Neural-Operator-Predicted Time-Dependent Reduced Subspaces for Projection-Based Simulation of Nonlinear PDEs
Abstract
Accurate numerical simulation of nonlinear partial differential equation initial value problems typically requires high-dimensional full-order discretizations at substantial computational cost. While purely data-driven surrogates can speed up prediction, these "black box" approximations do not necessarily satisfy the governing equations during inference. In this paper we present a hybrid framework, which combines neural operators with projection-based reduced-order modeling for nonlinear PDE simulation. The key idea is to train a neural operator to map the initial condition to a time-dependent reduced subspace, and then evolve low-dimensional reduced coordinates by solving the governing dynamical equations projected on this learned moving trial subspace. In this way, the network predicts an adaptive reduced representation rather than the full trajectory directly, while the online solver preserves a physics-based reduced evolution. We test this method on two representative nonlinear PDEs: the viscous Burgers' equation and the Fisher--KPP reaction--diffusion equation. In both cases, the proposed approach achieves successful reduced-order simulation with substantial wall-clock speedup relative to the corresponding full-order solver while maintaining relative errors on the order of 10−2 to 10−1 across the tested resolutions. These results demonstrate that a low-dimensional learned time-dependent reduced subspace can be used to predict solutions to nonlinear PDEs while retaining the advantages of projection-based reduced dynamics. Overall, this method provides a promising framework for fast and physically grounded data-assisted simulation of nonlinear PDEs.
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BibTeX
@misc{song2026neuraloperatorpredicted,
title = {{Neural-Operator-Predicted Time-Dependent Reduced Subspaces for Projection-Based Simulation of Nonlinear PDEs}},
author = {Boxi Song and Rennie Mirollo and Jan R. Engelbrecht},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08693},
eprint = {2610.08693},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08693},
}
RIS
TY - UNPB
TI - Neural-Operator-Predicted Time-Dependent Reduced Subspaces for Projection-Based Simulation of Nonlinear PDEs
AU - Boxi Song
AU - Rennie Mirollo
AU - Jan R. Engelbrecht
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08693
UR - https://arxiv.org/abs/2610.08693
AB - Accurate numerical simulation of nonlinear partial differential equation initial value problems typically requires high-dimensional full-order discretizations at substantial computational cost. While purely data-driven surrogates can speed up prediction, these "black box" approximations do not necessarily satisfy the governing equations during inference. In this paper we present a hybrid framework, which combines neural operators with projection-based reduced-order modeling for nonlinear PDE simulation. The key idea is to train a neural operator to map the initial condition to a time-dependent reduced subspace, and then evolve low-dimensional reduced coordinates by solving the governing dynamical equations projected on this learned moving trial subspace. In this way, the network predicts an adaptive reduced representation rather than the full trajectory directly, while the online solver preserves a physics-based reduced evolution. We test this method on two representative nonlinear PDEs: the viscous Burgers' equation and the Fisher--KPP reaction--diffusion equation. In both cases, the proposed approach achieves successful reduced-order simulation with substantial wall-clock speedup relative to the corresponding full-order solver while maintaining relative errors on the order of 10^−2 to 10^−1 across the tested resolutions. These results demonstrate that a low-dimensional learned time-dependent reduced subspace can be used to predict solutions to nonlinear PDEs while retaining the advantages of projection-based reduced dynamics. Overall, this method provides a promising framework for fast and physically grounded data-assisted simulation of nonlinear PDEs.
ER -