An article in the latest edition of Science magazine, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, describes an emerging branch of fluid dynamics at the intersection of scientific computing and deep learning. It starts with imaging data and discovers the underlying physics and/or properties of the fluid using the partial differential equations (PDEs) describing the flow behavior. This method, called inverse approach, is used for predicting the flow velocities and pressures for the entire transient range without directly measuring them and promises advances for a wide range of applications from preventing strokes to oil & gas exploration studies.

Another approach to AI can accurately predict phenomenon, such as lift and drag forces in aerodynamics, or the temperature and pressure drop in cooling of electronic devices and involves forward solution of the PDEs for the underlying physics. This is different from the data driven approach used by standard neural networks which need a large number of datasets for learning.

NVIDIA is already using physics-informed neural networks (PINNs) in SimNet, architected for problems requiring either inverse approach or forward solution like traditional numerical solvers with use cases such as the design of heat sinks for its DGX Systems powered by the revolutionary Volta GPU platform. These networks require no data, can work with single or parameterized geometries and solve single or multi-physics problems. At the backend, NVIDIA GPUs are used for both training and inference, with the cuDNN-accelerated TensorFlow deep learning framework.

The inverse approach in SimNet has its roots in the work done at Brown University and was first outlined in, Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations, on Science. For any kind of simulation, validation and verification are important for analysis to be trustworthy. For the flow in an Intracranial Cerebral Aneurysm, the predicted velocities and pressures using the physics driven neural networks (called learned) are in the same range as the results using a traditional open source CFD solver (called exact) as shown in Fig. 1. It is worth mentioning that two different CFD solvers can yield different results for the same use case due to different discretization schemes and numerical algorithms (and in fact, even the same solver with two different users can yield different results). Therefore, it is reasonable to expect that the neural network results would be similar to the traditional solvers but not exactly the same. This is demonstrated in Fig. 2, where SimNet results are compared to two different open source CFD solvers. The SimNet results have a closer correlation with the CFD solver that uses spectral h/p method than with the one using finite volume method.

As an extension of the current work in non-Newtonian fluids, the neural networks could also discover the relationship between the viscosity and shear strain rate. The resulting neural network could then be called from within the solver instead of using the existing empirical functions like Carreau or Casson models.

Companies such as HeartFlow are already reviewing the techniques to see how they can leverage them. The startup, part of NVIDIA’s Inception program, uses AI in a non-invasive alternative to angiograms to help physicians diagnose and treat coronary heart disease.

**From Heart Attacks to Heat Sinks**

As mentioned above, the inverse approach takes advantage of some measured sensor output, acoustic or imaging data for the neural networks to discover the physics using relevant PDEs. However, many use cases may not have any measured data. In such cases, one would need to do forward simulations. for multiple parameters or geometries to examine the best design.

Nvidia engineers have extended SimNet for forward solution of parameterized, multi-physics problems. Starting with only the geometry and other physical parameters like material properties, boundary conditions etc., SimNet evaluates the entire range of possibilities just like the traditional solver would. However, the big difference is while the traditional solver would require a separate run for each design, the neural networks can address the various different designs in one training cycle. Once the network is trained, the individual design can be evaluated through inference which computes in nearly real time. This approach holds advantage over both data driven neural networks (that require lots of data and do not generalize well) as well as traditional solvers (which are computationally expensive).

For the engineering physics problems, there are several considerations for the neural networks that must be accounted for in the neural network architecture, such as – sampling insensitivity, impact of the order of derivatives on the network structure, weighting the various PDEs for loss convergence acceleration, activation functions that do not reduce down to constants or vanish when differentiated, gradients and discontinuities due to geometrical effects and considerations of local versus global mass balance equations. Therefore, standard networks that are driven by the data alone are grossly inadequate for modeling physics on various geometries.

For validation, a forward solution of a Lid Driven Cavity was evaluated using SimNet. As shown in Fig. 3, SimNet results compare very well with a traditional solver with errors in the u and v components of velocity being 0.2% and 0.4% respectively.

SimNet is used to improve the design and effectiveness of heat sinks where thousands of configurations can be analyzed within hours as opposed to weeks with the traditional simulations.

Figs. 4A & B show good agreement of SimNet results with the traditional solver using conjugate heat transfer. Next, SimNet is used for the design optimization of heat sinks where the minimum value on the temperature surface (red & yellow) is chosen for a given value on the pressure drop surface (blue) as shown in Fig. 5.

As shown in Table 1, even 50 dimensional variations for each of the two parameters, edge fin height and center fin height, result in 2500 design evaluations. The computational time for the traditional simulations is intractable, but the neural networks offer the possibility to analyze a full spectrum of designs.

NVIDIA CEO Jensen Huang demonstrated the work in a talk at the SC19 supercomputing conference in November. “It has the potential to simulate much larger models and do so more accurately” than current techniques, said Huang.

*Look for the SimNet demo at Nvidia Booth at GTC 2020.*