Local neural operator for solving transient partial differential equations on varied domains

TL;DR

This work tackles the challenge of solving transient PDEs on varied domains with neural methods by introducing Local Neural Operator (LNO), which learns a local, shift-invariant time-marching operator $\mathcal{G}_L$ that maps $u_t$ to $u_{t+\Delta t}$ across different domains. The LNO architecture combines a lifting-projection framework with a dual path interior block—one physical path and one localized spectral path using Legendre transforms—while a boundary-treatment strategy decouples time marching from domain boundaries. Trained on randomized Navier–Stokes data, the pre-trained LNO successfully predicts flows on unseen domains (lid-driven cavities, cascaded airfoils) with substantial speedups over FEM (up to $\sim 10^3\times$) and robust performance across viscosities and boundary types. The approach also demonstrates potential to extend to other transient PDEs (e.g., Burgers, wave equations) and supports building a reusable library of pre-trained models for rapid, domain-general PDE solving, with a large CFL tolerance (e.g., $CFL\approx 3.2$).

Abstract

Artificial intelligence (AI) shows great potential to reduce the huge cost of solving partial differential equations (PDEs). However, it is not fully realized in practice as neural networks are defined and trained on fixed domains and boundaries. Herein, we propose local neural operator (LNO) for solving transient PDEs on varied domains. It comes together with a handy strategy including boundary treatments, enabling one pre-trained LNO to predict solutions on different domains. For demonstration, LNO learns Navier-Stokes equations from randomly generated data samples, and then the pre-trained LNO is used as an explicit numerical time-marching scheme to solve the flow of fluid on unseen domains, e.g., the flow in a lid-driven cavity and the flow across the cascade of airfoils. It is about 1000$\times$ faster than the conventional finite element method to calculate the flow across the cascade of airfoils. The solving process with pre-trained LNO achieves great efficiency, with significant potential to accelerate numerical calculations in practice.

Figures (10)

• Figure 1: Local neural operator (LNO) conception and method.a, Various flows of fluids. b, Mathematical description of the flow in different case-specific conditions. c, The concept of LNO to approximate local-related and shift-invariant time-marching operator $\mathcal{G}_L$ representing transient PDEs, while the case-specific conditions are imposed by specific boundary treatment. d, The specific architecture of the LNO, in which $\mathcal{C},\sigma,\mathcal{W}$ are convolutional layers, activation functions, and the linear layer, and $\mathcal{T}$ and $\mathcal{T}^{-1}$ are Legendre spectral transform and its inverse on local parts of the computational domain, respectively. e, Training LNO with samples on periodic domains while the boundary effect is excluded. After that, the pre-trained LNO can predict solutions on different domains by collaborating with case-specific boundary treatment. f, Procedure to solve transient PDEs on unseen domains with pre-trained LNO. The dotted lines are the artificial boundary of the domain, while the solid lines are ordinary ones. The areas shaded with gray or yellow in c and f are respectively the support set of the input $u_t$ and the output $u_{t+\Delta t}$ .
• Figure 2: Predicted velocity contours by trained LNO for 2-D incompressible N-S equations with three viscosities. a, $\mu=0.01$, b, $\mu=0.002$, c, $\mu=0.001$. The trained LNO predicts the solution function for each viscosity according to a random IC different from training data. LNO takes 5, 10, 20, and 40 cycles to predict these four frames at $t=0.2,0.5,1,2$. The FEM results are also presented for comparison.
• Figure 3: The pre-trained LNO solves the internal flow in a lid-driven cavity ($Re=1000$).a, Schematics of the problem and the domain division for LNO prediction. b, The time-marching workflow to predict the velocity fields with pre-trained LNO. c, Comparison of the LNO predicted streamlines with FEM numerical results. d and e, LNO predicted velocity contours and velocity profiles on the centerlines $x/L=0.5$ (the upper) and $y/L=0.5$ (the lower), respectively, in which the results from FEM and Ghia et al. Ghia1982 are also presented for comparison.
• Figure 4: The pre-trained LNO solves the external flow across the cascade of airfoils.a, Schematics of the problem with $b,d,\alpha,\beta,\gamma$ being the chord length, the interval, the angle of attack, the stagger angle, and the inflow angle, respectively. The domain is extended and divided to $\Omega_1$, $\Omega_2$ and $\Omega_3$ for LNO prediction. b, The time-marching workflow to predict the velocity fields with pre-trained LNO. c, Contours of velocity magnitude and absolute error at the steady state. The results are computed on the domain including one airfoil, and are periodically extended to three for better visualization. d, Evolving history of contours of velocity magnitude with velocity profiles at $x=0,1,2$. e, Steady-state streamlines predicted by the same pre-trained LNO for flow across different cascades. The FEM results are presented for comparison.
• Figure 5: Technical details in the present LNO architecture.a, Pointwise layers. b, Physical layers. c, Spectral layers. Schematics in a, b and c are basic layers in LNO categorized according to how they link function values at different positions. d, The present LNO architecture composed using layers in a, b and c. $\{v^{(i)}\}_{i=0}^n$ are intermediate hidden functions during LNO prediction. e, The interior blocks. f, Implementation of the spectral layers by using discretized convolutions. The spectral layer processes a single channel discretized function $v$ on an equidistant grid of size $N_w\times N_h$. This implementation brings an issue called ‘corrosion of the domain’ caused by insufficient coverage in near-boundary areas.