SIGGRAPH 2026 Technical Papers

Boundary-aware Neural Model Reduction for PDEs

Li Liao1,* Pengfei Shen1,* Yifan Peng1,†
1The University of Hong Kong

* Equal contribution. † Corresponding author.

Paper DOI

A single neural model learns reduced bases over changing shapes and boundary conditions.

Abstract

Eigenanalysis of partial differential operators is central to reduced-order physical simulation, but neural shape-space eigenanalysis has largely been limited to natural Neumann boundary conditions. This prevents direct control over supports, openings, heat-exchange boundaries, and other boundary effects that change the underlying operator.

We extend neural eigenanalysis for Laplace-type operators to Dirichlet, Robin, and mixed boundary conditions. Boundary placement and Robin coefficients are treated as model inputs, giving a joint shape-boundary space where eigenfunctions and spectra vary continuously with both geometry and boundary configuration.

The resulting boundary-aware bases support resonance tuning, reduced-order elastic simulation with changing supports, and transient heat analysis under controllable boundary exchange.

Highlights

Mixed boundary conditions

Neumann, Dirichlet, and Robin boundaries share one variational neural eigenanalysis framework.

Bases respect constraints

The reduced space is constructed to satisfy Dirichlet constraints instead of fighting them with penalties.

Designable boundary space

Boundary parameters act as inputs that can be searched, optimized, and reused across physical examples.

Method

Boundary-Aware Neural Eigenanalysis

Pipeline
Pipeline. Distance-aware lifting enforces Dirichlet constraints, while Neumann and Robin terms enter through variational energies.
Distance-gradient discontinuity
Distance-gradient discontinuity. The distance field is continuous, but its gradient contains switching curves that the lifted input exposes to the network.

Boundary-Space Control

Cavity Resonance Matching

Boundary-space cavity resonance matching
Boundary-space cavity resonance. Optimizing a sliding pressure-release opening moves the spectrum toward the target.
Boundary-space spectral tuning
Boundary-space spectral tuning. A moving Dirichlet patch induces smooth but significant low-frequency spectral variation.

Dirichlet Bases

Constraint-Aware Reduced Spaces

Dirichlet boundary-aware basis functions
Dirichlet boundary-aware basis functions. The basis functions smoothly vanish at the constrained boundary, so reduced simulation remains inside the admissible subspace.
Ground Truth
Ours, 12 Basis
Soft Barrier, Neumann Basis
Stiff Barrier, Neumann Basis

Elastic Boundary Space

Dirichlet Boundary-Space Elastic Simulation

Elastic simulation across one Dirichlet boundary-space trajectory.
Elastic simulation across another Dirichlet boundary-space trajectory.

Thermal Shape-Boundary Space

Heatsink Unit Basis

Heatsink shape-boundary space
Heatsink shape-boundary space. The Robin coefficient controls convective exchange in a mixed-boundary heat-transfer model.
Heatsink unit-cell eigenbasis under mixed Dirichlet, Neumann, and Robin boundaries.

BibTeX

@inproceedings{liao2026boundaryaware,
  author    = {Liao, Li and Shen, Pengfei and Peng, Yifan},
  title     = {Boundary-aware Neural Model Reduction for PDEs},
  booktitle = {SIGGRAPH 2026 Technical Papers},
  year      = {2026},
  pages     = {12},
  location  = {Los Angeles, CA, USA},
  publisher = {ACM},
  doi       = {10.1145/3799902.3811153}
}