用神经网络修正数值求解器误差，提升非线性色散方程求解精度与稳定性。
Hybrid Iterative Neural Low-Regularity Integrator for Nonlinear Dispersive Equations
- 融合经典求解器与轻量神经网络，学习并补偿系统性截断误差。
- 在粗糙数据下实验显示精度优于传统方法，且支持稳定的空间细化。
- 适合需要高精度、鲁棒性强的科学计算场景，如波动传播模拟。
我们提出 HIN-LRI，一种混合框架，通过训练神经算子来修正经典数值求解器的结构化截断误差。基础低正则积分器为非线性色散偏微分方程提供一阶一致近似，而一个作用于低维隐空间的轻量级神经网络，则学习分析方法无法闭合的残差缺陷。对神经修正项施加显式时间步长缩放，确保其 Lipschitz 贡献为 $ au$ 阶，从而获得与空间分辨率无关、步长独立的 Gronwall 稳定性因子。网络通过求解器闭环目标端到端训练，展开完整迭代过程，并在 Bourgain 型范数下惩罚轨迹误差，使学习对齐多步求解动态而非孤立单步目标。在给定假设下，全局误差满足 $C(\varepsilon_{net}+δ) au^γ ext{ln}(1/τ)$，其中 $ au$ 为时间步长，$ au^γ$ 表示收敛率，$ au^γ ext{ln}(1/τ)$ 反映对粗网格问题的适应性；$ au^γ$ 是核心收敛阶，$ ext{ln}(1/τ)$ 体现对弱奇异性解的处理能力，$ au^γ$ 代表基本收敛速度。实验在三个色散基准上验证：在粗糙初始数据下，相比解析积分器、分裂方法及神经微分方程代理模型，HIN-LRI 实现更高精度，具备稳定的空间精细能力、有效的分布外迁移性能，且在线开销小。
原文摘要 · Abstract (English)
We propose HIN-LRI, a hybrid framework that augments a classical numerical solver with a neural operator trained to correct the solver's structured truncation error. A base low-regularity integrator provides a consistent first-order approximation to nonlinear dispersive PDEs, while a lightweight neural network, operating on a low-dimensional latent manifold, learns the residual defect that analytical methods cannot close. An explicit time-step scaling on the neural correction ensures that its Lipschitz contribution remains $\mathcal{O}(τ)$, yielding a Gronwall stability factor bounded uniformly in the step size and independent of the spatial resolution. The network is trained end-to-end through a solver-in-the-loop objective that unrolls the full iteration and penalises trajectory error in a Bourgain-type norm, aligning learning with multi-step solver dynamics rather than isolated one-step targets. Under stated assumptions, the global error satisfies $C(\varepsilon_{net}+δ)\,τ^γ\ln(1/τ)$, where $\varepsilon_{net}$ measures the network approximation quality and $δ$ the training shortfall. Experiments on three dispersive benchmarks with rough data show that HIN-LRI improves accuracy over analytical integrators, splitting methods, and neural PDE surrogates, with stable spatial refinement, effective out-of-distribution transfer, and modest online overhead.
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