Deep learning of nonlinear flame instabilities

A typical example is the Darrieus-Landau (DL) instability of deflagration flame fronts which is a fundamentally important hydrodynamic phenomenon emerging in reacting system of combustion, inert confinement fusion and thermonuclear supernovae. Similar instability has also been found in advanced materials involving fronts transformation under electrical and magnetic fields, e.g., doping fronts in organic polymer semiconductor and spin-avalanches in nanomagnets crystals.

The DL instability arises due to thermal expansion across a flame front. A propagating flame alters the flow, leading to spontaneous wrinkling of the originally smooth flame front. The destabilized front continues to evolve in a nonlinear fashion, confined by the surrounding flow domain. The nonlinear behavior of the flame front can be demonstrated through a simple experiment of free flame propagation in a channel with periodic walls. If the channel width Λ is above a critical value Λc,  the DL instability becomes active and the flat front transforms into a steady cusp shape (Fig. 1.a). As the channel width increases, the curved front becomes prone to disturbance, resulting in small wrinkles traveling along the main curved front (Fig. 1.b). At sufficiently wide channels, the nonlinear evolution of the DL instability yields a cellular front structure with a fractal cascading pattern, as seen in Fig. 1.c where smaller cells reside on larger ones. At times, the complex front evolution results in pockets detaching from the main flame front.

In the study by Yu the data-driven approach using deep neural networks was applied to understand the nonlinear evolution of DL instability. Herein, operator-regression methods were utilized that learn the relationships between partial differential equations (PDEs) and their corresponding solutions. The key focus was to learn a time advancement operator that maps a flame front at one time to its new position at a future time, after a fixed time interval. By repeatedly applying this operator, it is possible to predict the sequence of flame fronts for an arbitrary length of time. The ultimate goal was to accurately reproduce the nonlinear behavior of the flame fronts through long-term predictions. This was accomplished by two state-of-the-art methods, the Fourier Neuron Operator (FNO) and the Deep Convolution Neural Network (CNN). The neural networks were able to accurately predict the front evolution in small channels, starting from random initial conditions Fig.2. For the noise-sensitive front evolution in wide channels, the networks were able to capture the pattern of minor cusps emerging along the main front of a giant cusp shape Fig.2.  Even in the case of complex front evolution in very large channels, the networks were able to reproduce intricate phenomena such as a fractal cascade of front structures and detached front pockets.

This work highlights the potential of deep neural network approaches in solving complex dynamical problems that can significantly reduce the computational demands associated with simulating the physical phenomena and demonstrates the potential of AI based approaches for tackling real-life problems in physical and engineering systems. In collaborative effort involving academia (Lund University), Center for applied AI at RISE and several industrial partners our aim is to investigate these AI-based approaches for industrial applicability.

Written by: Rixin Yu, Erdzan Hodzic, David Eklund

Top image: Characteristic temperature snapshot of a flame front after long time development in a periodic ‘channel’ of width Λ =3.5Λc (a) Λ =15.1Λc (b), and Λ =90.1Λc (c).  White arrows indicate characteristic steps in fractal cascade.  These results were obtained from a large-scale physical simulation costing ~2 million CPU-core hours. 

Small image: Comparison of temporal evolution of flame front slope: results from analytical references and neural network predictions. The figure displays the flame front slope as a function of time in both a wide channel (top two rows) and a small channel (bottom two rows). The two columns represent two separate simulations, each starting from a different random initial condition. The rainbow color map displays the slope values, with negative slopes represented in blue and positive slopes in red. The phenomenon of early branch merging (at times t/Δ_t < 500) is referred to as the inverse front cascade, and it is well known that the flame front evolution in wide channels is particularly sensitive to noise.