Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution

TL;DR

The paper addresses accelerating PDE simulations under varying parameters by learning time-advancement operators with parametric neural operators. It develops two main approaches, parametric CNN (pCNN) and parametric Fourier Neural Operator (pFNO), including a baseline, a parametric extension, and a simple variant (pFNO*), and tests them on 1D Michelson-Sivashinsky and Kuramoto-Sivashinsky equations as well as 2D DNS-flame data. Across 1D and 2D flames, the methods demonstrate strong short-term prediction accuracy and meaningful replication of long-term statistics, with KS dynamics favoring pFNO for extended predictions and MS dynamics showing regime-dependent performance. The results highlight the potential for parametric operator learning to reduce computational cost in engineering simulations while preserving essential dynamic and statistical properties, though challenges remain for long-term fidelity in noise-sensitive regimes and data-limited 2D scenarios.

Abstract

This study investigates the application of machine learning, specifically Fourier Neural Operator (FNO) and Convolutional Neural Network (CNN), to learn time-advancement operators for parametric partial differential equations (PDEs). Our focus is on extending existing operator learning methods to handle additional inputs representing PDE parameters. The goal is to create a unified learning approach that accurately predicts short-term solutions and provides robust long-term statistics under diverse parameter conditions, facilitating computational cost savings and accelerating development in engineering simulations. We develop and compare parametric learning methods based on FNO and CNN, evaluating their effectiveness in learning parametric-dependent solution time-advancement operators for one-dimensional PDEs and realistic flame front evolution data obtained from direct numerical simulations of the Navier-Stokes equations.

Figures (13)

• Figure 1: The parametric CNN is derived from the convolutional auto-encoder archetype by extending its encoder block, with the added components highlighted through linking with red lines. The sketch is demonstrated for a 2d input discretized as an image ($v(x_j)\in\mathbb{R} ^{1\times 256\times 256}$) with $L$=6 levels of encoding. The channel number $c_l$ is shown on top in a bracket and the image size $N_l$ is on the bottom. Dashed lines refer to skip connection. Max-Pooling and upsampling are used to shrink and increase image size, respectively. All gray blocks are implemented using a standard convolution layer (of filter size 3, stride 1, and periodic padding which enforces periodic boundary condition), and the magenta block is implemented by an Inception layerinception.
• Figure 2: Illustration of the parametric extension of the Fourier Neural Operator (pFNO), highlighting the extended components with red solid lines connecting to the original FNO parts. The top-right inset provides a zoomed view of the second map inside the function $D^*$.
• Figure 3: Long-term solutions of the 1D MS equation \ref{['eq:MS']} for front displacement $\phi(x,t)$ at four different parameters $\nu \in [0.025,0.035,0.07,0.15]$ (from top to bottom row). Reference solutions obtained using high-order numerical methods are represented by black dashed lines, while predictions from parametric operator learning methods pFNO* and pCNN are denoted by red solid and cyan long-dash lines, respectively. Each pair of solution sequences, displayed in the left and right columns, starts from differently randomized initial fronts. The sequences include eleven snapshots of $\phi(x,t_j)$ with $t_j=j\Delta_t$ at $j\in[0,50,125,250,500,750,1000,1250,1500,1750,2000]$ and a fixed time interval $\Delta_t$=0.015. A time shift ($t/100$) is applied to displayed fronts to reduce overlap.
• Figure 4: Front slope of $\partial_x \phi(x,t)$ calculated from a single instance reference solution (1st column) of the 1D MS equation \ref{['eq:MS']} (at four different parameters $\nu \in [0.025,0.035,0.07,0.15]$, from top to bottom row) is compared against the predictions by pFNO*, pFNO, and pCNN shown in the 2nd, 3rd, and 4th column, respectively. The rainbow color map represents values from negative to positive.
• Figure 5: Normalized total front length (i.e. $\int_\mathcal{D} (\phi_x^2 +1 )^{1/2} dx /\int_\mathcal{D} dx$) calculated from two random instances(shown in left and right column respectively) of reference solutions(black dash line) to the 1d MS equation (for each of six parameter values $\nu$ shown from top to bottom row) are compared against the corresponding predictions by three networks of pFNO* (red solid line), pFNO(blue dash-dot line) and pCNN (cyan dash line).