Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations

TL;DR

This work introduces Reference Neural Operators (RNO), a data-efficient framework for learning PDE solutions on deformable domains by predicting the change in the solution relative to a reference geometry. By leveraging a reference solution (u_r, Ω_r) and a smooth deformation φ to a query domain Ω_q, RNO learns the material derivative δu = u_q − u_r ∘ φ^{-1} using a geometry-aware encoder and Distance-Aware Cross-Attention to handle irregular meshes and multiple boundary components. The approach demonstrates strong, dataset-efficient performance across 2D and 3D PDEs (e.g., Navier–Stokes, Maxwell, heat transfer) and free-form geometric changes, outperforming traditional G-S neural operators and several baselines, with ablations confirming the importance of δu-targeting and DACA. The findings suggest a practical path for surrogate modeling in shape optimization and design workflows where simulation cost is prohibitive and geometry spaces are vast.

Abstract

For partial differential equations on domains of arbitrary shapes, existing works of neural operators attempt to learn a mapping from geometries to solutions. It often requires a large dataset of geometry-solution pairs in order to obtain a sufficiently accurate neural operator. However, for many industrial applications, e.g., engineering design optimization, it can be prohibitive to satisfy the requirement since even a single simulation may take hours or days of computation. To address this issue, we propose reference neural operators (RNO), a novel way of implementing neural operators, i.e., to learn the smooth dependence of solutions on geometric deformations. Specifically, given a reference solution, RNO can predict solutions corresponding to arbitrary deformations of the referred geometry. This approach turns out to be much more data efficient. Through extensive experiments, we show that RNO can learn the dependence across various types and different numbers of geometry objects with relatively small datasets. RNO outperforms baseline models in accuracy by a large lead and achieves up to 80% error reduction.

Figures (11)

• Figure 1: Comparison between two approaches. (Left) Neural operators map directly from geometries/functions $a$ to solutions $u$, which often requires large amount of data to cover various geometries. (Right) Alternatively, given a reference solution $u_r$ on $\Omega_r$, we hope to query solutions to various deformations of its geometry. Let $\varphi: \Omega_r\mapsto\Omega_q$ be a smooth deformation, a reference neural operator can predict the difference between the queried solution and the pushforward of the reference solution $\delta u = u_q - u_r\circ \varphi^{-1}$.
• Figure 2: (Left) Given geometry space $\mathcal{T}$ and solution space $\mathcal{U}$, a reference neural operator $\Psi_{\theta}$ learns the material derivative of a solution operator $\mathcal{G}:\mathcal{T}\rightarrow\mathcal{U}$ at a given reference geometry $\Omega_r$. It provides an approximation of the solution on a deformed geometry $\Omega_q=\varphi(\Omega_r)$ with $u_r\circ\varphi^{-1}+\Psi_{\theta}(u_r, \Omega_r, \varphi)$. (Right) The transform (gray vectors) from $\partial\Omega_q$ (orange circle) to $\partial\Omega_r$ (blue circle) is used to construct a vector field that represents a deformation $\varphi^{-1}: \Omega_q \mapsto \Omega_r$.
• Figure 3: Overview of model architecture which composes of two stages. The first stage is the preprocessing of input data. Input sequences $u_r$, $\Omega_r=\{x_{r_j}\}_j$ and $\Omega_q=\{x_{q_i}\}_i$ are tensors with shape $N_r\times d_s$, $N_r\times n$, $N_q\times n$, where $n, d_s$ are spatial dimension and dimension of target space. $\delta x_q = \varphi^{-1}(x_q)-x_q$ is the shift of every point $x_q\in \Omega_q$. The second stage is the forward passing of neural network. $\odot$ is element-wise product, $\otimes$ is matrix product, $+$ is element-wise sum. $\delta u$ is the predicted change of solution, and $\hat{u}_q=u_r\circ\varphi^{-1} + \delta u$.
• Figure 4: (Left) Mean error of all components versus the size of dataset. (Right) Error versus distance.
• Figure 5: (Left) Error versus number layers. (Right) Error versus $\gamma$ of DACA.