A domain decomposition-based autoregressive deep learning model for unsteady and nonlinear partial differential equations

TL;DR

The paper addresses the challenge of efficiently simulating unsteady and nonlinear PDEs at scale by introducing transient-CoMLSim, a domain-decomposition DL framework that operates on subdomain-local latent spaces. It combines a CNN-based autoencoder to learn low-dimensional representations with an autoregressive time integrator trained using curriculum learning, enabling stable long-horizon rollouts in latent space. Through four diverse datasets, it demonstrates superior interpolation and extrapolation accuracy compared to U-Net and FNO, and shows favorable generalization to larger domain sizes, including a 3D additive manufacturing scenario. The work offers a scalable, out-of-distribution-friendly approach for physics-informed DL surrogates and outlines concrete future improvements like Runge-Kutta time-stepping and latent-space boundary conditions.

Abstract

In this paper, we propose a domain-decomposition-based deep learning (DL) framework, named transient-CoMLSim, for accurately modeling unsteady and nonlinear partial differential equations (PDEs). The framework consists of two key components: (a) a convolutional neural network (CNN)-based autoencoder architecture and (b) an autoregressive model composed of fully connected layers. Unlike existing state-of-the-art methods that operate on the entire computational domain, our CNN-based autoencoder computes a lower-dimensional basis for solution and condition fields represented on subdomains. Timestepping is performed entirely in the latent space, generating embeddings of the solution variables from the time history of embeddings of solution and condition variables. This approach not only reduces computational complexity but also enhances scalability, making it well-suited for large-scale simulations. Furthermore, to improve the stability of our rollouts, we employ a curriculum learning (CL) approach during the training of the autoregressive model. The domain-decomposition strategy enables scaling to out-of-distribution domain sizes while maintaining the accuracy of predictions -- a feature not easily integrated into popular DL-based approaches for physics simulations. We benchmark our model against two widely-used DL architectures, Fourier Neural Operator (FNO) and U-Net, and demonstrate that our framework outperforms them in terms of accuracy, extrapolation to unseen timesteps, and stability for a wide range of use cases.

Figures (26)

• Figure 1: Schematic of the transient-CoMLSim framework for a 2-D problem setup. $\mathbf{E}^s$ and $\mathbf{E}^c$ are solution and condition encoders. $\mathbf{D}^s$ refers to the solution decoder. $\mathbf{TI}$ is the time-integrator network. $\eta^s$ and $\eta^c$ correspond to solution and condition latent embeddings. Model predicts upto $T$ timesteps into the future. For 3-D problem setup, the setup can be easily extended to include 7 neighbors (left, right, top, bottom, front, back).
• Figure 2: Schematic of the structured CNN-based autoencoder used to encode the physical fields to the latent space $\eta$ and to decode back from the latent space $\eta$ to the physical space. Encoding and decoding of condition variables are performed in a similar fashion. For 3-D datasets, a 3-D CNN is used.
• Figure 3: Schematic of the time integrator operating in the latent space $\eta$.
• Figure 4: Evolution of nRMSE as a function of timesteps for the four datasets. For panels $(a)-(c)$, all models are trained until $T=100$ timesteps and are inferred from $t=11$ to $100$ on the test samples. In panel $(d)$, models are trained for 400 timesteps and inferred until the end of the time horizon for the respective simulation. As mentioned in the text, we were unable to fit FNO or U-Net on the additive manufacturing dataset using our compute resources. Therefore, panel $(d)$ only shows the nRMSE evolution for the transient-CoMLSim.
• Figure 5: Randomly selected three test samples at $t=100$ randomly selected from 2-D shallow water dataset.