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 et al., 2022; Stachenfeld et al., 2022) . In the following, we are targeting a more challenging problem, where increasing spacial resolution and/or reducing time-scales of the incomplete PDE solver does not lead to a converged full solution. Rather, the incomplete and complete PDEs produce drastically different solutions due to the unknown physics. The central learning objective is to correct this behavior and retrieve the evolution that would be obtained with the complete PDE description. Our work expands on the combination of incomplete PDE solvers and neural networks (NNs) (Yin et al., 2021; Takeishi & Kalousis, 2021) to account for the effects of an incomplete physics model. The NN aims to complete the PDE description, where the differences in complete and incomplete PDE solutions are beyond the effects of spacial and temporal scales. We showcase that combining the trained NN model with a differentiable solver for the incomplete PDE can accurately reproduce the physical solutions of the complete, multi-physics PDE solver with stable long-term rollouts. We demonstrate the capabilities of this approach for an archetypal multi-physics system, namely a complex reactive flow which combines fluid mechanics and chemistry, where the latter is considered unknown. Reactive flow modeling has applications in numerous domains such as combustion processes in gas turbines (Lieuwen, 2012), climate modeling (Jacobson, 1999; Rolnick et al., 2022) and astrophysics simulations (Gamezo et al., 2003) . Resolving the Navier-Stokes equations lies at the core of these problems, where additionally the transport of different species of relevance must also be accounted for, together with their production or consumption often following complex reaction mechanisms (Poinsot & Veynante, 2005) . For chemically reacting flows, generation or consumption of multiple species via some chemical reaction are modeled using a net source term. It is a well-known fact that the incorporation of a detailed chemical kinetic mechanism in a reacting flow model can result in a stiff system of governing equations (Wanner & Hairer, 1996; Najm et al., 1998; Knio et al., 1999) . We consider the reactive flow simulation to be the complete PDE description, while the non-reactive flow simulation represents the incomplete PDE basis, where the chemical kinetic mechanisms are collected in the unknown physics term of equation 1. Figure 1 shows a visual example of the fundamental differences in system dynamics that can be caused by unknown reaction terms. We showcase the effectiveness of our approach for different cases of planar 2D premixed methane-air flames, and the varying transient evolution of Bunsen-type flames. We show that the proposed approach can handle large domains with highly resolved flames, which is closer to the practical flame domains used in many industrial applications. Specifically we concentrate on training a NN model to correct the spatio-temporal effects of energy and species transport source terms. We show that in addition to recovering the desired solutions, this approach overcomes inherent problems of temporal stiffness due to the complex reaction mechanism. Lastly, we demonstrate the applicability of neural network model to control the flow inlet velocities to arrive at desired flame shapes, when combined with a differentiable flow solver.

2. RELATED WORK

Deep learning methods have been widely used to model the solutions of partial differential equations (Lagaris et al., 1998; Long et al., 2018; Han et al., 2018; Bar-Sinai et al., 2019) and in particular, the Navier-Stokes equations (Thuerey et al., 2020; Fukami et al., 2019) . These models can be very fast and do not suffer from the time-step stability issues associated with traditional numerical solvers. Nevertheless, as these purely data-driven approaches lack the physical understanding of the system being modeled, they generally fail in generalizing to other operating conditions (Kim et al., 2019; Lapeyre et al., 2019) . To leverage the potential of deep learning in physical simulations, it is therefore necessary to incorporate some physical information within the deep learning framework.



Figure 1: The incomplete/non-reactive (top) and complete/reactive (bottom) PDE solvers we consider can yield fundamentally different evolutions, as shown here for a sample temperature field over time.

