INCOMPLETE TO COMPLETE MULTIPHYSICS FORE-CASTING -A HYBRID APPROACH FOR LEARNING UN-KNOWN PHENOMENA

Abstract

Modeling complex dynamical systems where only partial knowledge of their physical mechanisms is available is a crucial problem across all scientific and engineering disciplines. Purely data-driven approaches, which only make use of an artificial neural network and data, often fail to accurately simulate the evolution of the system dynamics over a sufficiently long time and in a physically consistent manner. Therefore, we propose a hybrid approach that uses a neural network model in combination with an incomplete PDE solver that provides known, but incomplete physical information. In this study, we demonstrate that the results obtained from the incomplete PDEs can be efficiently corrected at every time step by the proposed hybrid neural network -PDE solver model, so that the effect of the unknown physics present in the system is correctly accounted for. For validation purposes, the obtained simulations of the hybrid model are successfully compared against results coming from the complete set of PDEs describing the full physics of the considered system. We demonstrate the validity of the proposed approach on a reactive flow, an archetypal multi-physics system that combines fluid mechanics and chemistry, the latter being the physics considered unknown. Experiments are made on planar and Bunsen-type flames at various operating conditions. The hybrid neural network -PDE approach correctly models the flame evolution of the cases under study for significantly long time windows, yields improved generalization, and allows for larger simulation time steps.

1. INTRODUCTION

Modeling and forecasting of complex physical systems described by nonlinear partial differential equations (PDEs) are central to various domains with applications ranging from weather forecasting (Kalnay, 2003) , design of airplane wings (Rhie & Chow, 1983) , to material science (Wheeler et al., 1992) . Typically, a chosen set of PDEs are solved iteratively until convergence of the solution. Modeling complex physical dynamics requires a good understanding of the underlying physical phenomena. For cases where the complete physics information is missing, deep learning models can be employed to complete the physical description when additional data of the system is available. Deep learning methods have shown promises to account for these unknown components of the system (Yin et al., 2021) . We consider a set of partial differential equations with partially unknown physics represented. The corresponding PDE model for a general state ϕ is given by ∂ϕ ∂t = P i (ϕ, ∂ϕ ∂x , ∂ 2 ϕ ∂x 2 , ...) + S ϕ , where P i models the known but incomplete PDE description, S ϕ represents the unknown physics. Within the scope of this model, the influence of S ϕ can lead to fundamentally different scenarios. A commonly targeted case is when the governing equations of the complete PDE description are computationally too expensive to solve, turbulence modeling in computational fluid dynamics (CFD) being a good example. In CFD, a spacial filtering is performed on the original governing PDEs. This step introduces unclosed terms in the model equations that correspond to unrepresented physics in equation 1, due to the effects of the filtered scales. This is a widely studied problem, where the use of deep learning models is currently being explored (Lapeyre et al., 2019; Kochkov et al., 2021; List 

