We wish to work towards creating autonomous PDE solvers – solvers that will require zero to minimal interventions from humans in the loop. Three fundamental problems preclude current solvers from autonomy (a) Lack of robust discretization methods (b) Lack of automatic modeling schemes and (c) Lack of learning methods from existing solvers. Within the scope of this project, we wish to address the first problem in the context of PDE solvers for Navier-Stokes equations.
Modern day PDE solvers face severe challenges in robustness while solving fluid flows. Here, problems occur across several Mach number regimes for compressible flow solvers, there are eternal problems in modeling near wall regions in turbulence, interfacial phenomena create their own problems, etc.
Traditionally, three solutions have been offered for this – (a) Increase the number of mesh points, (b) Increase the order of the method (c) Add extra smoothening/stabilizing terms. The first two come at higher computational cost and only sometimes alleviate the problem at hand. The third (smoothening) is almost always ad-hoc and needs a “horses for courses” approach – there are separate tricks that a computational engineer needs to hand-craft for each flow situation.
The above situation is reminiscent of the situation with image recognition where specific hand-crafted filters were required to be designed for specific problems. Recently, Machine Learning techniques have proved to be extremely powerful in removing the requirement for hand crafted representations and being able to automatically compute features and filters that will work across a large range of problems.
We aim to exploit this and other parallels to create a framework to automatically generate robust high-order numerical methods for any given flow situation. Towards this end, we wish to evaluate four different approaches (a) Using Convolutional Neural Networks (CNN) based numerical methods (b) Using Gaussian Processes and the Field Inversion Machine Learning approach to find appropriate diffusion terms (c) Using Deep Neural Networks to determine Entropy Viscosity for compressible Navier-Stokes (d) Deriving directly a nonlinear, Deep RNN basis based nonlinear numerical method which gives the additional advantage of a meshless method. Within the scope of this project, the feasibility of the above approaches is to be evaluated for simple canonical problems.