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Learning solution operators of PDEs defined on varying domains via MIONet

Shanshan Xiao, Pengzhan Jin, Yifa Tang

TL;DR

This work addresses learning PDE solution operators when domains vary, by extending MIONet to metric spaces and formulating a polar-domain representation. It constructs a metric space $U$ of regions and maps the input data to a transformed space $U\times C(B(0,1))$ via $\sigma$, enabling learning of $\hat{\mathcal{G}}: U\times C(B(0,1))\to C(B(0,1))$ with MIONet. The learning uses a loss $L(\theta)$ and recovers the solution $u_{\Omega}$ through $u_{\Omega}=\sigma^{-1} \circ (\pi_1, \mathcal{M}) \circ \sigma(f_{\Omega})$, with a fully-parameterized extension described in the appendix. Numerical experiments on 2D Poisson problems with convex polygons and polar regions demonstrate meshless predictions with relative $L^2$ errors around $3\%$, and robustness to discretization choices, indicating practical applicability as a flexible PDE solver across varying geometries.

Abstract

In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces, establishing that MIONet can approximate mappings with multiple inputs in metric spaces. Subsequently, we construct a set consisting of some appropriate regions and provide a metric on this set thus make it a metric space, which satisfies the approximation condition of MIONet. Building upon the theoretical foundation, we are able to learn the solution mapping of a PDE with all the parameters varying, including the parameters of the differential operator, the right-hand side term, the boundary condition, as well as the domain. Without loss of generality, we for example perform the experiments for 2-d Poisson equations, where the domains and the right-hand side terms are varying. The results provide insights into the performance of this method across convex polygons, polar regions with smooth boundary, and predictions for different levels of discretization on one task. We also show the additional result of the fully-parameterized case in the appendix for interested readers. Reasonably, we point out that this is a meshless method, hence can be flexibly used as a general solver for a type of PDE.

Learning solution operators of PDEs defined on varying domains via MIONet

TL;DR

This work addresses learning PDE solution operators when domains vary, by extending MIONet to metric spaces and formulating a polar-domain representation. It constructs a metric space of regions and maps the input data to a transformed space via , enabling learning of with MIONet. The learning uses a loss and recovers the solution through , with a fully-parameterized extension described in the appendix. Numerical experiments on 2D Poisson problems with convex polygons and polar regions demonstrate meshless predictions with relative errors around , and robustness to discretization choices, indicating practical applicability as a flexible PDE solver across varying geometries.

Abstract

In this work, we propose a method to learn the solution operators of PDEs defined on varying domains via MIONet, and theoretically justify this method. We first extend the approximation theory of MIONet to further deal with metric spaces, establishing that MIONet can approximate mappings with multiple inputs in metric spaces. Subsequently, we construct a set consisting of some appropriate regions and provide a metric on this set thus make it a metric space, which satisfies the approximation condition of MIONet. Building upon the theoretical foundation, we are able to learn the solution mapping of a PDE with all the parameters varying, including the parameters of the differential operator, the right-hand side term, the boundary condition, as well as the domain. Without loss of generality, we for example perform the experiments for 2-d Poisson equations, where the domains and the right-hand side terms are varying. The results provide insights into the performance of this method across convex polygons, polar regions with smooth boundary, and predictions for different levels of discretization on one task. We also show the additional result of the fully-parameterized case in the appendix for interested readers. Reasonably, we point out that this is a meshless method, hence can be flexibly used as a general solver for a type of PDE.
Paper Structure (12 sections, 2 theorems, 51 equations, 7 figures, 2 tables)

This paper contains 12 sections, 2 theorems, 51 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem setup
  3. Theory and method
  4. MIONet for metric spaces
  5. Setting for varying domains
  6. Method
  7. Numerical experiments
  8. Polygonal regions
  9. Polar regions with smooth boundary
  10. Influence of mesh
  11. Conclusions
  12. Fully-parameterized solution operator

Key Result

Theorem 1

Let $X_{i}$ be metric spaces and $Y$ be a Banach space, assume that $X_{i}$ satisfies Assumption ass:projection with the projection mapping $P_q^i=\psi_q^i\circ\phi_q^i$, $K_{i}$ is a compact set in $X_{i}$. Suppose that is a continuous mapping, then for any $\epsilon>0$, there exist positive integers $p_{i}, q_{i}$, continuous functions $g^{i}_{j}\in C(\mathbb{R}^{q_{i}})$ and $u_{j} \in Y$ such

Figures (7)

  • Figure 1: The mapping $\alpha_{\Omega}$.
  • Figure 2: An illustration of the discretization mapping and the reconstruction mapping.
  • Figure 3: An illustration of the method.
  • Figure 4: Examples of predictions for Poisson equations defined on 4,5,6-polygons.
  • Figure 5: Examples of predictions for Poisson equations defined on polar regions with smooth boundary.
  • ...and 2 more figures

Theorems & Definitions (5)

  • Theorem 1: approximation theory
  • proof
  • Definition 1
  • Theorem 2
  • proof