Learning Flame Evolution Operator under Hybrid Darrieus Landau and Diffusive Thermal Instability

TL;DR

The paper addresses learning the time-advance operator for flame evolution under hybrid Darrieus-Landau and Diffusive-Thermal instabilities by formulating the problem as parametric operator learning with a two-parameter family $(\rho,\beta)$ in the Sivashinsky equation. It compares parametric Fourier Neural Operators (pFNO) and parametric CNNs (pCNN) for learning the forward operator across DL/DT blends, showing that $p$FNO variants deliver superior short-term accuracy and robust long-term statistics, while pCNN excels only in restricted parameter settings. The study demonstrates that the full two-parameter operator can be learned (with $p$FNO$^*$ variants) and that dispersion relations and auto-correlation metrics align well with ground truth, validating operator-learning as a viable framework for complex nonlinear PDEs in combustion. Despite success, both approaches tend to overestimate noise-induced wrinkles at low $\beta$, highlighting a key area for future improvement and model refinement.

Abstract

Recent advancements in the integration of artificial intelligence (AI) and machine learning (ML) with physical sciences have led to significant progress in addressing complex phenomena governed by nonlinear partial differential equations (PDE). This paper explores the application of novel operator learning methodologies to unravel the intricate dynamics of flame instability, particularly focusing on hybrid instabilities arising from the coexistence of Darrieus-Landau (DL) and Diffusive-Thermal (DT) mechanisms. Training datasets encompass a wide range of parameter configurations, enabling the learning of parametric solution advancement operators using techniques such as parametric Fourier Neural Operator (pFNO), and parametric convolutional neural networks (pCNN). Results demonstrate the efficacy of these methods in accurately predicting short-term and long-term flame evolution across diverse parameter regimes, capturing the characteristic behaviors of pure and blended instabilities. Comparative analyses reveal pFNO as the most accurate model for learning short-term solutions, while all models exhibit robust performance in capturing the nuanced dynamics of flame evolution. This research contributes to the development of robust modeling frameworks for understanding and controlling complex physical processes governed by nonlinear PDE.

Figures (10)

• Figure 1: The parametric CNN model adopted in this study is demonstrated for input function $v(x_i)$ discretized at 1D mesh of 256 points, with $L=6$ levels of encoding and decoding. Standard convolution layers are represented by gray rectangles, while Inception layers are depicted in magenta. The output data channels $c_l$ for each convolution layer are indicated within brackets.
• Figure 2: Dispersion relations (left) and relevant parameter values at prescribed values of $\rho$ and $\beta$.
• Figure 3: Left three columns: Comparison of reference dispersion relations (Eq. \ref{['eq:dRel_ana']}, solid lines) with those computed for the learned operators of all models (Eq. \ref{['eq:dRel_model']}, non-solid lines), where line colors indicate different $\rho$. Right column: Illustration of a learned operator Jacobian (Eq. \ref{['eq:Operator_Jacobian']}), with dark colors indicating small values.
• Figure 4: Long-term solutions of flame front displacement $\phi(x,t)$ at $\beta=10$ and $\rho = [0,1/4,1/2,3/4,1]$ (from top to bottom row). Black reference solutions to Eq. \ref{['eq:MKS']} obtained using high-order numerical methods are compared against predictions by the operator-learning methods of pFNO* (red) and pCNN (cyan). The left and right columns correspond to two randomly initialized solution sequences, each showing eleven snapshots of $\phi(x,t)$ at $t/0.15$ = 0, 50, 125, 250, 500, 750, 1000, 1250, 1500, 1750 and 2000. A time shift ($t/15$) is added to the displayed fronts to avoid overlap.
• Figure 5: Comparison of flame front displacement predicted by pFNO* and pCNN40 at $\beta=40$. Other details are the same as in Fig. \ref{['fig:disp_MKS_beta10']}.