发表机构
Boston College（波士顿学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种混合框架，用神经算子将初始条件映射到时间相关约化子空间，再投影求解控制方程，实现非线性PDE的快速约化模拟，在Burgers和Fisher-KPP方程上验证了加速与精度。
AI 中文摘要
非线性偏微分方程初值问题的精确数值模拟通常需要高维全阶离散化，计算成本高昂。纯数据驱动的替代模型虽能加速预测，但这些“黑箱”近似在推理过程中不一定满足控制方程。本文提出一个混合框架，将神经算子与基于投影的约化阶建模相结合，用于非线性PDE模拟。关键思想是训练一个神经算子，将初始条件映射到时间相关的约化子空间，然后通过求解投影在该学习到的移动试验子空间上的控制动力学方程，来演化低维约化坐标。这样，网络预测的是自适应的约化表示，而非直接预测完整轨迹，同时在线求解器保留了基于物理的约化演化。我们在两个代表性非线性PDE上测试了该方法：粘性Burgers方程和Fisher-KPP反应-扩散方程。在两种情况下，所提出的方法均实现了成功的约化阶模拟，相对于相应的全阶求解器获得了显著的墙钟时间加速，同时在测试的分辨率下保持了约10^{-2}至10^{-1}量级的相对误差。这些结果表明，低维学习的时间相关约化子空间可用于预测非线性PDE的解，同时保留基于投影的约化动力学的优势。总体而言，该方法为非线性PDE的快速且物理上有依据的数据辅助模拟提供了一个有前景的框架。
英文摘要
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.
