提出新型神经算子模型，解决多输入偏微分方程求解中的泛化差与离散敏感问题。
ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance
- 基于简化基方法与贪婪算法构建网络结构，离线自适应生成
- 在分布内和分布外测试中显著缩小泛化差距，且实现严格离散不变性
- 结构紧凑、计算成本低，适合物理信息强的科学计算场景
我们提出一种新型数据轻量级算子学习算法——简化基神经算子（ReBaNO），用于求解具有多种不同输入的偏微分方程组。受简化基方法和近期提出的生成式预训练物理信息神经网络启发，ReBaNO 采用数学严谨的贪婪算法，在离线阶段从零开始自适应构建网络结构。通过任务特定的激活函数知识蒸馏，使 ReBaNO 在在线计算时具备紧凑架构，计算开销极小，同时嵌入物理规律。相比 PCA-Net、DeepONet、FNO 与 CNO 等先进算子学习方法，数值结果表明，ReBaNO 在分布内与分布外测试中均显著缩小泛化差距，并是唯一实现严格离散不变性的算子学习算法。
原文摘要 · Abstract (English)
We propose a novel data-lean operator learning algorithm, the Reduced Basis Neural Operator (ReBaNO), to solve a group of PDEs with multiple distinct inputs. Inspired by the Reduced Basis Method and the recently introduced Generative Pre-Trained Physics-Informed Neural Networks, ReBaNO relies on a mathematically rigorous greedy algorithm to build its network structure offline adaptively from the ground up. Knowledge distillation via task-specific activation function allows ReBaNO to have a compact architecture requiring minimal computational cost online while embedding physics. In comparison to state-of-the-art operator learning algorithms such as PCA-Net, DeepONet, FNO, and CNO, numerical results demonstrate that ReBaNO significantly outperforms them in terms of eliminating/shrinking the generalization gap for both in- and out-of-distribution tests and being the only operator learning algorithm achieving strict discretization invariance.
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