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LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning

Jiahao Zhang, Yueqi Wang, Guang Lin

TL;DR

LegONet is introduced, a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations, and suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.

Abstract

Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. This makes them difficult to reuse when mechanisms change and it can limit stability under long-horizon rollout. We introduce Lego-like Operator Network (LegONet), a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations. LegONet separates boundary handling from mechanism learning, satisfying boundary conditions by construction. It also separates mechanism learning from time integration, enabling pretrained blocks to be assembled into new solvers without retraining. We also derive a finite-horizon error decomposition that separates block mismatch from splitting error and provides mechanism-level diagnostics for long-horizon predictions. Across ten time-dependent PDEs, LegONet delivers accurate closed-loop rollouts with improved stability under cross-PDE recombination and boundary reconfiguration. More broadly, this modular formulation suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.

LegONet: Plug-and-Play Structure-Preserving Neural Operator Blocks for Compositional PDE Learning

TL;DR

LegONet is introduced, a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations, and suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.

Abstract

Learned PDE solvers are often trained as monolithic surrogates for a specific equation, boundary condition and discretization. This makes them difficult to reuse when mechanisms change and it can limit stability under long-horizon rollout. We introduce Lego-like Operator Network (LegONet), a compositional framework that builds PDE solvers from plug-and-play, structure-preserving operator blocks defined on shared boundary-adapted spectral representations. LegONet separates boundary handling from mechanism learning, satisfying boundary conditions by construction. It also separates mechanism learning from time integration, enabling pretrained blocks to be assembled into new solvers without retraining. We also derive a finite-horizon error decomposition that separates block mismatch from splitting error and provides mechanism-level diagnostics for long-horizon predictions. Across ten time-dependent PDEs, LegONet delivers accurate closed-loop rollouts with improved stability under cross-PDE recombination and boundary reconfiguration. More broadly, this modular formulation suggests a path from task-specific neural solvers towards plug-and-play operator libraries for scientific computing.
Paper Structure (14 sections, 6 theorems, 77 equations, 15 figures, 1 algorithm)

This paper contains 14 sections, 6 theorems, 77 equations, 15 figures, 1 algorithm.

Table of Contents

  1. Proof of Theorem \ref{['thm:main_structure_error']}
  2. Block architectures and training details
  3. Coefficient prior $\mu_b$.
  4. 1D Dirichlet baseplate (Shen--Legendre)
  5. 2D periodic baseplate (Fourier)
  6. 2D Neumann baseplate (cosine/DCT)
  7. 3D periodic baseplate (Fourier/FFT)
  8. Additional numerical experiments
  9. 1D Dirichlet domains
  10. 2D periodic domains
  11. 2D Neumann domains
  12. 3D periodic domains
  13. Baseline configuration
  14. Baseline models

Key Result

Theorem 1

Fix $T>0$ and an integer $N_{\mathrm{steps}}\ge 1$, and set $\Delta t:=T/N_{\mathrm{steps}}$ and $t_n:=n\Delta t$. Let $\mathcal{K}\subset\mathbb{R}^K$ be compact and assume the standing conditions in Supplementary Information. Let $\mathbf{a}(t)$ solve the reference reduced dynamics eq:ref_dyn with and a conservative block $i$ uses a within-block update satisfying then the same inequalities and

Figures (15)

  • Figure 1: From monolithic neural solvers to modular operator blocks. Fig. \ref{['fig:compare']}: conceptual comparison of operator learning, physics-informed optimization and LegONet. Fig. \ref{['fig:workflow']}: LegONet workflow.
  • Figure 1: 1D Dirichlet benchmarks: block-level operator matching and structure diagnostics. Figs. \ref{['fig:1d_uxx_op']}--\ref{['fig:1d_uux_op']}: physical-space comparisons for pretrained blocks on a held-out sample; errors use a weighted $L^2$ norm on Gauss--Legendre nodes. Figs. \ref{['fig:1d_gl_energy']}--\ref{['fig:1d_heat_u0_energy']}: energy diagnostics for 1D Ginzburg--Landau and for the interior component $u_0$ in the heat equation after time-dependent boundary lifting.
  • Figure 2: Overview of LegONet benchmarks. We report four baseplates together with their structure-preserving operators, the pretrained operator blocks used on each coefficient interface, and the target PDEs solved in Fig. \ref{['fig:block']}. We also summarize cross-PDE rollout accuracy in Fig. \ref{['fig:sum']} and list the Strang-splitting composition order used for each experiment in Fig. \ref{['fig:strang']}.
  • Figure 2: 2D benchmarks: rollout accuracy and energy diagnostics across periodic and Neumann baseplates. Figs. \ref{['fig:ac2d_relerr']}--\ref{['fig:ac2d_energy']}: 2D Allen--Cahn: relative $L^2$ trajectory error and Allen--Cahn energy decay. Figs. \ref{['fig:burgers2d_u_relerr']}--\ref{['fig:burgers2d_v_relerr']}: 2D vector Burgers: relative $L^2$ trajectory error for both velocity components. Figs. \ref{['fig:sh2d_relerr']}--\ref{['fig:ch2d_relerr']}: 2D Neumann benchmarks in a cosine trial space: Swift--Hohenberg and Cahn--Hilliard rollout error curves.
  • Figure 3: 1D Ginzburg--Landau
  • ...and 10 more figures

Theorems & Definitions (15)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1: Structure-preserving rollout and total error bound
  • proof
  • proof
  • Lemma 5: Strang composition strang1968constructionhairer2006structure
  • Lemma 6: Exact subflow perturbation on $\mathcal{K}$
  • proof
  • Lemma 7: Substep defect
  • ...and 5 more