navier- Stokes equation, for the given case, using PINNs we have found the variation of u(x,y) , v(x,y) and p (x,y) as follows: Model 2: Fig. 4 Variation of velocity and pressure with x direction for model 2.

11. After solving the Navier- Stokes equation, for the given case, using PINNs we have found the variation of u(x,y) , v(x,y) and p(x,y) as follows: Model 3: Fig. 5 Variation of velocity and pressure with x direction for model 3.

12. After solving the Navier- Stokes equation, for the given case, using PINNs we have found the variation of u(x,y) , v(x,y) and p(x,y) as follows: Model 4: Fig. 6 Variation of velocity and pressure with x direction for model 4

13. References: Jin, X., Cai, S., Li, H., Karniadakis, G.E.: NSFnets (Navier-Stokes flow nets): Physics-informed neural networks for the incompressible Navier-Stokes equations. Journal of Computational Physics 426, 109951 (2021) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Physics informed learning machine (2021). US Patent 10,963,540 Raissi, M., Perdikaris, P., Karniadakis, G.E.: Machine learning of linear differential equations using Gaussian processes. Journal of Computational Physics 348, 683–693 (2017) Raissi, M., Perdikaris, P., Karniadakis, G.E.: Numerical Gaussian processes for time-dependent and nonlinear partial differential equations. SIAM Journal on Scientific Computing 40(1), A172– A198 (2018)

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