Physics Informed Machine Learning of SPH: Machine Learning Lagrangian Turbulence

Smoothed particle hydrodynamics (SPH) is a mesh-free Lagrangian method for obtaining approximate numerical solutions of the equations of fluid dynamics; which has been widely applied to weakly- and strongly compressible turbulence in astrophysics and engineering applications. We present a learn-able hierarchy of parameterized and "physics-explainable" Lagrangian based fluid simulators using both physics based parameters and Neural Networks (NNs) as universal function approximators. This hierarchy of parameterized Lagrangian models gradually introduces more SPH based structure, which we show improves interpretability, generalizability (over larger ranges of time scales and Mach numbers), preservation of physical symmetries (corresponding to conservation of linear and angular momentum), and requires less training data. Our learning algorithm develops a mixed mode approach, mixing forward and reverse mode automatic differentiation with local sensitivity analyses to efficiently perform gradient based optimization. We train this hierarchy on both weakly compressible SPH and DNS data, and show that our physics informed learning method is capable of: (a) solving inverse problems over the physically interpretable parameter space, as well as over the space of NN parameters; (b) learning Lagrangian statistics of turbulence (interpolation); (c) combining Lagrangian trajectory based, probabilistic, and Eulerian field based loss functions; (d) extrapolating beyond training sets into more complex regimes of interest; (e) learning new parameterized smoothing kernels better suited to weakly compressible DNS turbulence data.

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