Meta-Auto-Decoder: A Meta-Learning Based Reduced Order Model for Solving Parametric Partial Differential Equations

TL;DR

This paper addresses the challenge of efficiently solving parametric PDEs by moving beyond linear reduced-order models to nonlinear, mesh-free representations. It introduces Meta-Auto-Decoder (MAD), which learns a nonlinear trial manifold via an auto-decoder architecture and encodes heterogeneous PDE parameters into latent vectors, enabling fast adaptation to new tasks through MAD-L (latent search) or MAD-LM (joint fine-tuning). The work formalizes decoder width as a theoretical measure of the best possible nonlinear approximation and provides extensive numerical experiments on Burgers', Maxwell, Laplace, and Helmholtz equations, demonstrating faster convergence and competitive accuracy versus strong baselines. The approach offers practical benefits for real-time and many-query PDE solving, with robustness to heterogeneous domains and parameters and an interpretable connection to manifold learning concepts.

Abstract

Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc. Typical reduced order modeling techniques accelarate solution of the parametric PDEs by projecting them onto a linear trial manifold constructed in the offline stage. These methods often need a predefined mesh as well as a series of precomputed solution snapshots, andmay struggle to balance between efficiency and accuracy due to the limitation of the linear ansatz. Utilizing the nonlinear representation of neural networks, we propose Meta-Auto-Decoder (MAD) to construct a nonlinear trial manifold, whose best possible performance is measured theoretically by the decoder width. Based on the meta-learning concept, the trial manifold can be learned in a mesh-free and unsupervised way during the pre-training stage. Fast adaptation to new (possibly heterogeneous) PDE parameters is enabled by searching on this trial manifold, and optionally fine-tuning the trial manifold at the same time. Extensive numerical experiments show that the MAD method exhibits faster convergence speed without losing accuracy than other deep learning-based methods.

Figures (29)

• Figure 1: Architecture of Meta-Auto-Decoder.
• Figure 2: Illustration of how MAD-L works from the manifold learning perspective. The function space $\mathcal{U}$ is mapped to a 2-dimensional plane. The solid curve represents the solution set $G(\mathcal{A})$ formed by exact solutions corresponding to all possible PDE parameters, and each point on the curve represents an exact solution corresponding to one PDE parameter. The dotted curve represents the trial manifold $D_{\theta^*}(Z_B)$ obtained by the pre-trained model, and each point on the curve corresponds to a latent vector $\pmb{z}$. Given $\eta_{\text{new}} \in \mathcal{A}$, rather than searching in the entire function space $\mathcal{U}$, MAD-L only searches on the dotted curve to find an optimal $\pmb{z}$ such that its corresponding solution $u_{\theta^*}(\cdot,\pmb{z})$ is nearest to the point $u^{\eta_{\text{new}}}$.
• Figure 3: Visualization of the MAD pre-training process for the ODE problem.
• Figure 4: Visualization of the MAD fine-tuning process for the ODE problem.
• Figure 5: Illustration of how MAD-LM works from the manifold learning perspective. The trial manifold $D_{\theta^*}(Z_B)$ obtained in the pre-training stage is represented by a solid curve, and the solution set $G(\mathcal{A})$ lies within a neighborhood of $D_{\theta^*}(Z_B)$ that is represented by a gray shadow band. To find the solution $u^{\eta_{\text{new}}}$, we have to fine-tune $\theta$ (i.e., the dotted lines) and the latent vector $\pmb{z}$ (i.e., the points on the dotted lines) simultaneously to approach the exact solution. As the search scope is limited to a strip with a small width, the fine-tuning process can be expected to converge quickly.

Theorems & Definitions (4)

• Theorem 1
• proof
• Remark 1
• Remark 2